Arithmetic Conjectures Suggested by the Statistical Behavior of Modular Symbols

ARITHMETIC CONJECTURES SUGGESTED BY THE STATISTICAL BEHAVIOR OF MODULAR SYMBOLS

BARRY MAZUR AND KARL RUBIN

Abstract. Suppose E is an elliptic curve over Q and χ is a Dirich- let character. We use statistical properties of modular symbols to estimate heuristically the probability that L(E, χ, 1) = 0. Via the Birch and Swinnerton-Dyer conjecture, this gives a heuristic estimate of the probability that the Mordell–Weil rank grows in abelian extensions of Q. Using this heuristic we find a large class of infinite abelian extensions F where we expect E(F ) to be finitely generated. Our work was inspired by earlier conjectures (based on random matrix heuristics) due to David, Fearnley, and Kisilevsky. Where our predictions and theirs overlap, the predictions are consistent.

Contents

Introduction 2

Part 1. Modular symbols 4 1. Basic properties of modular symbols 4 2. Modular symbols and L-values 6 3. θ-elements and θ-coefficients 7 4. Distribution of modular symbols 8 5. The involution ιF 9 6. Distributions of θ-coefficients 10

arXiv:1910.12798v4 [math.NT] 6 Aug 2020 Part 2. Heuristics 14 7. The heuristic for cyclic extensions of prime degree 14 8. The heuristic for general cyclic extensions 16 9. Expectations 18 10. Conjectures 20 11. Analytic results 21

2010 Mathematics Subject Classification. Primary: 11G05, Secondary: 11F67, 11G40, 14G05. This material is based upon work supported by the National Science Foundation under grants DMS-1302409 and DMS-1500316. 1 2 BARRY MAZUR AND KARL RUBIN

References 25

Introduction By a number field we will mean a field of finite degree over Q, and any field denoted “K” below will be assumed to be a number field. Definition. A variety V over K is said to be diophantine stable for the extension L/K if V (L) = V (K); i.e., if V acquires no “new” rational points when extended from K to L. Depending on emphasis intended, we will also sometimes say L/K is diophantine stable for V . The “minimalist philosophy” leads us to the following question. Question A. If A is an abelian variety over K, ` is an odd prime number, and m is a positive integer, is it the case that the cyclic Galois extensions of K of degree `m that are diophantine stable for A are of density 1 (among all cyclic Galois extensions of K of degree `m, ordered by conductor)? The following result of [18] is an embarrassingly weak theorem in the direction of answering this question.

Theorem B. If A/K is a simple abelian variety such that EndK¯ (A) = EndK (A), then there is a set S of prime numbers of positive density such that for all ` ∈ S and for all positive integers m there are infinitely many cyclic Galois extensions of K of degree `m that are diophantine stable for A. Unfortunately we cannot replace the phrase “infinitely many” in the statement of Theorem B by “a positive proportion” (where “propor- tion” is defined by organizing these cyclic extensions by size of con- ductor). If we order the extensions that way and consider what we have proved for a given `m, among the first X of these we get at least X/ logα X as X → ∞ (for a small, but positive, α). Note that, for example, when K/Q is quadratic, the cyclic `m ex- tensions of K that are Galois dihedral extension of Q are potentially a source of systematic diophantine instability but are of density 0 in all cyclic `m extensions of K. The aim of this paper is to study the distributions of values of mod- ular symbols and of what we call θ-coefficients, in order to develop a heuristic prediction of the probability of diophantine stability (or in- stability) in the specific case where V = E is an elliptic curve, K = Q, and L/Q is abelian. For example, our heuristic leads to the following conjecture. ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 3

Conjecture 10.2. Let E be an elliptic curve over Q and F/Q any real abelian extension such that F contains only finitely many subfields of degree 2, 3 or 5 over Q. Then the group of F -rational points E(F ) is finitely generated.

For example, we can take the field F in Conjecture 10.2 to be the cyclotomic Z`-extension for any prime `, or the compositum of these Z`- extensions for all `, or the maximal abelian `-extension for any ` ≥ 7, or the compositum of all such extensions. Question A was inspired by conjectures of David, Fearnley and Kisi- levsky [9], who deal specifically with the case of elliptic curves over Q. They conjecture (cf. Conjecture 9.5 below) that for a fixed elliptic curve E over Q and a fixed prime ` ≥ 7, there are only finitely many Dirichlet characters of order ` for which L(E, χ, 1) = 0. Consequently, (assuming the Birch–Swinnerton-Dyer Conjecture) only finitely many cyclic extensions of Q of order ` are diophantine unstable for a fixed elliptic curve E over Q. Their conjectures are based on computations and random matrix heuristics. In this paper we will consider a more naive heuristic, that leads us to make conjectures first regarding the distributions of values of θ-coefficients (see Sections 3 and 6, and Conjecture 6.2). These conjec- tures, supported by numerical evidence, lead us to Conjecture 10.2. In the first part of the paper we introduce modular symbols, recall the properties we need, and define the θ-coefficients (which are sums of modular symbols). We use the known distribution properties of mod- ular symbols, along with numerical calculations, to make conjectures about the (more mysterious) distributions of θ-coefficients. We say “more mysterious” because (as suggested by random matrix heuristics) there may well be no useful limiting distribution, no matter how one normalizes the θ-coefficients. Nevertheless, our heuristic needs only quite weak estimates, that we label “upper bounds for heuristic likelihood.” These estimates seem to be in accord with the compu- tations that we have made—and (it seems to us) by a comfortable margin. In the second part of the paper we use these conjectures about θ- coefficients to develop a heuristic for the probability of vanishing of twisted L-values L(E, χ, 1) where χ is a Dirichlet character of order at least 3.

