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THE EFFECT OF CONVOLVING FAMILIES OF L-FUNCTIONS
ON THE UNDERLYING GROUP SYMMETRIES
EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Abstract. Let {FN} and {GM} be families of primitive automorphicL-functions for GLn(AQ)
and GLm(AQ), respectively, such that, as N,M → ∞, the statistical behavior (1-level den-
sity) of the low-lying zeros of L-functions in FN (resp., GM ) agrees with that of the eigen-
values near 1 of matrices in G1 (resp., G2) as the size of the matrices tend to infinity,
where each Gi is one of the classical compact groups (unitary U, symplectic Sp, or orthog-
onal O, SO(even), SO(odd)). Assuming that the convolved families of L-functions FN × GM
are automorphic, we study their 1-level density. (We also study convolved families of the
form f ×GM for a fixed f .) Under natural assumptions on the families (which hold in many
cases) we can associate to each family L of L-functions a symmetry constant cL equal to 0
(resp., 1 or −1) if the corresponding low-lying zero statistics agree with those of the unitary
(resp., symplectic or orthogonal) group. Our main result is that cF×G = cF · cG : the sym-
metry type of the convolved family is the product of the symmetry types of the two families.
A similar statement holds for the convolved families f × GM . We provide examples built
from Dirichlet L-functions and holomorphic modular forms and their symmetric powers. An
interesting special case is to convolve two families of elliptic curves with rank. In this case
the symmetry group of the convolution is independent of the ranks, in accordance with the
general principle of multiplicativity of the symmetry constants (but the ranks persist, before
taking the limit N,M →∞, as lower-order terms).
1. Introduction
1.1. Preliminaries.
Many questions in number theory, such as the study of the density of the primes or proper-
ties of class numbers, can be related to understanding the distribution of zeros of L-functions.
In the early 1970’s Dyson and Montgomery [Mon] discovered the agreement between the pair
correlation of zeros of the Riemann zeta-function ζ(s) and of eigenvalues of matrices in the
Date: May 27, 2008.
2000 Mathematics Subject Classification. 11M26 (primary), 11G05, 11G40, 15A52 (secondary).
Key words and phrases. low lying zeros, families of L-functions, Rankin-Selberg convolution, Satake
parameters.
We thank Jim Cogdell, Sol Friedberg, Gergely Harcos, Jeff Hoffstein, Wenzhi Luo, Stephen D. Miller, Steve
Rallis, Zee´v Rudnick, Peter Sarnak, Eitan Sayag and Joe Silverman for many enlightening conversations. The
first named author was partly supported by EPSRC grant N09176; the second named author was partially
supported by NSF grant DMS-0600848.
1

2 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Gaussian Unitary Ensemble (GUE). Two decades later Katz and Sarnak [KaSa1, KaSa2] of-
fered deeper insight into the connection between zero and eigenvalue statistics by studying
families of L-functions. Ever since, random matrix theory [CFKRS, KaSa2, KeSn, ILS] has
enjoyed remarkable success at modeling and predicting the behavior of L-functions.
Various pairs of eigenvalue/zero statistics can be shown, or at least are conjectured, to be
in perfect agreement. Among the early such statistics to be studied were n-level correlations
and nearest-neighbor spacings [Hej, Mon, Od1, Od2, RS]. These statistics pertain to the
whole (infinite) sequence of critical zeros of a single L-function, and are shown to agree with
the corresponding statistic of N ×N GUE matrices (in the limit N →∞). Neither of these
statistics reveals anything about the behavior of low-lying critical zeros of L-functions (that
is, of zeros near the arithmetically-crucial central point). The reason is that those statistics
are defined by averaging quantities defined using a large (but finite) subset of the zeros, most
of which will lie high up on the critical line—and thus the behavior of those few zeros that
lie near the central point is irrelevant in the limit as the number of zeros used to compute
the statistic tends to infinity.
The Katz-Sarnak philosophy has shifted the emphasis to the study of families of L-functions
and their low-lying zeros, whose statistics (upon averaging over the family) are well modeled
by the statistics of eigenvalues close to 1 of random matrices from the classical compact
groups. In the function field case these classical group statistics are explained by the mon-
odromy group of the family. For families of automorphic L-functions of number fields the
connection is quite a bit more mysterious. Usually, the corresponding classical compact group
is identified only by explicitly computing zero statistics. Our goal in this paper is to allow
predicting the group attached to a “convolved” family assuming only knowledge of the groups
describing the zero statistics of the two families being convolved. The relation turns out to
be very simple to describe and it will hopefully shed some light into the properties of the
(conjectural) correspondence between families of (number-field) automorphic L-functions and
classical groups.
We first describe the main statistic studied in this paper. In order to break away from
the universal global GUE statistics of the zeros of a single L-function, and to understand the
neighborhood of the central point, we study the n-level density in a family of L-functions; the
latter is a local statistic involving only critical zeros near the central point. Let F = ∪FN
be a family of L-functions ordered by their conductors (for example, FN might be Dirichlet
L-functions with conductor N or cuspidal newforms of weight-2 and level N) and write
the zeros of L(s, f) as 1/2 + iγj;f (assuming the General Riemann Hypothesis (GRH), each
γj;f ∈ R). Given an n-variable test function φ(t1, . . . , tn) = φ1(t1) · · · · ·φn(tn) (where each φk
is a Schwartz function on R), the n-level density of F is (by a slight abuse of language) the
measure on Rn with respect to which the integral of φ is
Dn,F(φ) = limN→∞
1
|FN |
∑
f∈FN
∑
j1,...,jn
ji 6=±jk
φ1
(
logQf
2π γj1;f
)
· · ·φn
(
logQf
2π γjn;f
)
, (1.1)

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 3
provided the limit exists, where Qf is the analytic conductor of L(s, f).1 For many families
of L-functions [DM, FI, Gu¨, HR, HM, ILS, KaSa2, Mil2, Ro, Rub, Yo2] (and, conjecturally
at least, for any natural such family, in accordance with the Katz-Sarnak philosophy), the
n-level density coincides with that of the (normalized) eigenvalues near 1 of matrices in one
of the infinite families of classical compact Lie groups, in the limit as the size N of the
matrix goes to infinity. In the context of matrices from classical Lie groups, the averaging
over FN in equation (1.1) is replaced by averaging over the whole group with respect to
its natural (Haar) probability measure—hence the terminology of “random matrices”. The
n-level densities for different classical compact groups are distinct—it is this feature that
allows “breaking” the universal GUE behavior observed when considering global statistics
such as n-level correlations or neighbor spacings. This one-to-one correspondence between
(infinite families of) classical groups and their n-level densities allows, at least conjecturally,
to assign a definite “symmetry type” to each family of primitive L-functions. For families of
zeta or L-functions of curves or varieties over finite fields, the corresponding classical compact
group is determined by the monodromy group of the family [KaSa1]. However, for families
of number-field automorphic L-functions there is no such thing as a monodromy group and
the underlying symmetry only manifests itself (in our current understanding) through the
zero statistics (although function field analogues of number-field families often suggest what
the symmetry type should be). Our goal in this paper is to determine the symmetry group
of the convolution of two families of number-field automorphic L-functions in terms of the
symmetry groups of the families being convolved together.
For families where the signs of the functional equations are all even and there is not an
obvious corresponding family with odd signs, a folklore conjecture (see for example page
2877 of [KeSn]) stated that the symmetry group should be symplectic. This was based on
the observation that SO(even/odd) symmetries in all known examples arose from splitting
orthogonal families according to the sign of the functional equation, while symplectic sym-
metries arose from a family with all even sign and no corresponding family with odd signs.
A priori the symmetry type of a family with all functional equations even is either symplec-
tic or SO(even). In [DM] we studied the family {L(s, φ × sym2f)}, where φ is a fixed even
Hecke-Maass eigenform on the modular group and f ranges over weight-k full level Hecke cusp
forms; see [LS] for applications of this family. All L(s, φ × sym2f) have even sign, and this
family does not arise from splitting sign within an orthogonal family. In [DM] it is shown (via
1- and 2-level densities) that the symmetry type agrees only with SO(even), thus disproving
the folklore conjecture mentioned above.
As a consequence of the counterexample to the folklore conjecture, the theory of low-lying
zeros is more than just a theory of the signs of functional equations. By analyzing Rankin-
Selberg convolutions of GL2 L-functions (and some of their lifts), we are led to attaching a
1It is a simple consequence of the Riemann-von Mangoldt zero-counting formula that the density of the
zeros near the central point s = 1/2 is roughly (logQf )/2π, so the rescaled (“normalized”) imaginary parts
γjn;f · (logQf )/2π have uniform (constant) density 1 in the large-conductor limit. Thus, for fixed A < B,
each L(s, f) has roughly B −A normalized zeros with imaginary parts in [A,B]. Also, critical zeros not near
s = 1/2 (on a scale of (logQf )/2π) are “rescaled away to infinity” in the large-conductor limit.

4 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
symmetry constant cF to each family F of L-functions. This constant depends only on the
second moment (i. e., the average over the family) of the Satake parameters at each unramified
prime. In all the cases investigated, the average is 0 (resp., 1 or −1) if the family has unitary
(resp., symplectic or orthogonal) symmetry. We are ready to set some notation and describe
our main result, namely that in many cases the symmetry constant of the convolution of two
families is the product of their symmetry constants.
1.2. n-Level Densities and NT-good Families.
We list four desirable properties for a family of primitive L-functions to have; a family
satisfying these properties is called NT-good. These properties are inspired by the families
that have been successfully investigated to date, and codify the conditions for which we can
calculate the one-level (and sometimes even the n-level) densities for a family of L-functions.
Though we could replace some of the bounds with slightly weaker conditions, these are the
conditions that are met in practice.
Definition 1.1 (NT-good). Let φ be an even Schwartz test function such that supp(φ̂) ⊂
(−σ, σ) for some σ > 0. A family F of primitive automorphic L-functions for GLn(AQ) is
NT-good with symmetry constant cF if F is a disjoint union of finite sets FN ⊂ F such that,
as N →∞:
(1) Cardinality:
(i) Bounded multiplicities: Members of a family can occur multiple times, say f ∈ FN
occurs µf times; however, we assume the multiplicities are bounded by a universal
constant, independent of N : µf ≤ µF .
(ii) Size of the family: |FN | → ∞, where each member is counted with its multiplicity:
|FN | =
∑
f∈FN µf .
(2) Conductors: The analytic conductors of f ∈ FN are essentially constant (say, logQf =
logRN + o(logRN ) for all f ∈ FN and some sequence {RN}. Further, there exists
δ0, δ′0 > 0 such that |FN |δ0 ≪ RN ≪ |FN |δ
′
0.
(3) Sums over primes and squares of primes:2
(i) Prime sums: For some rF ≥ 0,
− 2
∑
p
1√p
log p
logRN
φ̂
(
log p
logRN
)
1
|FN |
∑
f∈FN
bf (p) = rFφ(0) + o (1) ; (1.2)
2The numbers bf(n) are the Dirichlet coefficients of the logarithmic derivative:
L′(s, f)/L(s, f) =
∑∞
n=1 bf (n)/ns, cf., Definition 2.1 below.

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 5
we call rF the rank of the family. Often (1.2) is satisfied because ∃δ1 > 0, µ1, rF ≥ 0
such that
1
|FN |
∑
f∈FN
bf (p) = −
rF√p +O
(
|FN |−δ1pµ1
)
. (1.3)
It also suffices for this to hold for almost all primes, provided the contribution from
the bad primes is negligible.
(ii) Prime-square sums: For some cF ∈ {−1, 0, 1},
− 2
∑
p
1
p
log p
logRN
φ̂
(
2
log p
logRN
)
1
|FN |
∑
f∈FN
bf (p2) = −cF
φ(0)
2
+ o(1). (1.4)
Often (1.4) is satisfied because ∃δ2 > 0, µ2 ≥ 0, cF ∈ {−1, 0, 1} such that
1
|FN |
∑
f∈FN
bf (p2) = cF +O(|FN |−δ2pµ2). (1.5)
We call cF the symmetry constant of the family.
(4) Error terms: We have
1
|FN |
∑
f∈FN
∑
p
∞∑
ν=3
bf (pν)
pν/2
log p
logRN
φ̂
(
ν log p
logRN
)
= o(1). (1.6)
Many estimates on bf (pν) imply (1.6); we give three natural ones:
(i) Ramanujan Conjecture: For all p, |αf,j(p)| ≤ 1 implying bf (pν) = O(1).
(ii) ∃µ3(ν) > 0 with limν→∞ µ3(ν) = µ3 > 0 such that
bf (pν) ≪
pν/2
p1+µ3(ν)ν . (1.7)
(iii) ∃δ3 > 0, µ3 ≥ 0 such that∑
f∈FN
bf (pν) ≪ |FN |1−δ3pµ3 . (1.8)
The first condition ensures we have enough L-functions for averaging, and when we con-
volve two families, it is a needed ingredient in controlling the contribution of imprimitive
L-functions.3 The second condition allows us to handle the conductors, and ensures that
the number of L-functions is at least a power of the analytic conductor; this is often needed
in averaging to show certain terms are small. The fourth condition allows us to ignore the
contributions from ν ≥ 3 in the explicit formula (2.10). The point of condition (4.ii) is that
eventually (for ν large) we have b(pν)/pν/2 ≪ 1/p1+µ3ν , and this will be summable over ν
3We could weaken our assumptions and allow µf ≤ |FN |1−ǫ for some ǫ > 0. In the applications we have in
mind, all multiplicities are bounded, so for ease of exposition we assume the multiplicities are bounded. We
comment on this further in Corollary 1.6.

