Elsevier

Journal of Number Theory

Volume 224, July 2021, Pages 274-306
Journal of Number Theory

General Section
Elliptic curves and Thompson's sporadic simple group

https://doi.org/10.1016/j.jnt.2021.01.015Get rights and content

Abstract

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose graded traces are weight 32 weakly holomorphic modular forms satisfying certain special properties. We then use these modules to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups of quadratic twists of certain elliptic curves.

Introduction

In 1978, McKay and Thompson observed [59] that the first few coefficients of the normalized elliptic modular invariant J(τ)=q1+196884q+21493760q2+864299970q3+O(q4), a central object in the theory of modular forms, can be written as sums involving the first few dimensions of irreducible representations of the monster group M, e.g.,196884=211+2119688321493760=211+21196883+21296876864299970=121+12196883+21296876+842609326. This coincidence inspired Thompson's conjecture [58] that there is an infinite-dimensional M-module V=n1Vn whose graded dimension is J(τ) and whose McKay–Thompson seriesTg(τ):=n1tr(g|Vn)qn are distinguished functions on the upper half-plane. Conway and Norton [16] explicitly described the relevant McKay-Thompson series, and also christened this phenomenon “monstrous moonshine.” Their conjecture was proven by Borcherds [4] (building on work by Frenkel, Lepowsky and Meurman [28]) in 1992. In the few decades since the first observations of McKay and Thompson, it has become clear that monstrous moonshine is just the first of a series of similar phenomena encompassing several finite groups and their counterparts in the world of modular forms.

Generalized moonshine [9] (see also [41], [16], [45]), for example, relates various subquotients of the Monster to other weight zero modular forms. Umbral moonshine [12], [13] (see also [22], [29] and [14]), on the other hand, relates the 23 umbral groups (each of which is a quotient of the automorphism group of one of the 23 Niemeier lattices) to weight 12 mock modular forms. Thompson moonshine, conjectured by Harvey and Rayhaun [33] in 2015 and proven by Griffin and Mertens in [31], involves Thompson's sporadic simple group Th, and certain weight 12 modular forms. (We remark here that the Thompson group, being a subgroup of the Monster, also appears in the generalized moonshine setting mentioned above. For the purpose of this paper, “Thompson moonshine” refers to the Harvey and Rayhaun version.)

Recently, in [23], [24], Duncan, Mertens, and Ono discovered the first instance of moonshine for the O'Nan group, one of the so-called pariah groups (i.e., a sporadic simple group which is not a subquotient of the monster group), where the functions involved are modular forms of weight 32. Their work is not only a contribution to the theory of moonshine, it also serves another important purpose: In the same paper, they use their O'Nan-module to study properties of quadratic twists of certain elliptic curves and thus use moonshine to provide insight into objects that are central to current research in number theory.

While number theory's contribution to moonshine is ubiquitous and irrefutable, O'Nan moonshine is one of the first instances where we see moonshine's direct contribution to number theory. Such a role-reversal is our primary motivation for this work.

We begin this work by proving the existence of a family of infinite-dimensional graded Th-modules whose McKay–Thompson series are weight 32 modular forms that satisfy certain properties (see Theorem 3.2). The techniques we use to prove this are similar to ones used in Griffin and Mertens' work [31] to prove the Thompson moonshine conjecture [33]. (These techniques were first suggested by Thompson, and subsequently used by Atkin, Fong and Smith [26], [53] to prove monstrous moonshine abstractly.) On the other hand, our McKay–Thompson series are weight 32 modular forms (in contrast to the weight 12 forms of [33]) and the role played by theta functions in their paper is taken up by weight 32 cusp forms in ours. The involvement of weight 32 cusp forms allows us to employ an approach similar to Duncan, Mertens, and Ono (in [23], [24]): We exploit the existing relationship between these forms and elliptic curves to study geometric invariants of various elliptic curves. This is the content of Theorem 1.1, Theorem 1.3.

Our result regarding the existence of a family of Thompson modules is, in fact, a classification result. We classify all infinite-dimensional graded modules W=Wn (see Theorem 3.2) for the Thompson group whose McKay–Thompson series take the formFg(τ):=6q5+n>0tr(g|Wn)qn and satisfy the following properties (cf. Proposition 3.1):

  • (1)

    For each gTh, the corresponding McKay–Thompson series Fg(τ) is a weight 32 weakly holomorphic modular form of a specific level and multiplier system, and satisfies the Kohnen plus space condition.