Acknowledgements. We would like to thank Jon Keating and Asbjørn Nordencroft for helpful conversations about the random matrix theory conjectures. 4 BARRY MAZUR AND KARL RUBIN

Part 1. Modular symbols 1. Basic properties of modular symbols Fix once and for all an elliptic curve E defined over Q. Since E is fixed we will usually suppress it from the notation. Let N be the conductor of E, ¯ φE : H → X0(N)(C) → E(C) the (optimal) modular uniformization of E by H¯ := H ∪ P1(Q), the completion of the upper half-plane H := {z ∈ C | Im(z) > 0},

X 2πinz fE(z) = ane n the newform associated to E, and Z ± ± ΩE = Ω = ωE γ± the real and imaginary periods of E, where ωE is the N´erondifferential of (the N´eronmodel of) E, γ+ is the appropriately oriented connected component of the real locus of E, and γ− is, similarly, the appropriately oriented cycle in E(C) stabilized, but not fixed, by complex conjuga- tion. Note that we have two maps (one with “+”, one with “−”) that we can evaluate on closed curves γ in E(C), namely 1 Z Z  γ 7→ ± ωE ± ωE ∈ Z Ω γ −γ These induce linear maps

H1(E(C); Z) → Z.

The relationship of ωE to fE is given (cf. [1]) by ∗ φEωE = cE · 2πifE(z)dz, where cE is a nonzero integer (“Manin’s Constant”). For every r ∈ P1(Q) the image of the “vertical line” in the upper half-plane {z = r + iy | 0 ≤ y ≤ ∞} ⊂ H¯ in E is an oriented compact curve that “begins” at i∞ and terminates at some cusp in E (and any cusp is of finite order by the Manin–Drinfeld Theorem). ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 5

Definition 1.1. For every r ∈ Q define the (raw) modular symbols Z r {r} := 2πi fE(z)dz ∈ C i∞ and the plus/minus normalized modular symbols {r} ± {−r} [r]± := . 2Ω± We will also write simply Ω and [r] for Ω+ and [r]+, respectively. The modular symbols have the following well-known properties.

Lemma 1.2. For every r ∈ Q we have: ± −1 (i)[ r] ∈ δE Z for some positive integer δE independent of r (ii)[ r + 1]± = [r]± = ±[−r]± (iii) If A ∈ Γ0(N) ⊂ SL2(Z) then, viewing A as a linear fractional transformation we have [r]± = [A(r)]± − [A(∞)]±. If further A has a complex (quadratic) fixed point, then [A(∞)]± = 0, and therefore [A(r)]± = [r]± for all r ∈ Q ∪ {∞}. (iv) Atkin–Lehner relation: Let m ≥ 1 and write N = ef where f := gcd(m, N). Assume that e and f are relatively prime, and let We be the Atkin–Lehner Hecke operator [3]. Denote by we the eigenvalue of We on fE. If a, d ∈ Z and ade ≡ −1 (mod m), then ± ± [d/m] = −we · [a/m] .

(v) Hecke relations: Suppose ` is a prime, and a` is the `-th Fourier coefficient of fE. Then ± ± P`−1 ± (a) If ` - N, then a` · [r] = [`r] + i=0 [(r + i)/`] . ± P`−1 ± (b) If ` | N, then a` · [r] = i=0 [(r + i)/`] .

Proof. For (i), we can take δE to be (any positive multiple of) cE · λE where λE is the l.c.m. of the orders of the image of the cusps of X0(N) in E. Assertion (ii) follows directly from the definition. The first part of assertion (iii) is evident. For the second part, let z = A(z) be the fixed point of A, and γ a geodesic from ∞ to z. By invariance under A we get Z z Z A(z) Z z fE(z)dz = fE(z)dz = fE(z)dz. ∞ A(∞) A(∞)

R A(∞) ± ± Thus ∞ fE(z)dz = 0, and it follows that [A(∞)] = [∞] = 0. 6 BARRY MAZUR AND KARL RUBIN

For (iv), here is a construction of the Atkin–Lehner operator We. Let f := gcd(m, N) and e := N/f. The We operator is given by (any) matrix of the following form:  −ae b  (1.3) W := , e cN de with a, b, c, d ∈ Z and det(We) = e. Let c := −m/f and b := (ade + 1)/m ∈ Z. With these choices the matrix We of (1.3) has determinant e, ae a a W (∞) = − = − = , e cN cf m and (computing) We(d/m) = ∞.

Thus We takes the path {∞, d/m} to the path {a/m, ∞}. It follows that [d/m] = −we[a/m] where we is the eigenvalue of We acting on fE. This is (iv), and the proof of (v) is straightforward.  2. Modular symbols and L-values Definition 2.1. Suppose χ is a primitive Dirichlet character of con- ductor m. Define the Gauss sum m X τ(χ) := χ(a)e2πia/m a=1 P −s and, if L(E, s) = ann , the twisted L-function ∞ X −s L(E, χ, s) := χ(n)ann . n=1 If F/Q is a finite abelian extension of conductor m, we will identify characters of Gal(F/Q) with primitive Dirichlet characters of conductor dividing m in the usual way. Proposition 2.2. If F/Q is a finite abelian extension, and the Birch and Swinnerton-Dyer conjecture holds for E/Q and E/F , then X rank(E(F )) = rank(E(Q)) + ords=1L(E, χ, s). × χ:Gal(F/Q)→C χ6=1 Proof. This follows from the identity Y L(E/F , s) = L(E, χ, s). χ:Gal(F/Q)→C×  ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 7

Theorem 2.3 (Birch–Stevens). If χ is a primitive Dirichlet character of conductor m, then m X τ(χ)L(E, χ,¯ 1) χ(a)[a/m] = . Ω a=1 where the sign  := χ(−1) is the sign of the character χ.