6 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
and p (the small ν ≥ 3 terms can be handled individually by our assumptions); (4.iii) is
an alternate bound where the cancelation comes not from each individual L(s, f) but rather
from averaging over the family.
The third condition is the interesting one, especially (3.ii). It is here that we see family-
dependent behavior. In order to use (2.10) successfully, we need to be able to determine
family averages of bf (p) and bf (p2). In all the families of L-functions studied to date [DM,
FI, Gao, Gu¨, HM, HR, ILS, Mil1, Mil2, RR, Ro, Rub, Yo2], this condition holds; further rF
is zero except for families of elliptic curves with rank. The main term of the family averages
of bf (p2) do not depend on rF , which surfaces only in the averages of bf (p). Except for one-
parameter families of elliptic curves with constant j-invariant, in all other families studied to
date (1.5) holds. (Michel [Mic] proved that (1.5) holds for one-parameter families of elliptic
curves with non-constant j-invariant. If the j-invariant is constant one can often show by
direct computation that either (1.4) holds or (1.5) holds on average; see [Mil1, Mil2].)
We conclude this subsection by listing some NT-good families with constant analytic con-
ductors in each FN .
Unitary
• {L(s, χ) : χ a non-trivial Dirichlet character of prime conductorm},m→∞ (see [HR]);
Symplectic
• {L(s, χd) : d ranges over subsets of fundamental discriminants in [N, 2N ]}, N → ∞
(see [Gao, HR, Mil5, Rub]);
• {L(s, symrf) : r even and f ranges over weight-k full level cusp forms}, k → ∞ (see
[Gu¨, ILS]);
• {L(s, φ× f) : φ a fixed Maass form and f ranges over weight-k full level cusp forms},
k →∞ (see [DM]);
• L(s, ψ) with ψ a character of the ideal class group of the imaginary quadratic field
Q(√−D) with D > 3 square-free and congruent to 3 modulo 4 (see [FI]);
Orthogonal
• {L(s, f) : f ranges over weight-k level N cuspidal newforms with k, N or both tending
to infinity}; if we split by sign of the functional equations we get SO(even) or SO(odd)
symmetry ([ILS, Mil8, RR, Ro] for the 1-level and [HM] for the n-level density);
• {L(s, φ × sym2f) : φ a fixed Maass form and f ranges over weight-k full level cusp
forms}, k →∞ (see [DM]);
• {L(s, symrf) : r odd and f ranges over weight-k full level cusp forms}, with O sym-
metry if r ≡ 1, 5 mod 8, SO(even) symmetry for r ≡ 7 mod 8 and SO(odd) symmetry
for r ≡ 3 mod 8, k →∞ (see [Gu¨]);

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 7
With some more work, families with monotone increasing conductors can be handled.
This allows us to add an entry to each list. For unitary families we may consider non-
primitive Dirichlet characters with square-free conductor (see [Mil6]). For symplectic we may
consider primitive quadratic Dirichlet characters (see [Rub]). For orthogonal families we may
consider one-parameter (see [Mil2]) or two-parameter (see [Yo2]) families of elliptic curves. A
generic one-parameter family should have rank 0 and equidistribution of signs of functional
equations, giving O symmetry; however there are numerous families with positive rank, as
well as constant sign families (see [Mil2] for exact statements and details).
1.3. Main Results.
We adopt the following convention throughout this paper: If F and G are two families of
unitary automorphic cuspidal representations of GLn(AQ) and GLm(AQ) with trivial central
character, then by F ×G we mean the set of all the (conjectural) Rankin-Selberg automorphic
representations f × g of GLmn(AQ), where f ∈ F , g ∈ G. (Here the f × g are counted
with multiplicity µfµg.) For every purpose in this paper, this is equivalent to considering F
and G to be families of automorphic L-functions and F × G consists of the Rankin-Selberg
convolution L-functions L(s, f × g).
We occasionally remind the reader of this convention. We need some control over the
number of (f, g) where g is the contragredient of f . This is because the associated L-function
is imprimitive, and thus its zeros are the superposition of the zeros of at least two primitive
L-functions. In most cases the number and contribution of these L-functions to the 1-level
density is negligible.
Definition 1.2 (Symmetry constant, family constant). We denote the symmetry constant of
the family F by cF . It equals 0 (resp., 1 or −1) if the 1-level density of the family agrees with
unitary (resp., symplectic or any of the three orthogonal groups: O, SO(even) or SO(odd)).
As the three orthogonal groups all have cF = 1, to distinguish them we set ǫF equal to 0
(resp., 1 or −1) if F has half of the signs of its functional equation even (resp., all signs even
or odd); if F is not associated to an orthogonal group, then cF alone determines the group
and we simply put ǫF = 0. Finally, rF denotes the rank of F ; except for families of elliptic
curves, all other known families have rF = 0. We call c˜F = (cF , ǫF , rF) the family constant
of F .
Theorem 1.3. Let F and G be NT-good families of unitary automorphic cuspidal represen-
tations of GLn(AQ) and GLm(AQ) with trivial central character, with symmetry constants cF
and cG. Assume F × G is an NT-good family. Then the family F × G (which is the limit of
FN × GM , where N and M tend to infinity together) has symmetry constant
cF×G = cF · cG. (1.9)
If the family constants are c˜F = (cF , ǫF , rF) and c˜G = (cG, ǫG, rG) then the new family
constant is c˜F×G = (cF · cG, ǫF×G , 0).

8 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Remark 1.4. Note that the ranks of the two families do not enter in the determination of
the classical compact group associated to F × G; the new family has rank 0 (in the sense of
Definition 1.1). Determining the distribution of signs of the functional equations of F ×G is
often an involved calculation depending on fine properties of the two families; however, it is
not needed if we merely wish to classify the symmetry as unitary, symplectic or (non-specific)
orthogonal.
Remark 1.5. It is worthwhile to emphasize the meaning of the above theorem. Our notion
of a symmetry constant arises from one-level density expansions, though we expect to see the
same correspondence for any statistic (n-level correlations, central values, moments, . . . ).
Thus, an alternate way to phrase our results is that the one-level density of the convolution
(as the conductors tend to infinity) agrees with the scaling limit of either unitary, symplectic
or orthogonal matrices (depending on the value of the constant).
Corollary 1.6. The results of Theorem 1.3 still hold if we weaken the Bounded Multiplicities
condition (1.i) in Definition 1.1: Instead of assuming µf and µg are uniformly bounded, it
suffices to assume that
#{(f, g) : f ∈ FN , g ∈ GM , f = g˜} = O
(
|FN |1−δ|GM |+ |FN ||GM |1−δ
)
(1.10)
for some δ > 0 (in other words, that there is a power savings in the number of pairs where
a g is the contragredient of an f —these lead to imprimitive L(s, f × g)).
Instead of convolving F and G, we can instead fix an f ∈ F and consider the family f ×G
obtained by taking the limit as M →∞ of f × GM .
Theorem 1.7. Assume G and f ×G are NT-good and that G satisfies (1.3) and (1.5). The
symmetry type of f × G is controlled by the following two pieces of input: cG and bf (p2). If
f is a Dirichlet character, holomorphic cusp form or Maass form then we may associate a
symmetry constant cf to f such that cf×G = cf · cG. In particular, we have
(1) if f is a quadratic Dirichlet character then f × G has the same symmetry as G, and
if f is a non-quadratic Dirichlet character then f × G has unitary symmetry;
(2) if G has unitary (resp., symplectic, orthogonal) symmetry, then f × G has unitary
(resp., orthogonal, symplectic) symmetry if f is a Hecke holomorphic or Maass form.
Remark 1.8. If instead (1.2) and (1.4) hold then the result is probably still true (it can be
shown in special cases by partial summation); in general, though, more detailed knowledge
about sums of the coefficients of L(s, f) will be needed.
Remark 1.9. The universality in Theorems 1.3 and 1.7 is reminiscent of that found by
Rudnick and Sarnak [RS], where the universality in the n-level correlations of primitive auto-
morphic cuspidal L-functions is related to universality in the second moments of the Fourier
coefficients aπ(p).
An especially interesting case is when at least one of the two families is a one-parameter
family of elliptic curves over Q(T ) with positive rank rF . Miller [Mil2] showed that the one-
and 1-level densities of zeros of these families agree with those of subgroups of the orthogonal

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 9
group (in many cases unconditionally, in other cases assuming standard conjectures; see
[Yo2] for similar results involving special two-parameter families). As the conductors tend to
infinity, the random matrix ensemble modeling this situation (as N →∞) is
{(
IrF×rF
g
)
, g ∈ C
}
, (1.11)
where IrF×rF is the rF × rF identity matrix and C is O(N) (resp., SO(2N) or SO(2N + 1))
if half the signs of the functional equation are even (resp., all or none); the correct model is
not known for finite conductors (but see [Mil4] for numerical investigations of zeros near the
central point). Indeed, by Silverman’s specialization theorem and the Birch and Swinnerton-
Dyer conjecture, for all t sufficiently large each elliptic curve has (at least) rF zeros at the
central point; moreover, the ensemble (1.11) models these zeros as independent from the
remaining others. This independence is in agreement with function field analogues. We shall
see in Theorem 7.3 that if one convolves two families of elliptic curves with rank then, to first
order, one does not see any effects of this rank in the symmetry group of the new family!
What this means is that the symmetry group of the new family is just unitary, symplectic or
one of the full orthogonal groups. The rank seems to enter only as a lower-order correction
term, which is unfortunately difficult to isolate since it is smaller than bounds we can prove for
the error terms (though, conjecturally, it is larger than the actual bounds for these terms and
should, in principle, be detectable). In this regard our results are similar to Goldfeld’s [Go],
where he considered twists of a fixed elliptic curve by quadratic Dirichlet characters and
conjectured that the new family’s rank was independent of the rank of the fixed elliptic
curve.
In §2 and §3 we review the needed results from number theory and random matrix theory.
We discuss the properties and consequences of being an NT-good family of L-functions in
§4 and then in §5 prove Theorems 1.3 and 1.7. We then give examples of families where
these conditions are met: Convolving families of holomorphic cusp forms in Example 5.4,
symmetric powers of holomorphic cusp forms in §6, and one-parameter families of elliptic
curves in §7; these examples are all independent of each other and may be read in any order.
2. Number Theory Review
We quickly review the notion of automorphic L-functions. These are the L-functions at-
tached to automorphic representations of GLn(AQ). Our examples are built out of objects in
GL1 (Dirichlet L-functions) and GL2 (Maass forms and holomorphic modular forms, including
those attached to elliptic curves). We build more complicated L-functions by taking (Rankin-
Selberg) convolutions and other natural functorial operations (e. g., forming the symmetric
square L-functions). These constructions take us beyond GL2. It is impossible to cover here
but the barest facts about automorphic L-functions; see [Bor, Jac, JPS, RS] for more details.
We will focus on primitive L-functions; these are attached to cuspidal representations and
cannot be further factored as products of other L-functions, hence their critical zeros form
an irreducible set in this sense.

10 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Let π = ⊗ˆvπv be a unitary irreducible cuspidal automorphic representation of GLn(AQ)
with trivial central character. Here v is either a prime p or ∞, and each πv is an irreducible
admissible representation of Qv (where Q∞ := R). The finite part of the L-function attached
to π is an Euler product
L(s, π) =
∏
p
L(s, πp). (2.1)
Outside a finite set of primes, πp is unramified and
L(s, πp) = det(I − p−sAπ(p))−1 =
n∏
j=1
(1− απ,j(p)p−s), (2.2)
where {Aπ(p)} ∈ GLn(C) is a semi-simple conjugacy class parametrized by the eigenvalues
απ,j(p). The Satake correspondence is the bijection Aπ(p) ↔ πp between semi-simple conju-
gacy classes in GLn(C) and unramified irreducible admissible representations of GLn(Qp).
The complex numbers {απ,j(p)}nj=1 are called the Satake parameters of πp. In the context at
hand, the generalized Ramanujan conjecture is the statement that |απ,j| = 1 at the unramified
places (at a ramified prime p some of the απ,j(p) may vanish).
Definition 2.1. For π an automorphic representation, p a prime and πp with Satake param-
eters απ,1(p), . . . , απ,n(p), we define, for ν = 1, 2, 3, . . . ,
bπ(pν) := απ,1(p)ν + · · ·+ απ,n(p)ν . (2.3)
With this definition one has bπ(pν) = Trace(Aπ(p)ν) for unramified p, and
L′(s, π)
L(s, π) =
∑
p
∞∑
ν=1
bπ(pν)
pνs . (2.4)
The archimedean L-factor associated to π∞ is of the form
L(s, π∞) =
n∏
j=1
ΓR(s+ µπ,j), where ΓR(s) = π−s/2Γ
(s
2
)
. (2.5)
The numbers {µπ,j}nj=1 are analogs of the Satake parameters, and the analog of the Ramanujan
conjecture is in this case Selberg’s (generalized) eigenvalue conjecture, namely the statement
that the µπ,j have non-negative real part.
We define the completed L-function by
Λ(s, π) := N s/2π L(s, π∞)L(s, π), (2.6)
where Nπ is a positive integer called the arithmetic conductor. We have the functional
equation
Λ(s, π) = ǫ(π)Λ(1− s, π˜), (2.7)
where π˜ is the contragredient of π and ǫ(π) is a complex constant such that |ǫ(π)| = 1. In
the self-dual case, when π ≃ π˜, the functional equation relates L(·, π) to itself, and ǫ(s, π)
equals ±1.

