O'Nan moonshine and arithmetic
aa r X i v : . [ m a t h . N T ] M a r O’NAN MOONSHINE AND ARITHMETIC
JOHN F. R. DUNCAN, MICHAEL H. MERTENS, AND KEN ONO
Abstract.
Answering a question posed by Conway and Norton in their seminal 1979 pa-per on moonshine, we prove the existence of a graded infinite-dimensional module for thesporadic simple group of O’Nan, for which the McKay–Thompson series are weight / modular forms. The coefficients of these series may be expressed in terms of class numbers,traces of singular moduli, and central critical values of quadratic twists of weight 2 modular L -functions. As a consequence, for primes p dividing the order of the O’Nan group we obtaincongruences between O’Nan group character values and class numbers, p -parts of Selmergroups, and Tate–Shafarevich groups of certain elliptic curves. This work represents thefirst example of moonshine involving arithmetic invariants of this type. Introduction and Statement of Results
The sporadic simple groups are the twenty-six exceptions to the classification [7] of finitesimple groups: those examples that aren’t included in any of the natural infinite families. Itis natural to wonder where they appear, outside of the classification itself.At least for the monster , being the largest of the sporadics, this question has an interestinganswer. By the last decade of the last century, Ogg’s observation [72] on primes dividing theorder of the monster, McKay’s famous formula , and the much broader family of coincidences observed by Thompson [89, 90] and Conway–Norton [29], were proven by Borcherds [10] to reflect the existence of a certain distinguishedalgebraic structure. This moonshine module , constructed by Frenkel–Lepowsky–Meurman[44, 45, 46], admits a vertex operator algebra structure, has the monster as its full symmetrygroup, and has modular functions for traces. It is a cornerstone of monstrous moonshine , andindicates a pathway by which ideas from theoretical physics, and string theory in particular,may ultimately reveal a natural origin for the monster group and its curious connection tomodularity.In addition to the monster itself, nineteen of the sporadic simple groups appear as quotientsof subgroups of the monster. As such, we may expect that monstrous moonshine extends tothem in some form. This is consequent upon the generalized moonshine conjecture , whichwas formulated by Norton [71] following preliminary observations of Conway–Norton [29]and Queen [78], and which has been recently proven in powerful work by Carnahan [21].Certain more general analogues of monstrous moonshine have appeared in this century. In2010, Eguchi–Ooguri–Tachikawa [41] sparked a resurgence of interest in moonshine with theirobservation that the elliptic genus of a K3 surface—a trace function arising from a non-linear Mathematics Subject Classification. sigma model with K3 target—is, essentially, the sum of an indefinite theta function and a q -series whose coefficients are dimensions of modules for Mathieu’s largest sporadic group, M . In fact, this q -series is a mock modular form which, together with most of Ramanujan’smock theta functions, belongs to a family of distinguished examples [24] arising from a familyof finite groups. This is umbral moonshine [26, 27, 28], and the existence of correspondingumbral moonshine modules has been verified by Gannon [49] in the case of M , and in generalby Griffin and two of the authors of this work [37]. However, it must be noted that this theoryis not yet on the same footing as monstrous moonshine, as suitable umbral counterparts tothe moonshine module vertex operator algebra of Frenkel–Lepowsky–Meurman are not yetknown in general.We refer the reader to [46, 47] for fuller discussions of monstrous moonshine, and to Gan-non’s book [48] for a broad perspective on the theory. The more recent review [38] includessome umbral developments. We refer to [75, 76] for new work on the string theoretic interpre-tation of monstrous moonshine, and refer to [6, 25, 39, 40] for vertex algebraic constructionsof some of the umbral moonshine modules.Very recently, yet another form of moonshine has appeared in work of Harvey–Rayhaun[58] which manifests a kind of half-integral weight counterpart to generalized moonshine forThompson’s sporadic group. This is known as Thompson moonshine . The existence of acorresponding module has been confirmed by Griffin and one of the authors [54] (but in thiscase too, a vertex algebraic realization is yet to be found).All the umbral groups are involved in the monster in some way, so we are left to wonder ifthere are counterparts to monstrous moonshine for the remaining six pariah sporadic groups:the
Janko groups J , J , and J , the Lyons group Ly , the Rudvalis group Ru , and the O’Nangroup
O’N . Can moonshine shed light on these groups too? Conway and Norton asked thisquestion (cf. p. 321 of [29]) in their seminal 1979 paper: “Finally, we ask whether the sporadic simple groups that may not be involved in [the mon-ster]... have moonshine properties.”
This question is also Problem p -parts of Selmer groups and Tate–Shafarevich groups of elliptic curves. (See e.g. [84] or [88] for background on elliptic curvearithmetic.) This is the first occurrence of moonshine of this type. Since J is a subgroup of O’N it suggests that at least two pariah groups play an active part in some of the deepestopen questions in arithmetic.1.1.
Moonshine and Divisors.