3. θ-elements and θ-coefficients

Definition 3.1. Suppose m ≥ 1, and let Γm = Gal(Q(µm)/Q). Iden- × tify Γm with (Z/mZ) in the usual way, and let σa,m ∈ Γm be the Galois × a automorphism corresponding to a ∈ (Z/mZ) (i.e., σa,m(ζ) = ζ for ζ ∈ µm). Define ± X ± θm := δE [a/m] σa,m ∈ Z[Γm] a∈(Z/mZ)× where δE is as in Lemma 1.2(i). If F/Q is a finite abelian extension of conductor m, so F ⊂ Q(µm), define the θ-element (over F , associated to E) to be: ± ± θF := θm|F ∈ Z[Gal(F/Q)] ± ± where θm|F is the image of θm under the natural restriction homomor- phism

Z[Gal(Q(µm)/Q)] → Z[Gal(F/Q)]. By Lemma 1.2(i) we have

± X ± (3.2) θF = cF,γ · γ ∈ Z[Gal(F/Q)] γ∈Gal(F/Q) where ± X ± cF,γ := δE [a/m] . × a∈(Z/mZ) σa,m|F =γ ± We will refer to the cF,γ ∈ Z as θ-coefficients. Since we will most often be dealing with the “plus”-θ-elements, we will simplify notation + + + by letting θF := θF , cF,γ := cF,γ, and Ω := Ω . If F is a real field, then σ−1,m|F = 1, and [a/m] = [−a/m], so X (3.3) cF,γ = 2δE · [a/m]. × a∈(Z/mZ) /{±1} σa,m|F =γ With this notation, Proposition 2.3 can be rephrased as follows: 8 BARRY MAZUR AND KARL RUBIN

Corollary 3.4. Suppose F/Q is a finite real cyclic extension of con- × × ductor m and χ :(Z/mZ)  Gal(F/Q) ,→ C is a character that cuts out F . Then τ(¯χ)L(E, χ, 1) χ¯(θ ) = δ . F E Ω

In particular χ¯(θF ) vanishes if and only if L(E, χ, 1) does.

4. Distribution of modular symbols From now on, we assume that our elliptic curve E is semistable, so its conductor N is squarefree. The following fundamental result about the distribution of modular symbols was proved by Petridis and Risager (cf. (8.6) of [22]).

2 Q −1 −1 2 Definition 4.1. Let CE := 6/π `|N (1 + ` ) L(Sym (E), 1). Theorem 4.2 (Petridis & Risager [22], see also [21]). As X goes to infinity the values  +  [a/m] × : m ≤ X, a ∈ (Z/mZ) plog(m) approach a normal distribution with variance CE. Numerical experiments led to the following conjecture. Denote by Var(m) the variance 1 X Var(m) := ([a/m]+)2 ϕ(m) a∈(Z/mZ)× Conjecture 4.3. (i) As m goes to infinity, the distribution of the sets  +  [a/m] × : a ∈ (Z/mZ) plog(m) converges to a normal distribution with mean zero and variance CE. (ii) For every divisor κ of the conductor N, there is a constant DE,κ ∈ R such that lim (Var(m) − C log(m)) = D . m→∞ E E,κ (m,N)=κ Remark 4.4. We expect that a version of Conjecture 4.3 holds also for non-semistable elliptic curves, with a slightly more complicated formula for CE. ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 9

Note that Theorem 4.2 is an “averaged” version of Conjecture 4.3(i). Inspired by Conjecture 4.3, Petridis and Risager [22, Theorem 1.6] obtained the following result, which identifies the constant DE,κ and proves an averaged version of Conjecture 4.3(ii). Theorem 4.5 (Petridis & Risager [22]). For every divisor κ of N, there is an explicit (see [22, (8.12)]) constant DE,κ ∈ R such that 1 X lim ϕ(m)(Var(m) − CE log(m)) = DE,κ. X→∞ X ϕ(m) m

Remark 4.6. Petridis & Risager compute DE,κ in terms of the values L(Sym2(E), 1) and L0(Sym2(E), 1). They deal with non-holomorphic Eisenstein series twisted by modular symbols. Lee and Sun [17] more recently have proven the same result (for arbitrary N, averaged over m, but without explicit determination of the constants CE and DE,κ) by considering dynamics of continued fractions. A version of Conjecture 4.3(ii) with m ranging over primes was proved by Blomer, Fouvry, Kowalski, Michel, Mili´cevi´c,and Sawin in [5, Theorem 9.2]. For related results regarding higher weight modular eigenforms see [4]. See also [7] and [20] for other related results. Remark 4.7. The modular symbols are not completely “random” sub- Pβ ject to Conjecture 4.3. Specifically, partial sums a=α[a/m] behave in a somewhat orderly way. Numerical experiments led the authors and William Stein to conjecture the following result, which was then proved by Diamantis, Hoffstein, Kiral, and Lee (for arbitrary N, with an ex- plicit rate of convergence) [11, Theorem 1.2]. Theorem 4.8 ([11]). If 0 < x < 1 then mx ∞ 1 X X an sin(πnx) lim [a/m] = m→∞ m n2Ω a=1 n=1 P n where n anq is the modular form fE corresponding to E. Another proof in the case of squarefree N was given by Sun [23, Theorem 1.1].

5. The involution ιF Definition 5.1. Suppose F is a finite real cyclic extension of Q. Let m be the conductor of F , f := gcd(m, N), and e := N/f. Since N is assumed squarefree, e is prime to m. Let γF be the image of e under 10 BARRY MAZUR AND KARL RUBIN

× the map (Z/mZ)  Gal(F/Q). Define an involution ιF of the set Gal(F/Q) by −1 −1 ιF (γ) = γ γF

Let wF = −we where we is the eigenvalue of the Atkin–Lehner operator We (see Lemma 1.2(iv)) acting on fE. Recall from (3.2) that θ = P c γ. F γ∈Gal(F/Q) F,γ Lemma 5.2. Suppose F is a finite real cyclic extension of Q. 0 (i) We have cF,γ = wF cF,γ0 where γ = ιF (γ). −1 (ii) The fixed points of ιF are the square roots of γF in Gal(F/Q), so the number of fixed points is: • one if [F : Q] is odd, • zero if γF is not a square in Gal(F/Q), • two if [F : Q] is even and γF is a square in Gal(F/Q). (iii) If γ = ιF (γ) and wF = −1, then cF,γ = 0. Proof. Assertion (i) follows from the Atkin–Lehner relations satisfied by the modular symbols (Lemma 1.2(iv)). Assertion (ii) is immediate from the definition, and (iii) follows directly from (i).  Definition 5.3. If F/Q is a real cyclic extension, we say that γ ∈ + − Gal(F/Q) is generic, (resp., special , resp., special ) if γ 6= ιF (γ) (resp., γ = ιF (γ) and wF = 1, resp., γ = ιF (γ) and wF = −1). By Lemma − 5.2(iii), if γ is special then cF,γ = 0.