  • (2)

    Each McKay–Thompson series Fg(τ) has integer coefficients and is uniquely determined — up to the addition of certain cusp forms — by its polar parts at the cusps, which are specified in a uniform way. (See Section 3 for details.)

We note here that properties (1) and (2) listed above ensure that the functions Fg(τ) are, up to the addition of cusp forms, simply Rademacher sums projected to the Kohnen plus space (see Section 2 for background on Rademacher sums).

The connection between Rademacher sums and moonshine was first proposed in [21], where the McKay–Thompson series that appear in monstrous moonshine were characterized completely in terms of Rademacher sums of weight 0. In particular, it was shown that the so-called genus-zero property of monstrous moonshine is equivalent to the fact that the McKay–Thompson series of the Monster module coincide (up to a constant) with corresponding Rademacher sums of weight 0. It was later argued in [11], [12] that the correct analogue of the genus zero property in the case of Umbral (and Mathieu) moonshine is that the corresponding McKay–Thompson series must coincide with the relevant Rademacher sums in each case (see also [14], [20]). Here we take this perspective and hence consider it natural, from the point of view of moonshine, to ask for our McKay–Thompson series to satisfy the properties listed above.

To prove our classification result, we first construct spaces of weakly holomorphic modular forms of the appropriate level and multiplier for each gTh. We use Rademacher sums and eta-quotients to do this. Since we can explicitly compute the Fourier coefficients of these forms at various cusps, we can restrict our attention to the subspace of forms that satisfy properties (1) and (2). For a collection of these forms to be the McKay–Thompson series of a virtual module (as in Theorem 3.2), they must satisfy congruences modulo certain powers of primes that divide the order of the Thompson group (see Section 4). A complete description of these congruences can be obtained using Thompson's reformulation ([53]) of Brauer's characterization of generalized characters. We prove that our alleged McKay–Thompson series satisfy the congruences mentioned above in Section 4. We note here that it would be interesting to consider the analogous classification for the O'Nan group, building on the work already done in [24].

Once we have proven the existence of the Thompson modules, we use their properties to help detect the non-triviality of Mordell–Weil, Selmer, and Tate–Shafarevich groups of quadratic twists of certain elliptic curves (see Theorem 1.1, Theorem 1.3). To state our main results, we let E be an elliptic curve over Q, and for d<0 a fundamental discriminant, we let Ed denote the dth quadratic twist of E. (We refer the reader to [40], [50] and [55] for background on elliptic curves.) We further let E(Q) denote the set of Q-rational points on E, i.e., the set of points on E whose coordinates are rational numbers. Then, E(Q) has the structure of a finitely generated abelian group (by the Mordell–Weil theorem [50]), i.e., E(Q)=ZrE(Q)tor. Here rZ0 is called the (algebraic) rank of E, and E(Q)tor is a finite abelian group. Computing the rank of a general elliptic curve is considered a hard problem in number theory.

One way of approaching this problem is to study the L-function associated to E, defined for sC with Re(s)>32 byLE(s)=p prime(1apps+ϵ(p)p12s)1. Here, ap:=p+1#E(Fp) where E(Fp) is the group of Fp-rational points on the mod p reduction of E, and ϵ(p){0,1} (see [50] for a precise definition and further discussion on L-functions). The (weak) Birch and Swinnerton-Dyer Conjecture states that the order of vanishing of LE(s) at s=1 is equal to the rank r of E. Thus, studying the behavior of LE(s) near s=1 is one way of tackling the problem of computing the rank of E.

Note that a priori, the product in eq. (1.4) only converges for Re(s)>32, so part of the conjecture is that LE(s) can be analytically continued to a function that converges near s=1. This part follows from the Modularity Theorem (proven in [6] by extending the results of [62], [56]), which states that corresponding to every elliptic curve over Q, there exists a weight 2 newform of level equal to the conductor of E, whose L-function coincides with LE(s). The L-function associated to a cusp form f=n=1anqn of weight k is defined on Re(s)>1+k2 and extends analytically to a holomorphic function on C [2]. Hence it is possible to talk about the behavior of LE(s) as s1, for example. The Birch and Swinnerton-Dyer Conjecture has been proven in special cases (see for example [3]). In particular, it is known that if LE(1)0, then E(Q) is finite [32], [38]. We will use this result to prove our first theorem about elliptic curves.