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 11
For our applications, it is the analytic conductor (not the arithmetic conductor) that is
important for understanding the behavior of the zeros near the central point. The two are
related, and the analytic conductor may be taken as
Qπ = µπ,1 · · · µπ,n Nπ. (2.8)
We use the analytic conductor to rescale the low lying zeros, and then apply the explicit
formula to convert sums of an even Schwartz test function over the zeros of the L-function
to sums of the Fourier transform of the test function evaluated at prime powers. For such
calculations, it is the logarithm of the analytic conductor that normalizes the zeros; see for
example section 4 of [ILS]. In some other papers our factors of µπ,j are replaced with µ′π,j/2.
As we shall always be interested in situations where the analytic conductors tend to infin-
ity, both normalizations lead to the same results. Note that we have N s/2π in our functional
equation —other authors sometimes write this factor as (N ′π)s, which would lead to a factor
of (N ′π)2 in the analytic conductor.
Throughout the paper we make the following two assumptions, unless specified.
• We assume the Generalized Riemann Hypothesis for all automorphic L-functions.
Thus we may write the non-trivial zeros as 12 + iγ with γ ∈ R, and the correct scaling
for zeros near the central point is γ 7→ γ˜ = γ logQπ2π (low-lying γ˜’s have natural uniform
density 1 —see footnote 1). However, the results on 1-level densities may be inter-
preted, and remain true, even when the γ are allowed to be complex; see for example
[ILS]. In other instances (such as in §5.16), GRH is used to bound error terms and
thus enters in the argument in a more essential manner.
• We assume the Langlands functoriality conjectures for the automorphic representa-
tions under consideration. This is necessary in order to ensure that their attached
automorphic L-functions have good analytic properties. In some cases the analytic
properties of an L-function are known even without knowledge of its automorphic-
ity (e.g., for some symmetric-power L-functions attached to holomorphic modular
forms [KiSh1, KiSh2, K].) On the other hand, the automorphicity of all symmetric
powers of an automorphic representation f implies the Ramanujan-Selberg conjectures
for L(s, f), though bounds towards this goal are in some cases available uncondition-
ally [K] (e.g., Deligne’s proof of Ramanujan for holomorphic modular forms).
Remark 2.2. Automorphic L-functions associated to cuspidal representations are primitive
in the sense that one cannot write Λ(s, π) as Λ(s, π1)Λ(s, π2). However, a general non-cuspidal
automorphic L(s, π) factors as a product of primitive ones and its critical zeros are clearly
a union of the zeros of its primitive factors. While the low lying zeros of each primitive
factor will reveal a specific underlying symmetry (at least conjecturally), the low lying zeros of
the imprimitive function will in general not correspond to a definite symmetry. Here we are
using the word “symmetry” in the sense of §3. However, even the assumption of functoriality

12 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
does not ensure that lifts of cuspidal forms are cuspidal. The simplest counterexample is
the imprimitive L-function L(s, π × π˜) where π is a cuspidal automorphic representation of
GLn(AQ), n > 1, and π˜ is its contragredient.4 We will need to deal with this possibility on
occasion.
While we consider quite general families of L-functions, the building blocks for examples
which we can prove satisfy the necessary conditions are (the automorphic representations at-
tached to) either Dirichlet characters or modular forms. For their corresponding L-functions,
classical summation formulas for Fourier coefficients are available that make our approach
tractable.
Let φ be an even Schwartz test function on R whose Fourier transform
φ̂(y) =
∫ ∞
−∞
φ(x)e−2πixydx (2.9)
has compact support. Let F be a finite family of L-functions satisfying GRH. For example,
F might be the set of all L(s, χ) with χ a non-trivial Dirichlet character of conductor m, and
we would then investigate the limit as m → ∞. Other examples include weight-k level-N
cuspidal newforms (and let either k, or N , or both tend to infinity), as well as one-parameter
families of elliptic curves (where the parameter t varies over an interval [N, 2N ], and then we
let N →∞).
Consider a family F of L-functions L(s, f) and denote by Qf the analytic conductor of
L(s, f). Let FN be the finite subfamily of F consisting of those functions with Qf = N . Thus,
F = ∪NFN . To study the zeros of the functions in the family F , we use the Explicit Formula
to convert sums over zeros to sums over primes. For any L-function L(s, f) [ILS, RS]:
∑
ℓ
φ
(
γj;f
logR
2π
)
=
Af
logR φ̂(0)− 2
∑
p
∞∑
ν=1
φ̂
(ν log p
logR
) bf (pν) log p
pν/2 logR , (2.10)
where Af is an integral of gamma factors coming from the functional equation of L(s, f). We
have
Af = logQf + o(logQf ), (2.11)
and the little-oh implicit constant often depends only on F and not the individual f . We
shall also consider variations of the above family; for example, we may let FN be the set of
f in F with N ≤ Qf ≤ 2N . While the subject is considerably simplified if the conductors in
FN are constant, monotonically increasing conductors can be handled with additional work
(see [Mil2] for details for families of elliptic curves).
After averaging over the family, the resulting sums are often evaluated using the following
consequence of the Prime Number Theorem:
4If π1, π2 are automorphic unitary cuspidal representations as above, but not necessarily normalized to
have trivial central character, then L(s, π1 × π2) is imprimitive when π2 ≃ π˜1 ⊗ | det(·)|s is a twist of the
contragredient of π1.

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 13
Theorem 2.3. Let F̂ be an even Schwartz function of compact support. Then for any positive
integer ν,
∑
p
F̂
(
ν log p
logR
)
log p
logR
1
p =
1
2ν F (0) +O
(
1
logR
)
. (2.12)
3. Random Matrix Theory Review
Katz and Sarnak conjecture that to any infinite family F of L-functions one can associate
one of the (infinite families of) classical compact matrix groups (unitary, orthogonal, or
symplectic), say G(F), such that the large-conductor statistics of zeros near the central
point for L(s, f), f ∈ F , agree with those of the eigenvalues near 1 of matrices in G(F)
(as the matrix size N → ∞). Specifically, the n-level density of (rescaled) critical zeros
for the family F is the function Wn,F (to be more precisely, the important object is the
measure Wn,F dx1 . . . dxn on Rn) defined by its action on any test function φ(x1, . . . , xn) =
φ1(x1) · . . . · φ(xn) (where φ1, . . . , φn are Schwartz functions) by:
Dn,F(φ) = limN→∞
1
|FN |
∑
f∈FN
∑
j1,...,jn
ji 6=±jk
φ1
(
γj1;f
logQf
2π
)
· · ·φn
(
γjn;f
logQf
2π
)
=
∫
· · ·
∫
φ1(x1) · · ·φn(xn)Wn,F(x1, . . . , xn)dx1 · · · dxn
=
∫
· · ·
∫
φ̂1(u1) · · · φ̂n(un)Ŵn,F(u1, . . . , un)du1 · · · dun. (3.1)
(The Fourier transform Ŵn,F is most often used in proofs for technical reasons, e. g., using the
Explicit Formula (2.10).) The Katz-Sarnak philosophy posits that the n-level densities Wn,F
agree with the n-level densities Wn,G(F) of the matrix group G(F) associated to the family F .
This philosophical correspondence has been proved for many families when the Schwartz test
functions φi have Fourier transforms supported in a sufficiently small neighborhood of zero.
The n-level densities for the classical compact groups are (see [KaSa1]):
Wn,SO(even)(x) = det(K1(xi, xj))i,j≤n
Wn,SO(odd)(x) = det(K−1(xi, xj))i,j≤n +
∑n
k=1 δ(xk) det(K−1(xi, xj))i,j 6=k
Wn,O(x) = 12Wn,SO(even)(x) + 12Wn,SO(odd)(x)
Wn,Sp(x) = det(K−1(xi, xj))i,j≤n
Wn,U(x) = det(K0(xi, xj))i,j≤n,
(3.2)

14 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
where K(y) = sinπyπy , Kǫ(x, y) = K(x − y) + ǫK(x + y) for ǫ = 0,±1, and δ(u) is the Dirac
Delta functional.5 The Fourier transforms of the 1-level densities are
Ŵ1,SO(even)(u) = δ(u) + 12η(u)
Ŵ1,SO(odd)(u) = δ(u)− 12η(u) + 1
Ŵ1,O(u) = δ(u) + 12
Ŵ1,Sp(u) = δ(u)− 12η(u)
Ŵ1,U(u) = δ(u),
(3.3)
where
η(u) =



1 if |u| < 1
1
2 if |u| = 1
0 if |u| > 1.
(3.4)
When working with test functions φ whose Fourier transform φ̂ is supported in a small
neighborhood of 0, it is still possible to distinguish between unitary, symplectic and orthogonal
n-level densities; however, as long as φ̂ is supported in (−1, 1), all three flavors of orthogonal
symmetry (even, odd, or full) agree:
∫
φ̂(u)Ŵ1,SO(even)(u)du = φ̂(u) + 12φ(0)∫
φ̂(u)Ŵ1,SO(odd)(u)du = φ̂(u) + 12φ(0)∫
φ̂(u)Ŵ1,O(u)du = φ̂(u) + 12φ(0)∫
φ̂(u)Ŵ1,Sp(u)du = φ̂(u)− 12φ(0)∫
φ̂(u)Ŵ1,U(u)du = φ̂(u).
(3.5)
Let sign(G) = 0 (resp., 12 , 1) for G = SO(even) (resp., O, SO(odd)). For even functions
φ(x1, x2) = φ1(x1)φ2(x2) such that φ̂(u1, u2) = φ̂1(u1)φ̂2(u2) is supported in |u1|+ |u2| < 1,
∫ ∫
f̂1(u1)f̂2(u2)Ŵ2,G(u)du1du2 =
[
f̂1(0) + 12f1(0)
][
f̂2(0) + 12f2(0)
]
+ 2
∫
|u|f̂1(u)f̂2(u)du− 2f̂1f2(0)− f1(0)f2(0)
+ sign(G)f1(0)f2(0).
(3.6)
Thus, for arbitrarily small support, the 2-level density distinguishes the three orthogonal
groups; see [Mil1] for the calculation.
In studying families of elliptic curves [Mil2, Yo2], often the corresponding classical compact
group is a subgroup of one of the orthogonal groups. For one-parameter families of elliptic
curves over Q(T ) with rank rF , as remarked in (1.11), the correct model as the conductors
5While these determinant formulas hold for arbitrary support, in practice the resulting formulas for n ≥ 3
require some combinatorics when the support is large before agreement is seen with number theory. Hughes
and Miller [HM] derive an alternate formula for n-level statistics; while their formula holds for smaller support,
in the range where it is applicable it facilitates comparisons with number theory.