6. Distributions of θ-coefficients Consider real cyclic extensions F/Q of fixed degree d ≥ 3 and vary- ing conductor m. By (3.3), if γ is generic (resp., special+) then the θ-coefficient cF,γ is 2δE times a sum of ϕ(m)/(2d) modular symbols (resp., 4δE times a sum of ϕ(m)/(4d) modular symbols). If these were randomly chosen modular symbols [a/m], one would expect from Con- jecture 4.3(i) that the collection of data √  cF,γ d Σd := p : F/Q real, cyclic of degree d, ϕ(m) log(m)  m = cond(F ), γ ∈ Gal(F/Q) generic

(ordered by m) would converge to a normal distribution with variance 2 2δECE as m tends to ∞. Similarly the data defined in the same way except with γ special+ instead of generic, would converge to a normal 2 distribution with variance 4δECE. ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 11

Calculations do not support this expectation, at least not for small values of d. See Examples 6.7 below. In addition, random matrix heuristics (see [9, 15]) predict that the data above for d = 3 will not converge to a non-zero distribution. However, calculations do support the following much weaker Conjec- ture 6.2 below, which is strong enough for our purposes. Definition 6.1. For every F of degree d and conductor m, and every γ ∈ Gal(F/Q) define the normalized θ-coefficient √ cF,γ d c˜F,γ := . pϕ(m) log(m)

For d ≥ 3, α, β ∈ R, and X ∈ R>0, let Σd,α,β(X) be the collection of data (counted with multiplicity)

α β Σd,α,β(X) := {c˜F,γm log(m) : F/Q real, cyclic of degree d, + m = cond(F ) < X, γ ∈ Gal(F/Q) generic or special }.

Conjecture 6.2. There is a BE > 0 and for every d ≥ 3 there are αd, βd ∈ R such that (i) for every real open interval (a, b),

#{Σd,αd,βd (X) ∩ (a, b)} lim sup < BE(b − a), X→∞ #Σd,αd,βd (X)

(ii) {αdϕ(d): d ≥ 3} is bounded, and limd→∞ βd = 0. Remark 6.3. Random matrix theory heuristics (see [9, 10, 15]) suggest that for d = 3 and every real open interval (a, b),

#{Σd,αd,βd (X) ∩ (a, b)} (6.4) lim sup < Bd(b − a), X→∞ #Σd,αd,βd (X) with α3 = 0, β3 = 3/4 and a sufficiently large B3. Empirical data (see Examples 6.7) suggest (6.4) holds for all d, with αd = 0, βd converging to zero for large d and Bd bounded for large d. Taking BE to be the maximum of the Bd leads to the statement of Conjecture 6.2. Remark 6.5. Here is a heuristic shorthand description of Conjecture 6.2: There is a BE > 0 such that for • every cyclic extension F/Q of degree d ≥ 3, • every generic or special+ γ ∈ Gal(F/Q), and • every real interval (a, b), 12 BARRY MAZUR AND KARL RUBIN the “heuristic likelihood” thatc ˜F,γ lies in (a, b), which we will denote   by P c˜F,γ ∈ (a, b) , is bounded above by

αd βd BE(b − a)m log(m) ,

where m is the conductor of F (and αd, βd are as in Conjecture 6.2). By heuristic likelihood P[X] of an outcome X occurring we mean a real number that gives a (merely heuristically supported guess for an) upper bound for the number of times that the outcome X oc- curs within a specific range of events. The main use we will make of such (nonproved, but only heuristically suggested) estimates is that in various natural contexts we produce computationally supported upper bounds of heuristic likelihoods P[Xi] (for i in some index set I) where the {Xi}i∈I is a collection of possible outcomes for some specific range P of events. If i∈I P[Xi] is finite, we simply interpret our heuristic as suggesting that the totality of outcomes of the form Xi (for i ∈ I) that actually occur—within the specific range of events—is finite.

Remark 6.6. The smaller those exponents αd, βd are—and the faster they approach 0 as d grows—the stronger is the claim in Conjecture 6.2. It may well be that αd may be taken to be 0 for all d ≥ 3; the empirical data we have computed (see 6.7 below) which has led us to frame Conjecture 6.2, might almost suggest this. But we don’t believe we have computed far enough yet to say anything that precise, and— more importantly—the eventual qualitative arithmetic conjecture that we make (Conjecture 10.2) doesn’t need such precision for its heuristic support; we will only rely on the formulation of Conjecture 6.2.

Examples 6.7. For each of the three elliptic curves 11A1, 37A1, and 32A1 (in the notation of Cremona’s tables [8]), and five (prime) val- ues of d, we computed the first (approximately) 50,000 normalized θ-coefficientsc ˜F,γ, with F ordered by conductor and γ generic. The resulting distributions are shown in Figures 1 through 3. As d grows the distributions approach the “expected” normal distribution, shown as the dashed line in each figure. ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 13

1.5

d=3 d=7 1.0 d=19 d=31 d=101

0.5

-3 -2 -1 1 2 3

Figure 1. Distribution of normalized θ-coefficients for E = 11A1 and varying d.