To state our results, we letF(τ)=6(q5+85995q3565760q4+52756480q7190356480q8+O(q11)) be the unique weakly holomorphic modular form of weight 32 and level 4 in the plus space whose Fourier expansion is of the form F(τ)=6q5+O(q). (Note that F(τ) is relevant for us because for each of the graded Th-modules W=n>0Wn described in Theorem 3.2, the graded dimension of W is F(τ).) Let c(d) denote the |d|th Fourier coefficient in the expansion of F(τ). We denote by (mn) the usual Kronecker symbol [15], then we have the following theorem.

Theorem 1.1

Let d<0 be a fundamental discriminant which satisfies (d19)=1. Let E be an elliptic curve of conductor 19, and let Ed denote the dth quadratic twist of E. If c(d)0(mod19), then the Mordell–Weil group Ed(Q) is finite.

We can state a stronger result for elliptic curves of conductor 14, one which depends on a local version of the strong form of Birch and Swinnerton-Dyer Conjecture. To do this, we first establish some notation.

For E an elliptic curve of rank r, let #E(Q)tor denote the order of the torsion subgroup of the Q-rational points of E and let

be the order of the Tate–Shafarevich group of E. Then the strong form of the Birch and Swinnerton-Dyer Conjecture states the following [63].

Conjecture 1.2 The Birch and Swinnerton-Dyer Conjecture

The rank r of an elliptic curve E over Q equals the order of vanishing of LE(s) at s=1. Moreover, we have where LE(r)(s) is the rth derivative of LE(s).

The left-hand side of Equation (1.6) ties together the rank of the elliptic curve and the value of the rth derivative of the L-function at s=1. It has been proven to always be a rational number [1]. On the right-hand side, all quantities except
are effectively computable invariants of an elliptic curve (see Section 5 for details). The Tate-Shafarevich group
, however, is not even known to be finite except for special cases.

So we instead turn to a relatively simple object. Given p a prime, we define

to be the set consisting of elements of the Tate–Shafarevich group with order dividing p. Then,
is finite for all p. This follows from the finiteness of a related object: the p-Selmer group Selp(E) of E, which fits into a short exact sequence The Selmer group of an elliptic curve over Q is known to be finite for every p (see [50]), so we have that
is finite for every p. Also by the short exact sequence, we get that if Selp(E) is trivial for any prime p then E(Q) is finite, i.e., E has rank 0.

In this work, we will use the family of Thompson modules whose existence is proven in Theorem 3.2 to develop a criterion to check whether the p-Selmer groups of quadratic twists of elliptic curves of conductor 14 are trivial. More precisely, we let E be an elliptic curve over Q, and for each d<0 a fundamental discriminant, let Ed denote the dth quadratic twist of E. As we shall prove in Theorem 3.2, there exists an infinite-dimensional graded Th-module W=n>0Wn whose McKay–Thompson series Fg(τ) satisfy properties (1) and (2) as above. Then we have the following theorem.

Theorem 1.3

Let d<0 be a fundamental discriminant for which (d7)=1 and (d2)=1. Let E be an elliptic curve of conductor 14, and let g denote an element of order 14 in Th. If tr(g|W|d|)0(mod49), then the Mordell–Weil group Ed(Q) is finite and

is trivial. If, on the other hand, tr(g|W|d|)0(mod49) and tr(g|W4)43(mod56), then Sel7(Ed) is non-trivial, and if LEd(1) is non-zero then so is
.

This is akin to Theorem 1.4 of [24], and our proof will follow along similar lines. One notable difference is that there is no dependence on (generalized) class numbers in the corresponding congruences in our case. We can also write down an analogous statement for elliptic curves of conductor 19, but the techniques we use to prove Theorem 1.3 do not apply in this case (cf. Theorem 5.4), so it is conditional upon the (strong form of the) Birch and Swinnerton-Dyer Conjecture.