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 15
tend to infinity appears to be
{(
IrF×rF
g
)
, g ∈ C
}
, (3.7)
where IrF×rF is the rF×rF identity matrix and C is O (resp., SO(even) or SO(odd)) if half the
signs of the functional equation are even (resp., all or none), though see [Mil4] for a discussion
of the behavior for finite conductors. These rF independent zeros replace φ̂(u) + 12φ(0) with
φ̂(u) + 12φ(0) + rFφ(0) in the 1-level density expansion, and there is a similar modification in
the n-level density.
Because of this effect of rank, we attach a family constant to each family of L-functions F :
c˜F = (cF , ǫF , rF). (3.8)
Here cF is the symmetry constant of the family, equal to 0 (resp., 1 or −1) if the family is
unitary (resp., symplectic or orthogonal); we call any subgroup of O, SO(even) or SO(odd)
orthogonal. Since the three orthogonal groups all have cF = 1, we set ǫF equal to 0 (resp., 1
or −1) if F has half of the signs of its functional equation even (resp., all signs even or odd);
if F is not associated to an orthogonal group, then cF alone determines the precise group and
we define ǫF = 0. Finally, rF denotes the rank of F ; except for families of elliptic curves, all
other known families have rF = 0.
Our main result (Theorem 1.3) is that in order to determine the symmetry of the the
Rankin-Selberg convolution of two NT-good families, all that matters is cF and cG. Thus
we may interpret the symmetry constant as a convolution constant. Further, the new family
has rank zero. (This is unfortunate, since otherwise this would allow constructing families of
L-functions with high central vanishing.)
4. NT-good Families and n-Level Densities
As a warm-up to proving our main theorems in §5, in this section we investigate some
consequences of Definition 1.1 (NT-good).
It is worth commenting on the main terms in (1.2) and (1.3). Consider a one-parameter
family of elliptic curves over Q(T ) with rank rF ; FN is essentially just {Et : t ∈ [N, 2N ]}.
Then bt(p) = at(p)/
√p, where at(p) are the coefficients of the L-series of L(s, Et) (with
functional equation s→ 2−s, so the critical strip is ℜs ∈ [0, 2]). Rosen and Silverman [RoSi]
prove a conjecture of Nagao’s (unconditionally if the elliptic surface is rational; conditional
on Tate’s conjecture otherwise):
lim
X→∞
1
X
∑
p≤X
1
p
p−1∑
t=0
at(p) log p = −rF . (4.1)
Thus the at(p)’s give the rank of the family over Q(T ). For many families of elliptic curves
(see [ALM, Fe]), the main term of the average over Et ∈ FN of at(p)/p is independent of p,

16 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
and we have
− 1|FN |
∑
Et∈FN
bt(p)√p = −
1
|FN |
∑
Et∈FN
at(p)
p
=
1
|FN |
[
−|FN |p
∑
t mod p
at(p)
p
]
+O
( √p
|FN |
)
=
rF
p +O
( √p
|FN |
)
. (4.2)
If we have such a family we use (1.3); if not, we need to do a little more work and use (1.2)
and (2.12). The proofs follow similarly, the only real difference being a partial summation on
the primes to handle the test functions.
For ease of exposition we concentrate on cases where (1.3) holds, and remark that similar
arguments handle the case when we have (1.2).
Theorem 4.1. Let F be an NT-good family of automorphic L-functions for GLn. Then for
even Schwartz test functions φ such that φ̂ is supported in a sufficiently small (but explicitly
computable in terms of the constants δi, µi) neighborhood of 0, if rF = 0 then the 1-level
density of F agrees with unitary (resp., symplectic or orthogonal) if cF = 0 (resp., 1 or −1);
if rF > 0 then the corresponding classical compact group is modified by having an rF × rF
identity matrix as in (1.11).
Proof. Using the explicit formula to calculate the 1-level density, we have the expansion
D1,FN (φ) = φ̂(0)−
2
|FN |
∑
f∈FN
∞∑
ν=1
RσN∑
p=2
bf (pν) log p
pν/2 logRN
φ̂
(
ν log p
logRN
)
+ o(1). (4.3)
From (2.11), the factor of φ̂(0) above comes from the constancy of the main term of the
analytic conductors (and an analysis of the Γ-factor terms; in fact, this is what we use to
determine RN); the o(1) term arises from the correction factor in logQf = logRN+o(logRN ).
As our family is NT-good, there is no contribution from bf (pν) for ν ≥ 3 (either for all support,
or for support sufficiently small). Thus those terms may be absorbed into an error term.

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 17
We assume that (1.3) holds; the case when (1.2) holds follows similarly. The ν = 1 terms
contribute
S1 = −
2
|FN |
∑
f∈FN
RσN∑
p=2
bf(p) log p√p logRN
φ̂
(
ν log p
logRN
)
= 2
RσN∑
p=2
[
− 1|FN |
∑
f∈FN
bf(p)
]
log p√p logRN
φ̂
(
ν log p
logRN
)
= 2
RσN∑
p=2
[ rF√p +O
(
|FN |−δ1pµ1
)] log p√p logRN
φ̂
(
ν log p
logRN
)
= rF
RσN∑
p=2
log p
p logRN
φ̂
(
ν log p
logRN
)
+O

 1
|FN |δ1
RσN∑
p=2
pµ1− 12


= rFφ(0) +O
(
1
logRN
)
+O
(
R(µ1+
1
2 )σ
N
|FN |δ1
)
, (4.4)
where the main term in the last line is an immediate consequence of the Prime Number
Theorem (see Theorem 2.3 for a proof); as |FN | ≥ Rδ0N , for σ sufficient small (in terms of
µ1, δ1 and δ0), the last error term is negligible.
We are left with the contribution from the squares of the primes (the ν = 2 terms). As∑
f∈FN bf (p
2) = cF |FN | + O(|FN |1−δ2pµ2), for sufficiently small support, up to a negligible
term by Theorem 2.3 the resulting sum over primes is φ(0)2 . Thus the 1-level density satisfies
D1,F(φ) = φ̂(0)− cF ·
1
2
φ(0) + rFφ(0), (4.5)
which for small support agrees with the 1-level densities of (3.5) (trivially modified if there
are rF forced eigenvalues at 1). 
Remark 4.2 (Support of the test functions). The allowable support of φ̂ is determinable
from the constants δi, µi. In general, the support will not be large enough to distinguish the
three orthogonal densities, though it will suffice to distinguish unitary from symplectic from
orthogonal.
Remark 4.3 (General n-level density). It is natural to investigate the 2-level density to
distinguish the orthogonal groups. To do so requires two additional pieces of information: (1)
the distribution of signs of functional equations in the family; (2) being able to average over
the family bf (p1)bf (p22) and bf(p21)bf(p22). The presence of cross terms can seriously complicate
matters, though fortunately in all families considered to date these terms can be converted to
products of averages of single terms. For cuspidal newforms this follows from the Petersson
formula; for one-parameter families of elliptic curves, if p1, . . . , pk are distinct primes and

18 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
r1, . . . , rk are integers, simple counting (see [Mil2]) shows that
∑
t mod p1···pk
ar1t (p1) · · ·arkt (pk) =
k∏
i=1
∑
t mod pi
arit (pi), (4.6)
reducing the analysis to single product terms. In general (though see Remark 4.4 below), if
we know the distribution of signs we can determine the 2-level densities, as all that matters is
knowing the fraction of L(s, f) with even or odd sign; however, the story is markedly different
for third and higher level densities. There we need to know significantly more; we need to
know which of the L(s, f) have odd functional equation, and we need to execute sums over
just those L(s, f). Thus, for families of elliptic curves the third and higher level densities are
beyond current techniques (except for constant-sign families); see [Mil1] for more details.
Remark 4.4 (Variation in the analytic conductors). We concentrate on the 1-level density
in this paper. There are two ways to normalize the zeros of an L-function in a family FN : we
can use logQf or logRN . For the 1-level density, it does not matter which one is used in the
normalization; however, for the higher n-level densities we need to use the explicit formula
multiple times. While the normalization logRN greatly simplifies the 1-level computations
(because all test functions are scaled equally), in the general case we are forced to evaluate
sums of logQf against the coefficients of the L-functions. With additional work, these sums
can often be handled (see [Mil2] for the case of one-parameter families of elliptic curves).
5. Proof of the Main Results
5.1. Preliminaries.
Assuming F and G are NT-good families, by Theorem 4.1 we can determine their 1-level
densities, and associate a classical compact group to the family (uniquely in the case of unitary
and symplectic symmetry; for orthogonal symmetries, for small support the 1-level density
cannot distinguish SO(even) from O from SO(odd)). Assuming a few additional conditions,
we can determine the symmetry group of the Rankin-Selberg convolution of the two families
F and G.
Lemma 5.1. Assume π1, π2 are automorphic cuspidal representations of GLn(AQ), GLm(AQ),
respectively, and further that the functorial lift π1×π2 to GLmn(AQ) exists. Assume that, for
some prime p, both π1,p and π2,p are unramified, and their corresponding Satake parameters
are {απ1(i)}1≤i≤n and {απ2(j)}1≤j≤m. Then
bπ1×π2(pν) = bπ1(pν) · bπ2(pν). (5.1)
Proof. Let bπ1(pν) and bπ2(pν) be as in Definition 1.1. By the local Langlands correspondence,
the Satake parameters for π1,p × π2,p are
{απ1×π2(k)}nmk=1 = {απ1(i) · απ2(j)} 1≤i≤n1≤j≤m , (5.2)

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 19
which gives
bπ1×π2(pν) =
nm∑
k=1
απ1×π2(k)ν
=
n∑
i=1
m∑
j=1
απ1(i)ν · απ2(j)ν
=
n∑
i=1
απ1(i)ν ·
m∑
j=1
απ2(j)ν
= bπ1(pν) · bπ2(pν). (5.3)

Remark 5.2. In Lemma 5.1 above the assumption that π1 × π2 is automorphic is not really
necessary in order to define the Satake parameters at p nor the coefficients bπ1×π2(pν) of a
hypothetical π1 × π2. Recall that our viewpoint is that the L-function L(s, π1 × π2) should
be automorphic and primitive, which is the case if π1, π2 are cuspidal, except when π1 ≃ π˜2
(assuming unitary and of trivial central character).
If π is an automorphic cuspidal representation of GLn, then L(s, π) is primitive if and only
if π is primitive. For two unitary cuspidal automorphic representations π and π′ of GLn with
trivial central character, their convolution L(s, π× π′) is primitive if and only if π′ is not the
contragredient of π. This is equivalent to the lift π × π′ being a cuspidal representation of
GLn2.
We now prove our main results.
5.2. Convolving two families.
Proof of Theorem 1.3. We first prove the theorem under the assumption that all convolved
L-functions are primitive, and then handle the general case.
As we are assuming the convolved family is NT-good, in the new family FN × GM the
conductors are essentially constant, say logQf×g = logRN,M + o(logRN,M). By the multi-
plicativity assumptions, it will be relatively easy to evaluate
D1,FN×GM (φ)
= φ̂(0)− 2 · 1|FN | · |GM |
∑
f×g∈FN×GM
∞∑
ν=1
RσN,M∑
p=2
bf×g(pν) log p
pν/2 logRN,M
φ̂
(
ν log p
logRN,M
)
+ o(1).
(5.4)
Some care is required for the ν ≥ 3 terms. We need to show that these give a negligible
contribution. If condition (4.iii) holds for either of the NT-good families (namely, that if we
sum over the family, we have a power savings in the family cardinality), then this follows

20 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
immediately. If not, we need δ3(F), r3(F) (and similarly for the family G) to be such that
summing the ν ≥ 3 terms is negligible. This is always the case if we assume the Ramanujan
conjecture, condition (4.i).
We must determine the contributions from the ν = 1, 2 terms. As
bf×g(pν) = bf (pν) · bg(pν) (5.5)
(Lemma 5.1), we can execute the summations over f ∈ FN and g ∈ GM . The main term from
ν = 2 is
− 2 · 1|FN | · |GM |
RσN,M∑
p=2
cF |FN | · cG|GM | log p
p logRN,M
φ̂
(
2
log p
logRN,M
)
. (5.6)
If the zeros were normalized by logQf×g instead of logRN,M , it would not be as easy to
compute the contributions because the Schwartz functions would be evaluated at points
depending on Qf×g. This is the main reason we choose to normalize all zeros in a family by
the same quantity.
By the Prime Number Theorem (Theorem 2.3), the main term of the sum in (5.6) equals
− cF · cG
2
φ(0), (5.7)
and the error term is negligible. There are three other terms which contribute in the ν = 2
case:
|FN |1−δ2(F) · |GM | · pµ2(F), |FN | · |GM |1−δ2(G) · pµ2(G), |FN |1−δ2(F) · |GM |1−δ2(G) · pµ2(F)+µ2(G).
As we divide by |FN | · |GM |, each of the three terms leads to a negligible contribution for test
functions with suitably small support.
We are left with handling the ν = 1 terms. If rF or rG = 0 then we immediately see this
term does not contribute for suitably small support. For notational convenience we assume
(1.3) and not (1.2) holds, as the argument in each case is similar. We have
∑
f×g∈FN×GM
bf×g(p) =
[∑
f∈FN
bf (p)
]
·
[∑
g∈GM
bg(p)
]
=
rF · rG
p |FN | · |GM |+O
(
|FN |1−δ1,F · |GM |1−δ1,Gpµ1,F+µ1,G
)
+ O
(
|FN |1−δ1,F · |GM | · pµ1,F−
1
2
)
+ O
(
|FN | · |GM |1−δ1,G · pµ1,G−
1
2
)
. (5.8)
Summing over p, for test functions with small support the three error terms do not contribute.
The main term leads to
− 2
∑
p
rF · rG
p
log p√p logRN,M
φ̂
(
log p
logRN,M
)
. (5.9)