1.5

d=3 d=7 1.0 d=19 d=31 d=101

0.5

-3 -2 -1 1 2 3

Figure 2. Distribution of normalized θ-coefficients for E = 37A1 and varying d. 14 BARRY MAZUR AND KARL RUBIN

1.0

0.8 d=3 d=7 d=19 0.6 d=31 d=101

0.4

0.2

-3 -2 -1 1 2 3

Figure 3. Distribution of normalized θ-coefficients for E = 32A1 and varying d.

Part 2. Heuristics 7. The heuristic for cyclic extensions of prime degree Recall that we have assumed that E is semistable. Our heuristic is based on two assumptions:

(H1) the cF,γ are distributed randomly subject to the constraints imposed by Conjecture 6.2 and Remark 6.5, (H2) the only correlations among the cF,γ for fixed F and varying γ are the Atkin–Lehner relations of Lemma 5.2. See Remark 8.3 below for more about the issue of “intra-correlation”. For simplicity, we first consider the case of extensions of prime degree. Fix for this section a character χ of prime order p ≥ 3. As above we write m for the conductor of χ, F for the corresponding real cyclic extension of Q, and G := Gal(F/Q). Let BE, αp, βp ∈ R be as in Conjecture 6.2. Since p is odd, Lemma 5.2(ii) shows that the involution ιF has a unique fixed element σ ∈ G. Choose a subset S ⊂ G consisting of one element from each pair {γ, ιF (γ)} of generic elements. ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 15

Lemma 7.1. We have X χ(γ)cF,γ = 0 ⇐⇒ cF,γ = cF,σ for all γ ∈ S. γ∈G

Proof. The only Q-linear relation among the values of χ (i.e., the p-th roots of unity) is that their sum is zero. It follows that the sum over γ is zero if and only if all cF,γ are equal. The lemma now follows from Lemma 5.2(i) (note that aσ = 0 if wF = −1).  Heuristic 7.2. With notation as above,

 2αp (p−1)/4 2 pm P[L(E, χ, 1) = 0] < BE . ϕ(m) log(m)1−2βp Remark 7.3. The notation P[L(E, χ, 1) = 0] is meant to stand for (an upper bound for) the heuristic likelihood that the value L(E, χ, 1) vanishes (see Remark 6.5).

Justification for Heuristic 7.2. By Corollary 3.4 and Lemma 7.1, X L(E, χ, 1) = 0 ⇐⇒ χ(γ)cF,γ = 0 ⇐⇒ cF,γ = cF,σ for all γ ∈ S. γ∈G Let √ p rχ := , pϕ(m) log(m) so the normalized θ-coefficientsc ˜F,γ of Definition 6.1 are given byc ˜F,γ = cF,γrχ. Since the cF,γ are integers, we have

1 1 cF,γ = cF,σ ⇐⇒ cF,γ ∈ (cF,σ − 2 , cF,σ + 2 ) rχ rχ
⇐⇒ c˜F,γ ∈ c˜F,σ − 2 , c˜F,σ + 2 Using Conjecture 6.2, Remark 6.5 and our heuristic assumptions (H1) and (H2) above, we have

Y rχ rχ P[cF,γ = cF,σ ∀γ ∈ S] = P[˜cF,γ ∈ (˜cF,σ − 2 , c˜F,σ + 2 )] γ∈S (p−1)/4  pm2αp log(m)2βp  < (B r mαp log(m)βp )(p−1)/2 = B2 E χ E ϕ(m) log(m) as desired.  16 BARRY MAZUR AND KARL RUBIN

8. The heuristic for general cyclic extensions Fix for this section an even character χ of arbitrary degree d ≥ 3. As above we write m for the conductor of χ, F for the corresponding real cyclic extension of Q, and G := Gal(F/Q). Let BE, αd and βd be the constants in Conjecture 6.2. Heuristic 8.1. With notation as above,

 2αd ϕ(d)/4 2 dm P[L(E, χ, 1) = 0] < BE . ϕ(m) log(m)1−2βd The rest of this section is devoted to justifying Heuristic 8.1, using heuristic assumptions (H1) and H(2) of §7. Recall the sign wF = ±1, the element γF ∈ G, and the involution ιF : G → G of Definition 5.1. We extend ιF to a Z-linear involution of Z[G] (also denoted by ιF ). Fix a generator g of G. Let ζ ∈ µd denote the primitive d-th root of unity χ(g). Define a Z-linear involution ιχ of the cyclotomic ring Z[ζ] by −1 ιχ(ρ) = χ(γF ) ρ¯ where ρ 7→ ρ¯ is complex conjugation. Then we have a commutative diagram  ιF Z[G]ιF =wF / Z[G] / Z[G] χ χ χ        [ζ]ιχ=wF / [ζ] / [ζ] Z Z ιχ Z

ιF =wF where, for example, Z[G] denotes {ρ ∈ Z[G]: ιF (ρ) = wF ρ}.

ιF =wF Definition 8.2. For γ ∈ G, define vγ ∈ Z[G] by  γ + w ι (γ) if γ is generic,  F F + vγ := γ if γ is special , 0 if γ is special−.

+ Fix a subset G0 ⊂ G consisting of all special elements of G and one element from each pair {γ, ιF (γ)} of generic elements. Then the set ιF =wF {vγ : γ ∈ G0} is a Z-basis of Z[G] . Since χ : Z[G]ιF =wF → Z[ζ]ιχ=wF is surjective, we can choose a ιF =wF subset S ⊂ G0 such that, writing A ⊂ Q[G] for the Q-vector ιχ=wF space spanned by {vγ : γ ∈ S}, χ maps A isomorphically to Q(ζ) . Let λ : Q(ζ)ιχ=wF → A be the inverse isomorphism. In particular we have ιχ=wF #S = dimQ Q(ζ) = ϕ(d)/2. ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 17

Justification for Heuristic 8.1. Define X ρ := cF,γγ. γ∈G

Then ρ ∈ [G]ιF =wF , so ρ = P c v . Let Z γ∈G0 F,γ γ X X ρ1 := cF,γvγ, ρ2 := cF,γvγ,