We now describe a sketch of the proof of Theorem 1.1, Theorem 1.3. Let p{7,19} be the relevant prime in either statement and fix W=n>0Wn to be a virtual Thompson module whose McKay–Thompson series Fg(τ) satisfies the properties listed in Theorem 3.2. We then write each Fg(τ) for [g]{14A,19A} as a sum of traces of singular moduli (cf. Section 5.1) and weight 32 cusp forms. This expression combined with the condition on d in the statement of Theorem 1.1 gives us that the congruence in the statement holds if and only if the relevant cusp form coefficient is divisible by p=19. Thus, if the congruence in the statement of Theorem 1.1 does not hold, then the cusp form coefficient is not divisible by 19, and we can employ a corollary (cf. Lemma 5.3) of Kohnen's work [37] to show that this means LEd(1)0. Finally, Kolyvagin's work shows that Ed(Q) is finite. This completes the proof of Theorem 1.1. For Theorem 1.3, we first consider the case that tr(g14|W|d|)0(mod49). The expression for Fg(τ) in terms of traces of singular moduli and cusp forms implies that the relevant cusp form coefficient is not divisible by p=7. We can utilize Kohnen's work again to conclude that ordp(LEd(1)Ω(Ed))>0. At this point, we use work of Skinner and Urban (Theorem 5.4) which connects ordp(LEd(1)Ω(Ed)) to the non-triviality of the p-Selmer and Tate-Shafarevich groups of Ed to prove the theorem. A similar argument applies if we assume that tr(g14|W|d|)0(mod49) and tr(g14|W|4|)43(mod49).

The rest of this paper is organized as follows. In Section 2 we set up notation and define the terms that appear in Proposition 3.1 and Theorem 3.2. In Section 3 we uniquely characterize the modular forms Fg(τ) that satisfy properties (1) and (2) above and prove Proposition 3.1. In Section 4 we prove Theorem 3.2. Finally, in Section 5 we prove Theorem 1.1, Theorem 1.3.

Section snippets

Acknowledgments

The author would like to thank John Duncan for helping initiate this project and his invaluable guidance and support throughout the process. The author thanks Richard Borcherds, John Duncan, Jeff Harvey, Michael Mertens, Brandon Rayhaun, Robert Schneider, and the anonymous referee for helpful comments on an earlier draft.

Background and notation

Throughout this paper, we use the notation e(x)=e2πix and q=e(τ) with τ in the upper half-plane, which we denote H. We also use (mn) to denote the Kronecker symbol [15, Algorithm 1.4.10]. We will use the ATLAS [17] notation for conjugacy classes of Th, and understand nAB to mean nAnB.

McKay–Thompson series

To state our main theorem of this section, we associate to each rational conjugacy class [g] of the Thompson group Th, the following data:

  • (1)

    Integers vg and hg as specified in Table A.1. We use these to define the character ψg:=ψ4|g|,vg,hg (see Equation (2.3)), where |g| denotes the order of g in Th.

  • (2)

    The space of cusp forms Sg:=S32+(Γ0(4|g|),ψg) of weight 32 in the plus-space which transform under Γ0(4|g|) with character ψg. We define dg to be the dimension of this space and let fg be the dg-tuple f

Proof of Theorem 3.2: integer multiplicities

To prove Theorem 3.2, we have to show that the Fgλ(τ)'s we described in Section 3 are indeed the McKay–Thompson series of a virtual module of the Thompson group.

This is equivalent to proving that there exist integers m1λ(n),...,m39λ(n) such that if Fgλ(τ)=6q5+n3αgλ(n)qn, then for each n3 the Fourier coefficient αgλ(n) can be written in the form,αgλ(n)=j=139mjλ(n)χj(g), where χ1,,χ39 are the irreducible rational characters of Th (see Section 2.1 for a definition of rational character). We

Elliptic curves

The family of Th-modules that we get from Theorem 3.2 encodes arithmetic information about quadratic twists of elliptic curves with conductors 14 and 19. This is the content of Theorem 1.1, Theorem 1.3. We will prove these theorems in this section, but first, we have to develop some background. We recall here some basic notation and facts about traces of singular moduli, which were studied by Zagier in [64] and have since been examined extensively.

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