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 21
If F ×G were to have rank, this sum would have to contribute. Comparing to equation (1.2),
the difference is that in the sum above we have rF ·rGp instead of something like
r√p times
log p√p logR φ̂
(
log p
logR
)
. The presence of p rather than √p in the denominator means this sum is
of size (logRN,M)−1 rather than of size 1. This leads to a lower order correction term to the
1-level density of size rF ·rGlogRN,M .
We now remove the assumption that all the convolutions are primitive; i.e., we now allow
a contragredient of an f ∈ F to be in G. This can only happen if m = n. All we require
is some control on the number such pairs (f, f˜) and their contribution. As the families are
NT-good, the multiplicities are bounded: µf ≤ µF and µg ≤ µG. Thus the number of such
pairs is trivially bounded by min(µG|FN |, µF |GM |) = O(min(|FN |, |GM |)).
For any such (f, f˜) the convolution L(s, f × f˜) is not primitive and has a simple pole
at s = 1, contributing two additional terms, φ
(
± logR4π i
)
, to the explicit formula (see, for
example, [RS]). If supp(φ̂) ⊂ (−σ, σ), then
φ(t+ iy) =
∫ ∞
−∞
φ̂(ξ)e2πi(t+iy)ξdξ ≪ e2π|y|σ. (5.10)
Thus
φ
(
± logR
4π i
)
≪ Rσ/2; (5.11)
as we divide by |FN | · |GM | and there are only O(min(|FN |, |GM |)) such pairs, for σ sufficiently
small these two terms have a negligible contribution.
We now show the contribution to the prime sums from these pairs is also negligible if
σ is sufficiently small. Any improvement of the exponent 1/2 in the Jacquet-Shalika [JS]
bound for the Satake parameters suffices; we use the Rudnick-Sarnak6 [RS] bound: if π is an
automorphic representation of GLr(AQ), then |απ,j(p)| ≤ p
1
2−
1
r2+1 . Thus each pair contributes
at most
∑
ν
∑
p≤Rσ/ν
p
ν
2−
ν
n4+1
pν/2 ≪n R
σ. (5.12)
to the prime sums in the explicit formula. As there are only O(min(|FN |, |GM |)) pairs,
for σ sufficiently small these lead to negligible contributions upon dividing by the family’s
cardinality, |FN | · |GM |. 
Remark 5.3. The universality in Theorem 1.3 can be surprising at first. In determining
the underlying classical compact group of the convolution of two families, all that matters
are the distribution of signs of functional equations, the rank of the family and the family
averages of the bf (p)’s and bg(p)’s (i.e., the family averages of the second moments of the
Satake parameters at each unramified prime). Upon convolving two such nice families, the
6The Rudnick-Sarnak bound is stated only for Satake parameters of cuspidal representations, and it trivially
extends to isobaric sums of cuspidal representations, hence to arbitrary (not necessarily cuspidal) automorphic
representations.

22 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
main term is independent of the family ranks; however, there is a lower order correction term
which can often be isolated and which does depend on the ranks. Unfortunately the bounds
for the errors from the ν ≥ 3 terms, even assuming Ramanujan, will be of the same size, as
could the other error bounds from the ν = 1 and ν = 2 terms. Conjecturally, however, it is
reasonable to expect there to be cancelation in these errors upon summing over the families,
and hence that there could be corrections to the 1-level density. For other examples of lower
order corrections, see [FI, Mil3, Mil5, Mil7, Mil8, St, Yo1].
Example 5.4. We give an interesting example of Theorem 1.3. Consider families Fi of
weight-ki holomorphic cuspidal newforms of prime level N (ki fixed, N → ∞); perhaps we
might want to take the sub-families of even or odd sign. These families Fi are NT-good,
as is F1 × F2 when k1 6= k2. By [ILS] each Fi has orthogonal symmetry. As these are
GL2 holomorphic cuspidal newforms, we know the Ramanujan conjectures and thus condition
(4.i) holds. From the Petersson formula, (1.3) holds with rFi = 0. Thus these families
have orthogonal symmetries and hence their symmetry constants are cFi = −1. Therefore
cF1×F2 = cF1 · cF2 = 1, implying that F1 × F2 has symplectic symmetry. In particular, all
elements should have even sign (which we do get from Rankin-Selberg). Note this is a GL4
family of L-functions. Is there a larger natural GL4 family containing it (similar to the
quadratic Dirichlet characters sitting inside all Dirichlet characters)? Further, if k1, k2 and
k3 are distinct then cF1×F2×F3 = cF1 · cF2 · cF3 = −1, implying F1 × F2 × F3 has orthogonal
symmetry.7
5.3. Convolving by a Fixed Form.
Proof of Theorem 1.7. We may assume all the convolutions are primitive. As G is NT-good,
from our cardinality assumption there are at most µG = O(1) imprimitive convolutions.
Arguing as in the proof of Theorem 1.3 we can show they have a negligible contribution for
sufficiently small support.
Using the explicit formula to calculate the 1-level density, we have the expansion
D1,f×GM (φ) = φ̂(0)−
2
|GM |
∑
g∈GM
∞∑
ν=1
RσM∑
p=2
bf (pν)bg(pν) log p
pν/2 logRM
φ̂
(
ν log p
logRM
)
+ o(1).
(5.13)
There will be no contribution from the ν ≥ 3 terms if we have sufficiently good bounds for
bf (pν)bg(pν)
pν/2 , or if we have some power savings (relative to |GM |) in
∑
g∈GM bg(p
ν). For example,
if we take f to be any nice L-function on GL2 (say a holomorphic cuspidal newform of weight k
and level N or an even Maass form), then we have good bounds on bf (pν). In the holomorphic
7It is known that f1 × f2 is automorphic. While it is not known that f1 × f2 × f3 is automorphic, we do
know that L(s, f1× f2× f3) is entire; the automorphicity follows from standard functoriality conjectures, and
would imply the L-function is primitive. See [Bu, Ga, Ram] for details.

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 23
case, we know Ramanujan and bf (pν) ≪ 1; in the Maass case we have bf(pν) ≪ p7/64 (see
[K]). We can quantify exactly what bounds we need on bg(pν) for each g ∈ GM (ν ≥ 3), and
these bounds are available in many cases of interest. We then execute the summation over p,
which gives (logRM)−1, and then we trivially handle the sum over GM .
As G satisfies (1.3), for ν = 1 a simple calculation shows there is no contribution in the
limit as M → ∞ for sufficiently small support. We are left with the crucial case of ν = 2;
note that this is the term that determines the symmetry type: If it is 0 (resp., 1 or −1), we
have unitary (resp., symplectic or orthogonal). From our assumption that G satisfies (1.5),
we can execute the summation
∑
g∈GM bg(p
2), and we find that the main contribution from
ν = 2 is just
− 2|GM |
RσM∑
p=2
[
cG · |GM |+ |GM |1−δ2pµ2
]
bf (p2) log p
p logRM
φ̂
(
2
log p
logRM
)
. (5.14)
For sufficiently small support, as bf (p2) is bounded by some power of p, the second term
doesn’t contribute. We are left with
− 2cG
RσM∑
p=2
bf (p2) log p
p logRM
φ̂
(
2
log p
logRM
)
. (5.15)
Thus the symmetry will be the product of cG and the above sum. If f is a Dirichlet character
χ, then bf (p2) = χ(p)2. If χ is quadratic than χ(p)2 = 1 and the symmetry constant will be
cG again; if χ is not quadratic than the sum of χ(p)2 (times the other factors) over the primes
is o(1), yielding unitary symmetry. If f is a nice GL2 L-function (say holomorphic cuspidal
Hecke newform or Hecke-Maass), then the prime sum is −14φ(0) because
bf (p2) = α2f,1(p) + α2f,2(p)
= (αf,1(p) + αf,2(p))2 − 2
= af(p)2 − 2
=
[
af (p2) + 1
]
− 2
= af(p2)− 1, (5.16)
where we have used the fact that f is a Hecke eigenform to say af(p)af (p) = af(p2)+1 (at least
for p relatively prime to the conductor). The af(p2) will be related to the symmetric square
L-function associated to f , and by GRH for that L-function, its sum over primes is negligible
(see [ILS] for details). Thus the ν = 2 terms contribute (−2cG) · (−14φ(0)) = cG · 12φ(0).
Setting cf = 1 if f is a quadratic Dirichlet character, 0 if f is a non-quadratic Dirichlet
character, and cf = −1 if f is a Hecke holomorphic or Maass form, we find that the 1-level
density of f × G is
φ̂(0)− cf · cG ·
1
2
φ(0). (5.17)


24 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Remark 5.5. These results are similar to those obtained by Rubinstein in his thesis, where
he considered the convolution of the family of quadratic Dirichlet L-functions with a fixed
GLn form; see [Rub]. In our notation, if f is self-dual, then cf = +1 (resp., cf = −1) if
L(s, sym2f) (resp., L(s,∧2f)) has a pole at s = 1. If f is not self-dual then cf = 0.
6. Convolving Families Of Symmetric Powers Of Modular Forms
Families of L-functions attached to holomorphic modular forms and their functorial liftings
are often NT-good, at least under the assumption of standard conjectures. The main purpose
of this section is to provide further examples illustrating Theorems 1.3 and 5.16. Additional
examples (independent of this section) involving elliptic curves are given in §7.
Let Hk be a Hecke eigenbasis of the space of modular cusp forms of weight k for the full
modular group SL2(Z). Then |Hk| = k12 +O(1). We denote the average over Hk by
〈Af〉Hk :=
1
|Hk|
∑
f∈Hk
Af . (6.1)
We normalize f ∈ Hk so its leading Fourier coefficient is one, viz.,
f(z) =
∞∑
n=1
af (n)
nk−12
exp(2πinz) (6.2)
L(s, f) =
∞∑
n=1
af (n)n−s =
∏
p
(1− af(p)p−s + p−2s)−1 (6.3)
=
∏
p
(1− αf (p)p−s)−1(1− αf(p)−1p−s)−1, ℜs > 1, (6.4)
Λ(s, f) = 2(2π)−(s+ k−12 )Γ
(
s + k − 1
2
)
= (−1)k/2Λ(1− s, f). (6.5)
Here αf(p), αf(p)−1 are the Satake parameters at p. Since we never need to look at Satake pa-
rameters simultaneously for two different primes, we usually omit p and write simply αf . It is
well known that f uniquely determines an automorphic cuspidal unitary self-dual representa-
tion π of GL2 with trivial central character. Moreover π∞ is the discrete series representation
of weight k.8 In what follows we will implicitly use this identification and rarely bother to
talk about the representation π per se. From the completed L-function in equation (6.5) (in
particular from its gamma factor) it follows that the analytic conductor of Fk is Rk ≍ k2.
Because Hk consists of forms of full level, πp is unramified for all p. The following orthog-
onality relations for the Fourier coefficients {af(n)} are crucial:
8Some authors prefer to say that this π∞ has weight k − 1. We follow the convention in [CM].

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 25
Lemma 6.1. We have
1
|Hk|+O(1)
∑
f∈Hk
ζ(2)
L(1, sym2f) af (m)af (n) = δ(m,n) + E , (6.6)
where
δ(m,n) =
{
1 m = n
0 m 6= n (6.7)
and
E =



Oℓ
(
(mn)1/4 logmn
k5/6
)
if m,n have no more than ℓ factors
O
(√mn
2k
)
if 12π√mn ≤ k.
(6.8)
Formula (6.6) is a consequence of the Petersson formula (see equation 2.12 of [ILS]). Note
that the left-hand side of (6.6) is just the average 〈af(m)af (n)〉Hk , except for the presence
of the weights ζ(2)/L(s, sym2f). This is called the harmonic averaging of af(m)af (n) and
often makes the analysis more tractable (see [DM, ILS, Mil8, Ro]). If we were interested in
bounding the order of vanishing at the central point in the family then the harmonic weights
would cause difficulty (see Remarks 2.11 and 6.1 in [HM]).
Following [ILS], by additional work we can remove the harmonic weights in the 1-level
density. The cost is a slight worsening the constants δ1, δ2, δ3 in the definition of NT-good.
Alternatively, we can simply redefine the average 〈af(m)af (m)〉Hk to be given by the left-hand
side of (6.6).
Note that
af (pn) = αnf + αn−2f + · · ·+ α−n+2f + α−nf , (6.9)
so from Definition 2.1 it follows immediately that
bf(p) = af (p)
bf (p2) = af (p2)− 1. (6.10)
These formulas, together with the orthogonality relations (6.6), already suffice to prove
conditions (3.i) and (3.ii) of Definition 1.1 with δ1 = δ2 = 1/6, any µ1 > 1/4, µ2 > 1/2, rank
zero and, most importantly, with symmetry constant −1 (note that af(pn) = af (pn)af(1) and
δ(1, pn) = 0 for n = 1, 2, whereas −1 = −af (1)af (1)). Conditions (1) and (2) are obvious,
and the Ramanujan conjecture (condition (4)) is known for these f by Deligne. We therefore
recover the result from [ILS, Ro] that the family {Hk} as k → ∞ has orthogonal 1-level
density (at least for small support).
For small support of test functions, one cannot in general pinpoint the exact underlying
symmetry in the orthogonal case. However, with the help of the root number (sign of the
functional equation), the symmetry should be SO(even) if all the functional equations have
positive sign and SO(odd) if all have negative sign. Determining the sign of the functional
equation is most easily done through the local Langlands correspondence. Since we will
be building automorphic representations starting from modular forms of full level, all finite
places (p prime) contribute local root numbers equal to +1, and we only need the archimedean