γ∈S γ∈G0−S so ρ = ρ1 + ρ2 and ρ1 ∈ A. We have X χ(γ)cF,γ = 0 ⇐⇒ χ(ρ) = 0 γ∈G

⇐⇒ χ(ρ1) = −χ(ρ2) ⇐⇒ ρ1 = −λ(χ(ρ2)). P Since λ(χ(ρ2)) ∈ A we can write −λ(χ(ρ2)) = bγvγ with bγ ∈ . √ γ∈S Q d Hence, writing rχ := √ as in §7, ϕ(m) log(m) X χ(γ)cF,γ = 0 ⇐⇒ cF,γ = bγ for every γ ∈ S γ∈G 1 1 ⇐⇒ cF,γ ∈ (bγ − 2 , bγ + 2 ) for every γ ∈ S rχ rχ ⇐⇒ c˜F,γ ∈ (bγrχ − 2 , bγrχ + 2 ) for every γ ∈ S so using Conjecture 6.2, Remark 6.5 and the heuristic assumptions (H1) and (H2) of §7 we have     X Y rχ rχ P χ(γ)cF,γ = 0 = P c˜F,γ ∈ (bγrχ − 2 , bγrχ + 2 ) γ∈G γ∈S #S/2  dm2αd log(m)2βd  < (B r mαd log(m)βd )#S = B2 E χ E ϕ(m) log(m)

(note that the bγ for γ ∈ S depend only on the cF,γ for γ ∈ G0 − S). P Since #S = ϕ(d)/2, and L(E, χ, 1) = 0 if and only if γ∈G χ(γ)cF,γ = 0 by Corollary 3.4, this gives the desired heuristic.  Remark 8.3. The upper bound of Heuristic 8.1, specifically the expo- nents ϕ(d)/4 and ϕ(d)/2, depends on the assumption (H2) of no intra- correlations, i.e., no statistical correlations connecting ϕ(d)/2 different θ-coefficients of a given θ-element, these being chosen to have the prop- erty that no two are brought into one another by the involution ιF (see §5). We might refer to the exponent ϕ(d)/2 as the intra-correlation ex- ponent. We will see that a qualitative version of our heuristic will still 18 BARRY MAZUR AND KARL RUBIN seem viable even if we assume significantly smaller intra-correlation exponents, e.g., if we have an upper bound of the form:

 2αd t(d) 2 dm P[L(E, χ, 1) = 0] < BE ϕ(m) log(m)1−2βp where t(d)  log(d) (see Proposition 11.15 below).

9. Expectations In this section we suppose that Conjecture 6.2 holds, and we fix data BE, αd, βd for all d ≥ 3 as in that conjecture.

Definition 9.1. For every d let Cd denote the set of even primitive Dirichlet characters of order d. Define

Fd(m) := #{χ ∈ Cd : conductor(χ) = m}, and for every X

Cd(X) := {χ ∈ Cd : conductor(χ) ≤ X},

 2αd ϕ(d)/4 X 2 dm (9.2) Ed(X) := Fd(m) BE . ϕ(m) log(m)1−2βd m≤X Our Heuristic 8.1 suggests the following. Heuristic 9.3. We expect

#{χ ∈ Cd(X): L(E, χ, 1) = 0} ≤ Ed(X).

In particular if Ed(∞) is finite, then the number of χ of order d with L(E, χ, 1) = 0 should be finite. More generally, if D ⊂ Z≥3, then we expect X #{χ ∈ ∪d∈DCd(X): L(E, χ, 1) = 0} ≤ Ed(X). d∈D Proposition 9.4. Suppose p is prime, and let

σ := (1 − 2αp)(p − 1)/4, τ := βp(p − 1)/2.

As X ∈ R goes to infinity we have the following upper bounds on Ep(X). (i) If σ > 1, or σ = 1 and τ < −1, then Ep(∞) is finite.

(ii) If σ = 1 and τ = −1, then Ep(X) E,p log log(X). τ+1 (iii) If σ = 1 and τ > −1, then Ep(X) E,p log(X) . 1−σ τ (iv) If 0 < σ < 1, then Ep(X) E,p X log(X) . ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 19

Proof. This follows from Lemma 11.7 below. Precisely, since m  ϕ(m) log(m) (see for example [14, Theorem 328]), we have X log(m)τ E (X)  F (m) . p E,p p mσ m≤X Let g(x) := log(x)τ /xσ. Then on a suitable interval [r, ∞), g satisfies the hypotheses of Lemma 11.7, so that lemma shows that Z X Ep(X) E,p g(t)dt. r Now the proposition follows.  The following conjectures and predictions are suggested by the ran- dom matrix heuristic. This involves work of Conrey, Keating, Rubin- stein, and Snaith [6] for p = 2, and David, Fearnley, and Kisilevsky [9, 10] and Fearnley, Kisilevsky, and Kuwata [13] for p ≥ 3. For more precise references see Remark 9.6 below. If p(X), q(X): R≥r → R>0 are functions for some real number r, we write p(X) ∼ q(X) to mean that limX→∞ p(X)/q(X) exists and is nonzero. Conjecture 9.5 ([9, 10, 13]). For a prime p ≥ 3 let

np(X) := #{χ ∈ Cp(X): L(E, χ, 1) = 0}. Then there are nonzero constants c such that √ E,p cE,3 (i) n3(X) ∼ X log(X) , cE,5 (ii) n5(X) ∼ log(X) , (iii) np(X) is bounded independently of X if p ≥ 7. Remark 9.6. Assertion (iii) is part of [10, Conjecture 1.2]. The statements (i) and (ii) are implicit in [10, p. 256] but not ex- plicitly stated as conjectures. The authors of [10] comment: The exact power of log X that is obtained with the random matrix approach depends subtly on the dis- cretisation, and is difficult to predict. They state a weaker conjecture: Conjecture 9.7 (Conjecture 1.2 of [10]).

(i) limX→∞ log n3(X)/ log(X) = 1/2.  (ii) n5(X) is unbounded and  X as X tends to infinity, for any  > 0. Remark 9.8. See [13] for further interesting results related to the above conjecture for cubic characters. Fearnley and Kisilevsky [12] 20 BARRY MAZUR AND KARL RUBIN exhibit one example of L(E, χ, 1) = 0 with χ a character of order ` = 11 and E the elliptic curve 5906B1 (using Cremona’s classification [8]). The character χ is of conductor 23 (i.e., χ has the smallest possible conductor for characters of its order). In this example rank(E(Q)) = 2, + and rank(E(Q(µ23) )) = 12.

Remark 9.9. Suppose p = 3, α3 = 0, and β3 = cE,3. Then Proposition 9.4(iv) gives

1/2 cE,3 E3(X)  X log(X) .

Now suppose p = 5, α5 = 0, and β5 = (cE,5 −1)/2. Then Proposition 9.4(iii) gives cE,5 E5(X)  log(X) .

Now suppose p ≥ 7 and αp < (p − 5)/(2(p − 1)). Then Proposition 9.4(i) shows that Ep(∞) is finite. In other words, for appropriate values of α and β (that are consistent with computational data), Heuristic 9.3 is consistent with Conjecture 9.5. Proposition 9.10. We have X Ed(∞) < ∞.

d : ϕ(d)(1−2αd)>4 Proof. This follows from Proposition 11.15 below (and comparing equa- tions (9.2) and (11.16)) taking 2 t(d) := ϕ(d)/4, σd := 2αd, τd := 2βd,M := BE. 

10. Conjectures Proposition 9.10, Heuristic 9.3, and Conjecture 6.2 lead to the fol- lowing “analytic” and “arithmetic” conjectures. Conjecture 10.1. Let [ C = Cd ϕ(d)>4 (the set of all even Dirichlet characters of order at least 7 and different from 8, 10, or 12). Then the set {χ ∈ C : L(E, χ, 1) = 0} is finite.

Conjecture 10.2. Suppose F/Q is an (infinite) real abelian extension that has only finitely many subfields of degree 2, 3, or 5. Then E(F ) is finitely generated. ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 21

Conjecture 10.2 would be a consequence of Conjecture 10.1 and the Birch and Swinnerton-Dyer conjecture. As in Proposition 2.2, if the Birch and Swinnerton-Dyer conjecture holds then it is enough to show that there are only finitely many characters χ of Gal(F/Q) such that L(E, χ, 1) = 0. But the hypotheses in Conjecture 10.2 imply that Gal(F/Q) has only finitely many characters of order 2, 3, or 5, and consequently it has only finitely many characters of order 6, 8, 10, or 12 as well. Now the conjecture follows from Conjecture 10.1.  Remark 10.3. As mentioned in Remark 8.3, our heuristic—following Proposition 11.15—would suggest qualitatively similar conjectures (pos- sibly with a larger set of exceptional degrees d) even if there were sig- nificant “intra-correlation”.

11. Analytic results

Definition 11.1. For every d ≥ 1, recall (Definition 9.1) that Fd(m) denotes the number of even primitive Dirichlet characters of order d and conductor m. We are interested in sums of the form X X Sd(g; X) := Fd(m)g(m) m=r where g : R≥r → R≥0 is a function, and we let Sd(g) = Sd(g; ∞). ˜ Let Fd(m) denote the number of all primitive Dirichlet characters of order dividing d and conductor m. The following theorem is essentially due to Kubota [16]. Theorem 11.2 (Kubota). For every d ≥ 1, define Dirichlet series ∞ X ˜ −s Y Ld(s) := Fd(m)m ,Zd(s) := ζQ(µt)(s) m=1 t|d,t>1 where ζQ(µt) denotes the Dedekind ζ-function of the cyclotomic field Q(µt). Then Ld(s)/Zd(s) has an analytic continuation to the half plane <(s) > 1/2 and is nonzero at s = 1. Proof. The key ideas of the proof are in [16, Theorem 6], although Kubota does not state a result this strong. See also [9, Proposition 5.2] for a proof in the case d = 3 that works for all primes d. We prove the theorem in general by showing that Ld(s)/Zd(s) is given by an Euler product that converges on the half plane <(s) > 1/2. Q Suppose m ∈ Z>0 factors into powers of distinct primes as m = i qi. × ∼ Q × Using the isomorphism (Z/mZ) = i(Z/qiZ) , we see that every Dirichlet character of conductor m and order dividing d can be written 22 BARRY MAZUR AND KARL RUBIN uniquely as a product of Dirichlet characters of conductor qi and order ˜ dividing d. It follows that the function Fd(m) is multiplicative in m, and thus Ld(s) can be written as an Euler product ∞ Y X ˜ k −ks Ld(s) = Ld,p(s) where Ld,p(s) := Fd(p )p . p k=0 It is straightforward to verify that: −s • Ld,p(s) is a (finite degree) polynomial in p , ˜ • Fd(p) = gcd(d, p − 1) − 1, ˜ k • if p - d and k ≥ 2 then Fd(p ) = 0. In particular −s (11.3) Ld,p(s) = 1 + (gcd(d, p − 1) − 1)p if p - d. −1 Let Ld,p(s) denote the Euler factor at p of the Dirichlet series Zd(s) . × If t ∈ Z>0 and p - t let ν(p, t) denote the order of p in (Z/tZ) . Then if p - d we have

−s Y −ν(p,t)s ϕ(t)/ν(p,t) (1 − p )Ld,p(s) = (1 − p ) t|d Y P ϕ(t)/n = (1 − p−ns) t|d,ν(p,t)=n . n|d P The exponent of the n = 1 term is t|d,t|(p−1) ϕ(t) = gcd(d, p − 1), so −s gcd(d,p−1)−1 Y −ns P ϕ(t)/n (11.4) Ld,p(s) = (1 − p ) (1 − p ) t|d,ν(p,t)=n . n|d,n>1 Q It follows from (11.3) and (11.4) that the product p Ld,p(s)Ld,p(s) converges on the half plane <(s) > 1/2 and is not zero at s = 1. This completes the proof of the theorem. 