26 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
local correspondence. Moreover, since the only archimedean place of Q is Q∞ = R we can
simplify the notation a bit. The reader who wants an authoritative survey of the archimedean
Langlands correspondence should read Knapp’s article [Kn].
The archimedean local correspondence for GLn(AQ) is a bijection ρ ↔ π∞ between ad-
missible representations ρ : WR → GLn(C) and irreducible admissible representations π∞ of
GLn(R). Here WR := C× ∪ jC× (disjoint union) is a multiplicative group with j2 = −1, and
j acts on C× by jzj = z¯. WR is the Weil group of R; it can also be identified with an obvious
multiplicative subgroup of the quaternions. We will not discuss the meaning of admissibility
here.
Irreducible admissible representations of WR are one or two dimensional. There are two
families of inequivalent one-dimensional representations, each parametrized by a complex
number t ∈ C. They are denoted {[+, t]} and {[−, t]}. Additionally, there are two-dimensional
representations; they are parametrized by an integer k ≥ 2 and a complex number t ∈ C.
They are denoted [k, t]. There are no irreducible admissible representations of dimension
greater than two, and any (finite-dimensional) admissible representation of WR is fully re-
ducible (decomposes as a direct sum of irreducible ones).
The correspondence assigns [+, 0] to the trivial representation and [−, 0] to the “sign”
representation x 7→ sgn(x) = x |x|−1 of GL(1,R). The discrete-series representation of weight
k ≥ 2 corresponds to [k, 0]. The parameter t ∈ C parametrizes twists: either by the character
|x|t of GL(1,R) or by |det(x)|t of GL(2,R).
In order to characterize the archimedean components of functorial liftings of automorphic
representations, we need to understand the effect of certain operations on representations of
WR.
Lemma 6.2. Let (−)κ be ‘+’ for κ even, ‘−’ for κ odd. Then, for all m ≥ 1, k > k′ ≥ 2,
t, t′ ∈ C:
∧2[k, t] ≃ [(−)k, 2t] (6.11)
symm[+, t] ≃ [+, mt] (6.12)
symm[−, t] ≃ [(−)m, mt] (6.13)
sym2m+1[k, t] ≃
m⊕
ℓ=0
[(2ℓ+ 1)(k − 1) + 1, (2m+ 1)t] (6.14)
sym2m[k, t] ≃ [(−)m(k−1), 2mt]⊕
m⊕
ℓ=1
[2ℓ(k − 1) + 1, 2mt] (6.15)
[+, t]⊗ [+, t′] ≃ [−, t]⊗ [−, t] ≃ [+, t+ t′] (6.16)
[+, t]⊗ [−, t′] ≃ [−, t′]⊗ [+, t] ≃ [−, t+ t′] (6.17)
[+, t]⊗ [k, t′] ≃ [−, t]⊗ [k, t′] ≃ [k, t+ t′] (6.18)
[k, t]⊗ [k′, u] ≃ [k′, u]⊗ [k, t] ≃ [k + k′ − 1, t+ t′]⊕ [k − k′ + 1, t+ t′] (6.19)
[k, t]⊗ [k, t′] ≃ [2k − 1, t+ t′]⊕ [+, t+ t′]⊕ [−, t+ t′] (6.20)

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 27
The proof is easy and we omit it. Cogdell and Michel prove (6.14) and (6.15) in [CM].
The archimedean ε- and L- (gamma) factors are as follows 9 (see [Kn]):
ΓR(s) : = π−s/2Γ(s/2) (6.21)
ΓC(s) : = ΓR(s)ΓR(s+ 1) = 2(2π)−sΓ(s) (6.22)
L(s, [+, t]) = ΓR(s+ t) ε([+, t]) = 1 (6.23)
L(s, [−, t]) = ΓR(s+ t+ 1) ε([−, t]) = i (6.24)
L(s, [k, t]) = ΓC
(
s+ t+ k−12
)
ε([k, t]) = ik. (6.25)
Finally, ε- and L-factors are multiplicative with respect to direct sums of representations of
WR (which, via the archimedean Langlands correspondence, are associated to isobaric sums
of irreducible admissible representations of GLni(R)), and if ρ↔ π∞ then L(s, π∞) = L(s, ρ),
and similarly for ε-factors.
With these results in hand we can easily determine the underlying symmetry type of var-
ious families obtained by functorial operations starting from {Hk}. However, we introduce
one last bit of notation: since, for f ∈ Hk, the automorphic representation πf is unitary and
has trivial central character, the parameter t ∈ C is always zero in our applications and we
will adopt the following:
Convention: We write [k] for [k, 0], [+] for [+, 0], and [−] for [−, 0]. Also define [1, t] :=
[+, t]⊕ [−, t] and [1] := [1, 0]. Then equation (6.20) is the special case k = k′ of (6.19). Note
that [1, t] is a reducible two-dimensional representation of WR.
6.1. Families of Symmetric Powers. We begin with the family G(M)k = symMHk for
a fixed M ≥ 1, and study the limit as k → ∞. It is conjectured, and we assume this
as a hypothesis, that every f ∈ Hk has a self-dual automorphic cuspidal functorial lift
g = symMf ≃ ⊗′(symMfv) whose local factors symMfv are defined through the local Lang-
lands correspondence (by composition with the M-th symmetric-power of the defining rep-
resentation of GL2(C)). This is known for M = 1, 2, 4 by work of Hecke, Gelbart-Jaquet,
Kim-Shahidi, and Kim [GJ, KiSh1, K]. Under this hypothesis, the family G(M)k = symMHk
consists of primitive L-functions for GLM+1.
Theorem 6.3. With the assumptions above, the family G(M) is NT-good with symmetry
constant cG = (−1)M .
Proof. Firstly, |G(M)k | = |Hk| = k +O(1) →∞ as k →∞, so the cardinality condition holds.
Now, f∞ ≃ [k] for all f ∈ Hk, so g∞ ≃ symMf∞ are all isomorphic admissible represen-
tations of GLM+1(R) as f varies over Hk. In addition, all the non-archimedean places are
unramified, so the analytic conductors Qg are completely determined by g∞, and hence are
constant in Gk.
9With respect to standard Lebesgue measure on R and provided the additive character used to define the
Fourier transform is x 7→ e2πix.

28 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
To compute the ε- and Γ-factor L∞(s, symMf) we use the Langlands correspondence,
Lemma 6.2, and equations (6.23)–(6.25). Recall that f∞ ≃ [k].
We split into two cases: M = 2m and M = 2m+ 1.
• M = 2m+ 1.
Γ(s, sym2m+1f) =
m∏
ℓ=0
ΓC
(
s+ (2ℓ+1)(k−1)2
)
(6.26)
ε(s, sym2m+1f) =



ik m ≡ 0 (mod 4)
−1 m ≡ 1 (mod 4)
−ik m ≡ 2 (mod 4)
1 m ≡ 3 (mod 4).
(6.27)
• M = 2m.
Γ(s, sym2mf) = ΓR
(
s+ 1−(−1)m(k−1)2
) m∏
ℓ=1
ΓC
(
s + 2ℓ(k−1)2
)
(6.28)
ε(s, sym2mf) = 1. (6.29)
As explained in [DM], the contribution of the archimedean places to the analytic conductor
Qg can be read off from the gamma factors (cf., equations (2.10) and (2.11) at the end of
§2): each factor ΓR(s + T ) contributes a factor of T/2 to the analytic conductor, and each
factor ΓC(s+T ) contributes T (T +1)/4 ≍ T 2/4. Equations (6.26) and (6.28) reveal that the
analytic conductor is Qg ≍ (k/2)M+1 if M is odd, Qg ≍ (k/2)M if M is even. This verifies the
conductors condition. The error terms are handled using the Ramanujan bounds of Deligne.
To analyze the crucial conditions on prime sums, let us write αp, α−1p for the Satake pa-
rameters of fp. Then those of (symMf)p are
αMp , αM−2p , . . . , α−M+2p , α−Mp . (6.30)
Writing a(pn), b(pn) for af (pn), bf(pn) and B(pn) for bsymMf(pn), we have:
B(p) = αMp + αM−2p + · · ·+ α−M+2p + α−Mp
= a(pM) (6.31)
B(p2) = α2Mp + α2M−4p + · · ·+ α−2M+4p + α−2Mp
= a(p2M )− a(p2M−2) + a(p2M−4)− a(p2M−6) + · · ·+ (−1)Ma(1). (6.32)
Once again the orthogonality relations of (6.6) prove the condition on the prime sum (with
rank zero) and the prime square sum, with symmetry constant csymMHk = (−1)M . This reveals
underlying symplectic symmetry when M = 2m is even and orthogonal when M = 2m + 1
is odd. In the latter case, by looking at the ε-factor (root number), we expect that the
symmetry is SO(even) for m ≡ 3 mod 4, SO(odd) for m ≡ 1 mod 4, and full orthogonal when
m is even. Furthermore, in this last case the symmetry is SO(even) (resp., SO(odd)) when
k/2 is even (resp., k/2 is odd). 

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 29
Remark 6.4. Gu¨log˘lu has obtained results for larger support for symmetric-power fami-
lies [Gu¨].
6.2. Convolutions of Symmetric Powers.
Theorem 6.5. Fix M,N ≥ 1 and consider the families F (M)k = symMHk and G
(N)
k′ =
symNHk′. Assume that the convolutions f × g, f ∈ F (M)k , g ∈ G
(N)
k′ are automorphic. Let
H(M,N)k,k′ = F
(M)
k × G
(N)
k′ (where, as usual, we discard the non-cuspidal f × f when M = N
and k = k′). Then, as k, k′ → ∞ in such a way that log k′/ log k → 1, the family H(M,N) =
F (M) × G(N) is NT-good with symmetry constant cH(M,N) = (−1)M+N = cF(M) · cG(N) .
Remark 6.6. The automorphicity of the convolutions f×g is known when M+N ≤ 3 [Ram,
KiSh1].
Proof. For simplicity we will only consider the case when k = k′ → ∞; the proof of the
general case differs from this case only in trivial details.
As in the previous section, all non-archimedean places of f ∈ F (M)k , g ∈ G
(N)
k are unramified.
We will once more split into cases when M,N are even or odd.
Using Lemma 6.2 we obtain10:
sym2m+1[k]⊗ sym2n+1[k] ≃
⊕
0≤ℓ≤m
0≤λ≤n
(
[2(ℓ+ λ+ 1)(k − 1) + 1]⊕ [2 |ℓ− λ| (k − 1) + 1]
)
(6.33)
sym2m[k]⊗ sym2n[k] ≃ [(−)m+n]⊕
⊕
1≤ℓ≤m
[2ℓ(k − 1) + 1]⊕
⊕
1≤λ≤n
[2λ(k − 1) + 1]
⊕
⊕
1≤ℓ≤m
1≤λ≤n
(
[2(ℓ+ λ)(k − 1) + 1]⊕ [2 |ℓ− λ| (k − 1) + 1]
)
(6.34)
sym2m+1[k]⊗ sym2n[k] ≃
⊕
0≤ℓ≤m
[(2ℓ+ 1)(k − 1) + 1]
⊕
⊕
0≤ℓ≤m
1≤λ≤n
(
[(2ℓ+ 2λ+ 1)(k − 1) + 1]⊕ [|2ℓ− 2λ+ 1| (k − 1) + 1]
)
. (6.35)
The ε-factors are as follows:
ε(sym2m+1[k]⊗ sym2n+1[k]) = ε(sym2m[k]⊗ sym2n[k]) = +1 (6.36)
ε(sym2m+1[k]⊗ sym2n[k]) =
{
(−1)(m+1)(n−m)+(m+1)2 k2 m < n
(−1) (m−n)(m+n+1)2 +(m+1)2 k2 m ≥ n.
(6.37)
We omit explicitly writing down the Γ-factors, but observe that every term [a(k−1)+1] with
a > 0 contributes a factor ≍ a24 · k2 to the analytic conductor Qf×g. Hence, up to an additive
constant, the analytic log-conductors are logQf×g ∼ 2(2m + 1)(n + 1) log k, resp. 2m(2n +
10Recall [1] := [+]⊕ [−], and observe that [+]⊗ [1] ≃ [−]⊗ [1] ≃ [1].