Corollary 11.5. The Dirichlet series Ld(s) of Theorem 11.2 converges on the half plane <(s) > 1, and has a pole of order τ(d) − 1 at s = 1, where τ(d) is the number of divisors of d.

Proof. This is immediate from Theorem 11.2, since each ζQ(µt) has a simple pole at s = 1.  ˜ Since Fd(m) ≤ Fd(m), the following is an immediate consequence of Corollary 11.5. −s Corollary 11.6. If g(X)  X for some s > 1, then Sd(g) is finite.

Recall that if p(X), q(X): R≥r → R>0 are functions for some real number r, we write p(X) ∼ q(X) to mean that limX→∞ p(X)/q(X) exists and is nonzero. ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 23

Lemma 11.7. Suppose r ∈ R and g : R≥r → R>0 is decreasing, continuously differentiable, and satisfies −g0(X)  g(X)/X. Then for every prime p we have

X X Z X Sp(g; X) p,g g(m) ∼ g(t)dt. m=r r PX ˜ Proof. Fix a prime p, and let A(X) := m=r Fp(m). Since p is prime, Corollary 11.5 shows that the Dirichlet series Lp(s) of Theorem 11.2 has a simple pole at s = 1. Hence a standard Tauberian theorem (for example [19, Theorem I, Appendix 2]) that (11.8) A(X) = cX + o(X) where c is the residue at s = 1 of Lp(s). Let X ˜ X ˜ Sp(g; X) := Fp(m)g(m). m=r Abel summation (see for example [2, Theorem 4.2]) gives Z X ˜ 0 (11.9) Sp(g; X) = A(X)g(X) − A(r)g(r) − A(t)g (t)dt. r Integration by parts gives Z X Z X (11.10) g(t)dt = Xg(X) − rg(r) − tg0(t)dt. r r Let B(X) := A(X) − cX with c as in (11.8). Subtracting c times (11.10) from (11.9) gives

Z X ˜ (11.11) Sp(g; X) − c g(t)dt r Z X = B(X)g(X) − B(r)g(r) − B(t)g0(t)dt. r Since g is decreasing and −g0(X)  g(X)/X, we have Z X Z X Z X 0 0 |B(t)| (11.12) B(t)g (t)dt ≤ − |B(t)|g (t)dt  g(t)dt. r r r t In addition, since B(X) = o(X) and g is decreasing, Z X (11.13) B(X)g(X) = o(Xg(X)) and Xg(X) ≤ g(t)dt + rg(r). r 24 BARRY MAZUR AND KARL RUBIN

Combining (11.11) with (11.12) and (11.13) shows that

( X X Z X oR g(t)dt
if R g(t)dt is unbounded, S˜ (g; X) − c g(t)dt = r r p R X r O(1) if r g(t)dt is bounded. ˜ Since Sp(g, X) ≤ Sp(g, X), this proves the lemma.  Lemma 11.14. There is an absolute constant C such that for every m ≥ 3 we have ϕ(m) log log(m) > m/C.

Proof. This follows from [14, Theorem 328]. 

Proposition 11.15. Suppose M ∈ R>0, t : Z≥3 → R is a function satisfying t(d)  log(d), and σd, τd are sequences of real numbers such that {t(d)σd : d ≥ 3} is bounded and limd→∞ τd = 0. Then

∞ t(d) X X Mdmσd log(m)τd  (11.16) F (m) d ϕ(m) log(m) d : t(d)(1−σd)>1 m=1 converges.

Proof. Let ∞ t(d) X Mdmσd log(m)τd  T := F (m) . d d ϕ(m) log(m) m=1

We will show below that Td converges if t(d)(1 − σd) > 1. We want to show that in fact P T converges. Note that F (m) = 0 d : t(d)(1−σd)>1 d d unless m > d. Fix k such that if d ≥ k, then

• t(d)(1 − σd) > 3, (11.17) • τd < 1/2, • log log(x)/ log(x)1/2 is decreasing on [d, ∞).

Let C be the positive constant from Lemma 11.14. Using Lemma 11.14, we have for every d

 σd t(d) t(d) X m (11.18) Td = (Md) Fd(m) ϕ(m) log(m)(1−τd) m>d  t(d) t(d) X C log log(m) < (Md) Fd(m) . m1−σd log(m)1−τd m>d ARITHMETIC CONJECTURES INSPIRED BY MODULAR SYMBOLS 25

× We have Fd(m) ≤ |Hom((Z/mZ) , µd)| < m, and using (11.17) we get  t(d) X X t(d) X C log log(m) Td < (Md) m m1−σd log(m)1/2 d≥k d≥k m>d t(d) XMCd log log(d) X < m1−t(d)(1−σd) log(d)1/2 d≥k m>d t(d) XMCd log log(d) d2−t(d)(1−σd) < 1/2 log(d) t(d)(1 − σd) − 2 d≥k t(d) XMC log log(d) (11.19) ≤ d2+b log(d)1/2 d≥k where b = sup{t(d)σd : d ≥ 3}. Fix an integer n ≥ b + 4. Since t(d)  log(d), there is a positive constant C0 such that for all large d we have

 1/2 t(d) log(d) 0 ≥ log(d)C log(d) MC log log(d) = eC0 log(d) log log(d) = dC0 log log(d) > dn. Thus the sum (11.19) converges. For the finitely many d < k with t(d)(1 − σd) > 1, (11.18) and Corollary 11.6 applied with  log log(x) t(d) g(x) = x1−σd log(x)1−τd shows that Td converges. This completes the proof of the proposition. 

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Department of Mathematics, Harvard University, Cambridge, MA 02138, USA E-mail address: [email protected]

Department of Mathematics, UC Irvine, Irvine, CA 92697, USA E-mail address: [email protected]