30 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
1) log k, resp. 2(m+1)(2n+1) log k corresponding to the cases (6.33), resp. (6.34), resp. (6.35)
above (we assumed m ≥ n in the first two cases).
By Theorem 1.3, it only remains to show that F (M)×G(N) is NT-good. The argument above
shows that the conductor condition is satisfied. When M = N and k = k′ the representations
f × f are not cuspidal; hence we must discard O(k) of them. Note that |F (M)k × G
(N)
k | =
k2+O(k) (so the cardinality condition holds) and that possibly shrinking the family introduces
error terms of size O(1/k), which are quite admissible. Properties of cardinality and the
handling of error terms (by Ramanujan) are thus valid. We need not verify the conditions
on prime sums explicitly: the reason is that there are no ramified primes and conductors are
essentially constant. Hence Lemma 5.1 and the argument in the proof of Theorem 1.3 suffice
to prove that F (M) × G(N) is NT-good with symmetry constant cF(M)×G(N) = cF(M) · cG(N) =
(−1)M+N .
Therefore, for small support, the 1-level density of the family F (M) × G(N) agrees with
symplectic for M + N even, whereas for M = 2m + 1, N = 2n the symmetry is orthogonal
and the root number, as read off from equations (6.36) and (6.37), determines whether the
underlying symmetry is SO(even) or SO(odd). Equation (6.36) holds even when k 6= k′, but
the form of equation (6.37) is specific to the case k = k′. 
7. Convolving Families Of Elliptic Curves
We now consider the interesting case of convolving two families of elliptic curves. Specifi-
cally, consider the one-parameter families
FN : y2 = x3 + A1(T )x+B1(T ), T ∈ [N, 2N − 1]
GM : y2 = x3 + A2(S)x+B2(S), S ∈ [M, 2M − 1], (7.1)
where the polynomials A1(T ) through B2(S) have integer coefficients. If we specialize T to t
we obtain an elliptic curve EF(t) with discriminant ∆F (t) and conductor CF(t); similarly if
we specialize S to s we obtain an elliptic curve EG(s) with discriminant ∆G(s) and conductor
CG(s). The conductors are products of powers of primes dividing the discriminants. It is
known (see [BCDT, TW, Wi]) that the L-function of an elliptic curve of conductor C agrees
with a weight-2 cuspidal newform of level C. Thus if Ei are elliptic curves with conductors
Ci and associated newforms fi, by L(s, E1 × E2) we mean the Rankin-Selberg convolution
L(s, f1 × f2), which is a GL4 L-function. The arithmetic conductor Q(f1 × f2) of such
L(s, f1 × f2) is an integer satisfying
(C1C2)2/(C1, C2)4 ≤ Q(f1 × f2) ≤ (C1C2)2/(C1, C2), (7.2)
where (C1, C2) is the greatest common divisor of C1 and C2; see for example [HaMi]. We
often write Q(C1, C2) for Q(f1 × f2).
We are interested in the behavior of FN × GM as N and M tend to infinity. The gamma
factors for these GL4 L-functions depend neither on the specific curve nor on the family. As

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 31
such, since we need only identify the analytic conductor up to a constant, we may use the
integer Q(C1, C2) as the analytic conductor.
We normalize the low lying zeros for the convolution L-function by the average of the
logarithms of the analytic conductors. Thus, we set
logRN,M :=
1
NM
2N−1∑
t=N
2M−1∑
s=M
logQ(CF(t), CG(s)). (7.3)
We need RN,M to tend to infinity with N and M . A weak estimate on the size of
Q(CF(t), CG(s)), namely that the average log-conductor in (7.3) tends to infinity with N
and M , suffices for our purposes.
To show this requires a few basic facts about elliptic curves. An elliptic curve E : y2 =
x3 + a4x+ a6 has discriminant ∆ = −16(4a34 +27a26) and j-invariant j = 3a34/(4a34 +27a26); it
is also convenient to set c4 = −48a4 and c6 = −864a6. Let R be the ring of integers for some
local field K; K is a local field which is complete with respect to a discrete valuation v. Let
M = {x ∈ K : v(x) > 0} be the maximal ideal of R, and let k = R/M be the residue field.
If ai ∈ R and v(c4) < 4 or v(c6) < 6, then the equation for the elliptic curve is minimal with
respect to the valuation v.
Theorem 7.1. Notation as above, assume that there are non-constant monic integral poly-
nomials f1(x) and g1(x) such that f1(x) divides ∆F(x) and g1(x) divides ∆G(x). To simplify
the analysis, assume f1(x) does not divide either cF ,4(x) or cF ,6(x) (and similarly for g1(x)).
Define the average log-conductor by (7.3). If jF(T ) and jG(S) are both non-constant, then
for some a > 0
logNM
(log logmin(N,M))a ≪F ,G logRN,M ≪F ,G logNM. (7.4)
The proof follows from basic facts on solutions to Diophantine equations and properties of
elliptic curves, and is given in Appendix A.
The following observation ensures that, except for a negligible fraction of the time, the
L-functions in the convolved family are good (i.e., primitive).
Lemma 7.2. Assume jF(T ) and jG(S) are non-constant. The Rankin-Selberg convolution of
EF (t) and EG(s) is imprimitive for at most O(min(N,M)) of the NM pairs (s, t).
Proof. Without loss of generality assume N ≤ M . If for some pair (s, t) we have EF (t)
and EG(s) are associated to the same weight-2 cuspidal newform, then the Rankin-Selberg
convolution will be imprimitive (and divisible by ζ(s)); call such a pair bad. If two elliptic
curves are isomorphic, then they have the same j-invariant. Thus for a bad pair,
jF (t) =
3A1(t)3
4A1(t)3 + 27B1(t)2
=
3A2(s)3
4A2(s)3 + 27B2(s)2
= jG(s). (7.5)
As we are assuming jF(t) and jG(s) are non-constant, for each fixed t there are only finitely
many solutions to jG(s) = jF (t) (the number is bounded by the degrees of A2(s)3 and

32 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
4A2(s)3 +27B2(s)2). Thus of the NM pairs (s, t), at most O(N) of the pairs have a Rankin-
Selberg convolution divisible by the Riemann zeta function. As the only non-primitive L-
functions L(s, f × g) for f, g primitive weight-2 cuspidal newforms of levels N1 and N2 arise
when f = g, the remaining pairs yield primitive L-functions. 
We now prove our main result about convolving two families of elliptic curves.
Theorem 7.3. Consider two one-parameter families of elliptic curves (elliptic surfaces over
Q):
EF : y2 = x3 + A1(T )x+B1(T )
EG : y2 = x3 + A2(S)x+B2(S). (7.6)
Let FN be the specialization of EF with t ∈ [N, 2N − 1], GM be the specialization of EG with
s ∈ [M, 2M − 1], and set F = ∪FN and G = ∪GM . Assume logN ≪ logM ≪ logN and
(1) the first family is an elliptic curve over Q(T ) of rank rF and non-constant jF(T );
(2) the second family is an elliptic curve over Q(S) of rank rG and non-constant jG(S);
(3) the average log-conductor of FN × GM satisfies (7.4);
(4) the Fourier coefficients of each family satisfy either (1.2) or (1.3).
Then Theorem 1.3 holds for the family F ×G; the symmetry is symplectic and the rank is 0.
Remark 7.4. Rosen and Silverman [RoSi] show that (1.2) is a consequence of Tate’s conjec-
ture [Ta]: Let E/Q be an elliptic surface and L2(E , s) be the L-series attached toH2e´t(E/Q,Ql).
L2(E , s) has a meromorphic continuation to C and −ords=1L2(E , s) = rank NS(E/Q), where
NS(E/Q) is the Q-rational part of the Ne´ron-Severi group of E . Further, L2(E , s) does not
vanish on the line Re(s) = 1.
Tate’s conjecture is known for rational surfaces11. Theorem 7.3 should be true for families
with constant j-invariants; however, for such families Michel’s result on the average second
moments of the Fourier coefficients is not available, and one must show by direct calculation
that (1.4) holds.
Proof. For the L-function attached to EF(t)× EG(s), the explicit formula (2.10) becomes
∑
ℓ
φ
(
γEF(t)×EG(s),ℓ
logRN,M
2π
)
=
AEF (t)×EG(s)
logRN,M
φ̂(0)
− 2
∑
p
∞∑
ν=1
φ̂
( ν log p
logRN,M
) bEF (t)×EG(s)(pν) log p
pν/2 logRN,M
, (7.7)
where
AEF (t)×EG(s) = logQ(EF(t), EG(s)) + o(1). (7.8)
The o(1) error follows from Theorem 7.1, where we showed RN,M cannot be too small. Note
(7.7) may be slightly off in that, if EF (t) = EG(s), then the L-function associated to EF (t)×
11An elliptic surface y2 = x3 +A(T )x+ B(T ) is rational if and only if one of the following is true: either
0 < max{3 degA, 2 degB} < 12 or 3 degA = 2degB = 12 and ordT=0T 12∆(T−1) = 0.

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 33
EG(s) is imprimitive. We would have a superposition of zeros of two primitive L-functions,
one of which has a pole. Fortunately, by Lemma 7.2, this occurs for at most O(min(N,M))
of the NM pairs; as we divide by NM this contribution is negligible.12
Thus summing (7.7) over t ∈ [N, 2N − 1] and s ∈ [M, 2M − 1], and recalling the definition
of the 1-level density and the average log-conductor, we find
D1,FN×GM (φ) = φ̂(0) + o(1)−
2
NM
2N−1∑
t=N
2M−1∑
s=M
∑
p
∞∑
ν=1
φ̂
( ν log p
logRN,M
) bEF (t)×EG(s)(pν) log p
pν/2 logRN,M
.
(7.9)
By Lemma 5.1, bEF (t)×EG(s)(pν) = bEF (t)(pν)·bEG(s)(pν) when L(s, EF(t)×EG(s)) is primitive.
We use this for all EF(t)× EG(s), as the O(min(N,M)) instances where this is false lead to
a difference that is o(1).
There is trivially no contribution in (7.9) for ν ≥ 3. As for each EF (t)×EG(s) the conductor
is at most (NM)b for some b, at primes dividing the conductor if necessary we may adjust
the coefficients at p and p2 and introduce an error at most o(1). This is because the worst
case is if (NM)b is the product of the first ℓ primes, where pℓ ≪ log(NM)b. This would lead
to a sum bounded by
1
logRN,M
∑
p≤log(NM)b
1√p ≪
√
log((NM)b)
logRN,M
= o(1), (7.10)
where the last inequality follows from the lower bound for the average log-conductor.
The proof is completed by showing our family is NT-good. We must check the four con-
ditions of Definition 1.1. The first is straightforward (with Lemma 7.2 a key ingredient),
the second (on the size of the log-conductors) follows from our assumption that the average
log-conductor satisfies (7.4). The fourth is an easy consequence of the Hasse bound. We are
left with the third condition, which concerns the sums over primes and squares of primes. We
handle the prime sums first.
12Here we are using Corollary 1.6, which says it suffices to show there is a power savings in the number of
bad pairs. Alternatively, instead of using Lemma 7.2 we could show that the multiplicity of any elliptic curve
in our parametrizations is O(1).

34 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
The needed result for the sum of the Fourier coefficients at the primes is true because (1.3)
is satisfied with r = 0. To see this, note
1
NM
2N−1∑
t=N
2M−1∑
s=M
∑
p
φ̂
(
log p
logRN,M
) bEF (t)×EG(s)(p) log p√p logRN,M
=
1
NM
2N−1∑
t=N
2M−1∑
s=M
∑
p
φ̂
(
log p
logRN,M
) bEF (t)(p)bEG(s)(p) log p√p logRN,M
=
∑
p
φ̂
(
log p
logRN,M
)[
1
N
2N−1∑
t=N
bEF (t)(p)
][
1
M
2M−1∑
s=M
bEG(s)(p)
]
log p√p logRN,M
.
(7.11)
We analyze the t-sum; the s-sum follows similarly. Let aEF (t)(p) = bEF (t)
√p; by Hasse’s
bound we have |aEF(t)(p)| ≤ 2
√p, and these correspond to the associated L-function having
functional equation u→ 2− u. Let Ap(EF) = 1p
∑
t mod p aEF (t)(p). We have
1
N
2N−1∑
t=N
bEF (t)(p) =
1
N
(
N
p
∑
t mod p
aEF (t)(p)√p +O(p)
)
=
Ap(EF)√p +O
( p
N
)
.
(7.12)
The O(p/N) term (and the corresponding O(p/M) term from the s-sum) lead to o(1) con-
tributions if φ̂ has suitably restricted support. We are left with the Ap(EF)Ap(EG)/p term.
Thus (7.11) becomes
∑
p
φ̂
(
log p
logRN,M
) Ap(EF)Ap(EG) log p
p3/2 logRN,M
+ o(1). (7.13)
As Ap(EF) and Ap(EG) are bounded independent of p (see [De], or [Mic] for an explicit bound
in terms of the curves), the above sum is O(1) and hence negligible upon division by NM .
We are left with showing that (1.5) (the second part of the third condition of Definition
1.1) holds, i.e., analyzing the prime square sums (the sums of bEF (t)(p2) over t and bEG(s)(p2)
over s). As we have assumed jF (T ) and jG(S) are non-constant, this follows immediately
from work of Michel [Mic], who showed that for a one-parameter family F over Q(T ) with
non-constant jF (T ) that
∑
t mod p
aEF (t)(p)2 = p2 +O(p3/2). (7.14)
The exponent in the error term cannot be improved in general, and may be related to family
specific lower order correction terms to the 1-level density; see [Mil3]. From (5.16) and our

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 35
normalizations13 we have bEF (t)(p2) = p−1aEF (t)(p)2 − 2, which implies
2N−1∑
t=N
bEF (t)(p2) =
N
p
∑
t mod p
bEF (t)(p2) +O(p)
=
N
p
∑
t mod p
aEF (t)(p)2
p − 2N +O(p)
= −N +O(p2) = −|FN |+O(p). (7.15)
Thus (1.5) holds with cF = −1; an analogous result holds for sums of bEG(s)(p2).
Therefore the two families have orthogonal symmetry (as was already known), but the
convolution family has symplectic symmetry (cF×G = cF · cG = (−1)2 = 1). 
Remark 7.5. The conditions of Theorem 7.3 are quite weak, and are easily seen to be satisfied
in many cases of interest (for example, by many of the families studied in [ALM, Fe]).
Appendix A. Average Log-Conductors for Elliptic Curve Families
We prove Theorem 7.1. The upper bound follows trivially from (7.2) and bounds relating
the discriminant of an elliptic curve to its conductor. We prove the lower bound through
a series of lemmas. We first introduce some notation. Let f1(x), . . . , fk1(x) be the distinct
monic irreducible factors of ∆F(x)cF ,4(x)cF ,6(x), and let g1(x), . . . , gk2(x) be the distinct
monic irreducible factors of ∆G(x)cG,4(x)cG,6. By relabeling if necessary, we may assume
f1(x)|∆F(x), and similarly g1(x)|∆G(x).
Further, we may assume all of the k1+k2 polynomials are relatively prime. If some fi(x) and
gj(x) were not relatively prime, then we could find a fixed x′ such that g1(x+x′), . . . , gk2(x+x′)
are relatively prime (as functions of x) to the fi(x)’s. Thus instead of considering the interval
[M, 2M ] we would consider the interval [M−c, 2M−c], which forM large is still approximately
[M, 2M ]. Hence we may assume f1(x)|r∆G(x) and g1(x)|r∆F (x), and without loss of generality
we may assume N ≤ M .
The proof is completed by showing that, for some constants ǫ, δ, a > 0, at least ǫNM/(log logN)a
of the L(u,EF(t)×EG(s)) have arithmetic conductor at least (NM)δ. We do this by showing
for at least ǫNM/(log logN)a of the pairs (t, s) that we can find a number at least N δ dividing
f1(t) and the conductor CF(t) but not any other fi(t), and a number at least M δ (relatively
prime to the number at least N δ) dividing g1(s) and CG(s) but not any other gi(s). As the
arithmetic conductor is an integer, it will then be divisible by at least (NM)δ, which implies
the lower bound in (7.4). We do this through the following series of lemmas.
Lemma A.1. There exists an integer c (a product of distinct primes) and an integer r such
that, for i 6= j, for a positive fraction of t ∈ [N, 2N ] we have (fi(t), fj(t))|cr (and similarly
for the g’s).
13Remember bEF(t)(p)
√p = aEF (t)(p). We must be careful in our normalizations, since we wish our elliptic
curve L-functions to have a functional equation as u → 1− u (not as u → 2− u).

36 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Proof. Let i 6= j. By the Euclidean algorithm, there exists a cij (independent of x) such that
if p|(fi(x), fj(x)) then p|cij. Let cf be the product of 6 and the prime divisors p ≥ 5 of the
cij ’s. Choose an x0 such that fi(x0) 6= 0 for all i. Let rf be the largest integer such that
if p|cf then pr+1 divides none of the fi(x0). Then for all i 6= j the greatest common divisor
of fi(crf+1f x + x0) and fj(c
rf+1
f x + x0) divides c
rf
f . We similarly construct cg and rg so that
the greatest common divisors of the gi’s divides crgg . Let c equal the product of the prime
divisors of cfcg and r = max(rf , rg). We change variables, sending t→ cr+1t+x0. A positive
fraction of t ∈ [N, 2N ] satisfy this condition. We similarly change s → cr+1s + s0. For ease
of exposition we denote these polynomials by f˜i and g˜j; thus f˜i(x) = fi(cr+1x+ x0). 
It is possible that f˜i(x) is divisible by a fixed square (or higher power) for all x; for example,
x4 − x2 + 20 is always divisible by 4. Further, if ∆F(x)cF ,4(x)cF ,6(x) = aF
∏
i fi(x), for some
x an f˜i(x) could share a factor with aF . The following lemma handles such primes.
Lemma A.2. Notation as in Lemma A.1, let C be the product of all numbers that divide an
f˜i(x) for all x, a g˜j(x) for all x, or aFaG. Then there exists an integer m such that, for a
positive fraction of t ∈ [N, 2N ], Lemma A.1 holds and if p|C then pm does not divide fi(t)
for all i (and similarly for the g’s).
Proof. Let f˜i and g˜j be as in Lemma A.1. Choose an x1 such that f˜i(x1) and g˜j(x1) are non-
zero for all i and j. Arguing as before, after a simple linear change of variables we can ensure
that at most a fixed power of C divides our polynomials for any x. Specifically, consider
f˜i(Cmx + x1); for m sufficiently large, if p|C then pm |r f˜i(Cmx + x1) (and the same is true
for the g˜j’s). Let f̂i(x) = f˜i(Cmx+ x1) (and similarly for ĝj). 
We have shown that for a positive fraction of all t ∈ [N, 2N ] and s ∈ [M, 2M ]: (i) the
greatest common divisor of the f̂i(t)’s is at most cr and the greatest common divisor of the
ĝj(s)’s is at most cr; (ii) the product of all the squares or factors of aFaG that divide a
f̂i(t) for all t is at most Cm (and similarly for ĝj(s)). We would like to say the arithmetic
conductor is at least f̂1(t)ĝ1(s), except there are two problems: (i) we must show f̂1(t) divides
the conductor of EF(t) (and similarly for ĝ1(s) and EG(s)); (ii) we must show (f̂1(t), ĝ1(s)) is
small. We handle (i) first.
Lemma A.3. Let d = max(∑i deg fi,
∑
j deg gj) + 2. For a positive fraction of t ∈ [N, 2N ]
and s ∈ [M, 2M ], the results of Lemmas A.1 and A.2 hold, the conductor of EF (t) is ≫ N1/d
and the conductor of EG(s) is ≫M1/d
Proof. Notation as in Lemmas A.1 and A.2 and conditions as in Theorem 7.1, we show
that for a positive fraction of the time that the conductor of EF (t) is ≫ N1/d. Recall the
following basic facts (see for example [Nag]) for an integral polynomial D(t) of degree k and
discriminant δ:
(1) Let p be a prime not dividing the coefficient of xk. Then D(t) ≡ 0 mod p has at most
k incongruent solutions.

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 37
(2) Suppose p |r δ. Then the number of incongruent solutions of D(t) ≡ 0 mod p equals
the number of incongruent solutions of D(t) ≡ 0 mod pα.
Note that if the discriminant of h(x) is δ, then the discriminant of h(ax+b) is anδ for some
n. Let D be the product of the prime divisors of the discriminants and leading coefficients
of all the f̂i’s and ĝj’s, as well as any missing primes at most d. We make one last change of
variables: for sufficiently large n consider
Fi(x) = f̂i(Dnx+ x2), Gj(x) = ĝj(Dnx+ x2), (A.1)
where x2 is chosen so that all f̂i(x2) and ĝj(x2) are non-zero. The advantage is that the degree
of divisibility of Fi(x) (resp., Gj(x)) by primes dividing the discriminants, cF ,4(x) and cF ,6(x)
(resp., cG,4(x) and cG,6(x)), leading coefficients or at most d is bounded independent of x, say
by k. It is now immediate that, for a positive fraction of x, Fi(x) has a dth power free factor
≫ Ndeg Fi/d. To see this, let νFi(pd) denote the number of solutions to Fi(x) ≡ 0 mod pd. For
p |rD, p does not divide the discriminant of Fi and thus νFi(pd) = νFi(p) ≤ deg Fi. Thus the
fraction of t giving Fi(t) d-power free (except for divisors of D) is at least
∏
p|rD
(
1− νFi(p
d)
pd
)
≥
∏
p|rD
(
1− deg fipd
)
≥
∏
p|rD
(
1− 1pd−1
)
. (A.2)
As d was chosen to be at least 3, this last factor is larger than
∏
p(1− p−2) = 6/π2. By our
linear change of variables (how we defined the Fi), the number of times a p|D divides Fi(x)
is bounded independent of x and i, say by k. Thus, for a positive fraction of t, Fi(t) has a d
power free part at least Fi(t)1/d/Dk. As the greatest common divisors of any two of the Fi is
at most cr, for a positive fraction of t we have F1(t) has a d power free factor of size at least
F1(t)1/d/crDk that is relative prime to the Fi(t) for i 6= 1.
We need only show this factor (which is at least F1(t)1/d/crDk) divides the conductor. This
follows by showing the conductor of the elliptic curve y2 = x3 + A1(t)x + B1(t) is minimal
for each p |r cCD that divides F1(t)1/d. This follows from our assumption that jF (T ) is not
constant, as this implies that cF ,4(x) and cF ,6(x) are not identically zero. Thus neither are
CF ,4(x) or CF ,6(x) (where we have used the obvious notation to represent the linear change of
variables). By assumption (see the conditions of Theorem 7.1), as the irreducible polynomial
factors of cF ,4(x) and cF ,6(x) were included in our list of the fi’s, and we assumed f1(x) is
relatively prime to either cF ,4(x) or cF ,6(x), for p |r cCD with p|F1(t), p cannot divide both
CF ,4(x) and CF ,6(x). Thus the elliptic curve is minimal for such primes p, implying the
conductor is at least F1(t)1/d/crCmDk.
Thus as N →∞, for a positive fraction of t the conductor of EF (t) is ≫ Ndeg F1/d ≫ N1/d;
an analogous statement holds a positive fraction of the time for the conductor of EG(s). We
call such t and s good. 
Remark A.4. We chose d = max(
∑
i deg fi,
∑
j deg gj)+2 and not max(maxi deg fi, maxj deg gj)
+2 because of Lemma A.5.
The following lemma completes the proof of Theorem 7.1.

38 EDUARDO DUEN˜EZ AND STEVEN J. MILLER
Lemma A.5. Notation as in Theorem 7.1, for some ǫ, a > 0 for at least ǫNM/(log logN)a
of the pairs (t, s) ∈ [N, 2N ]× [M, 2M ] the results of Lemmas A.1 through A.3 hold, and the
greatest common divisor of
∏
i Fi(t) and
∏
j Gj(s) is bounded independent of t and s.
Proof. Consider the positive fraction of t and s that are good. We must make sure that each
such Gj(s) is essentially relatively prime to the Fi(t). If so, then since the arithmetic conductor
is an integer it would have to be ≫ N1/dM1/d (remember the arithmetic conductor comes
from the arithmetic conductors of EF(t) and EG(s), and these are ≫ N1/d and ≫ M1/d).
For a good t, the worst case for common factors of
∏
i Fi(t) and
∏
j Gj(s) is when
∏
i Fi(t)
is the product of the first ℓ primes. We can easily handle the bounded contributions from c
(Lemma A.1), C (Lemma A.2) or D (see the proof of Lemma A.3), and thus we need only
investigate primes p such that p > D and p |r cC. Letting µ = deg∆F (x)CF ,4(x)CF ,6(x), the
product of the Fi(t)’s is ≪ Nµ. Thus
∏
p≤pℓ
p ≪ Nµ
∑
p≤pℓ
log p ≪ µ logN ⇒ pℓ ≪ µ logN. (A.3)
We may need to discard some good s because a Gj(s) is not relatively prime to p1 . . . pℓ.
Thus, even though we are going to use F1(t) and G1(s), we must make sure that there are no
large common factors of
∏
i Fi(t) and
∏
j Gj(s), as otherwise the conductor of the Rankin-
Selberg convolution could be reduced (see the division by the greatest common divisor in the
left hand side of (7.2)). As d >
∑
j deg gj, we have
∑
j µGj (p) < d for p|rD; this allows us to
obtain the needed estimate.
We have already handled p ≤ d and p dividing a discriminant in our construction of the
good s. For each j, the product in (A.2) for G1(s) is modified by a factor no worse than
∏
d<p≤pℓ
p|rcCD
(
1−
∑
j νGj (p)
p
)
≥
∏
d<p≤pℓ
(
1− dp
)
≫ exp
(
−2009d · log
∑
d<p≤pℓ
1
p
)
≫ (log pℓ)−2009d ≫ (log logN)−2009d (A.4)
(the last bound is Mertens’ theorem, see for example [Da]). We therefore obtain a d power
free factor of the conductor of EG(s) whose common factors with F1(t) · · ·Fk1(t) is bounded
by crDk. 
This completes the proof of Theorem 7.1. 

THE EFFECT OF CONVOLVING ON THE UNDERLYING GROUP SYMMETRIES 39
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E-mail address : eduenez@math.utsa.edu
Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX
78249
E-mail address : Steven.J.Miller@williams.edu
Department of Mathematics, Brown University, Providence, RI 02912 and Department of
Mathematics and Statistics, Williams College, Williamstown, MA 02167