Before describing our results in more detail we offer aconceptual number theoretic perspective which ties together some of the recent developmentsmentioned above. Suppose that G is one of the finite groups appearing in the aforementionedcases of moonshine. Then we have an infinite-dimensional graded G -module, say V G , whichmanifests a collection of modular forms, one for each conjugacy class. For monstrous, umbral, ’NAN MOONSHINE AND ARITHMETIC 3 and Thompson moonshine we have V G = M m V Gm moonshine −−−−−−−−−−−→ ( f [ g ] ) ∈ L [ g ] ∈ Conj ( G ) M !0 (Γ [ g ] ) monstrous L [ g ] ∈ Conj ( G ) H (Γ [ g ] ) umbral, Thompson . The defining feature of the f [ g ] is that their m th coefficients equal the graded traces tr( g | V Gm ) .In monstrous moonshine, the f [ g ] are Hauptmoduln for genus 0 groups Γ [ g ] (essentiallylevel o ( g ) congruence subgroups). At the cusp ∞ , they have Fourier expansion f [ g ] = q − + O ( q ) (note q := e πiτ throughout), and are holomorphic at other cusps. In particular, this meansthat div ( f [ g ] ) = cz − ∞ for some z ∈ X (Γ [ g ] ) and some integer c . In contrast, the f [ g ] in umbral and Thompson moonshine are not functions on modular curves, so it does notgenerally make sense to consider their divisors. Instead, they are weight 1/2 harmonic Maassforms (with multiplier) for Γ [ g ] , which means that the McKay–Thompson series are generally mock modular forms , the holomorphic parts of the f [ g ] . Although they are not functions onthese modular curves, it turns out that they actually encode even more information aboutdivisors on X (Γ [ g ] ) . For each discriminant D , there is a map Ψ D for which V G = M m V Gm moonshine −−−−−−−−−−−→ ( f [ g ] ) Ψ D −−−−−→ (Ψ D ( f [ g ] )) ∈ M [ g ] ∈ Conj ( G ) K (Γ [ g ] ) , where K (Γ [ g ] ) is the field of modular functions for Γ [ g ] . The Ψ D ( f [ g ] ) are generalized Borcherdsproducts as defined by Bruinier and one of the authors [20]. They are meromorphic modularfunctions with a discriminant D Heegner divisor, and their fields of definition are dictatedby the Fourier coefficients of the f [ g ] .As the preceding discussion illustrates, monstrous, umbral, and Thompson moonshine are(surprising) phenomena in which a single infinite-dimensional graded G -module organizesinformation about divisors on products of modular curves that are indexed by the conjugacyclasses of G . Moreover, the levels of these modular curves are (essentially) the orders ofelements in these classes. In the case of monstrous moonshine, the divisors are simple: theyare of the form cz − ∞ . In umbral and Thompson moonshine, we obtain Heegner divisorson X (Γ [ g ] ) .The appearance of Heegner divisors recalls the seminal work of Zagier [99] on traces ofsingular moduli on X (1) . Loosely speaking, Zagier proved that the generating function forsuch traces in D -aspect can be weight 3/2 weakly holomorphic modular forms. One of hismotivations was to offer a classical perspective on special cases of Borcherds’ work [11] oninfinite product expansions of modular forms with Heegner divisor.Although Zagier’s paper has inspired too many papers to mention, we highlight an im-portant note by Gross [55]. Gross observed that these types of theorems could be recast interms of generalized Jacobians with cuspidal moduli. In particular, the generalized Jacobianof X (1) with respect to the cuspidal divisor ∞ ) is isomorphic to the additive group, and JOHN F. R. DUNCAN, MICHAEL H. MERTENS, AND KEN ONO so the sum of the conjugates of Heegner points in the generalized Jacobian is equal to thetrace of their modular invariants.Here we adopt this perspective. Although we do not directly apply these results in thiswork, our view is that the McKay–Thompson series presented here should be viewed in thisway, as generating functions for traces of singular moduli and as functionals on Heegnerdivisors. This interpretation is an extension of the celebrated theorem of Gross–Kohnen–Zagier [56] which asserts that the generating function for Heegner divisors on X ( N ) areweight 3/2 cusp forms with values in the Jacobian of X ( N ) . By work of Waldspurger [93, 94]this earlier theorem can be thought of as a result on central critical values of quadratic twistsof weight 2 modular L -functions.1.2. Main Results.
In view of these developments, it is natural to seek weight 3/2 moon-shine. One can loosely think of this as the moonshine obtained by summing weight 1/2moonshine in D -aspect (e.g. umbral and Thompson moonshine), where the resulting McKay–Thompson series are generating functions for the arithmetic of Heegner divisors. Namely,we seek moonshine of the form V G = M m V Gm moonshine −−−−−−−−−−−→ ( f [ g ] ) ∈ M [ g ] ∈ Conj ( G ) H (Γ [ g ] ) ⊗ Jac ( X (Γ [ g ] )) , where Jac ( X (Γ [ g ] ) denotes a suitable generalized Jacobian of X (Γ [ g ] ) . In such moonshine,the f [ g ] will be generating functions for suitable functionals over Heegner divisors. Theircoefficients will be sums of class numbers, traces of singular moduli, and square-roots ofcentral critical values of L -functions of quadratic twists of weight 2 modular forms.Here we establish the first example of moonshine of this type, and it is pleasing that pariahsporadic groups appear. We prove moonshine for the O’Nan group O’N , a group discoveredin 1973 as part of the flurry of activity related to the classification of finite simple groups[73] and shown not to be involved in the monster by Griess [52, Lemma 14.5]. This groupwas first constructed by Sims (cf. [73, p. 421]), and Ryba [81] later gave an alternativeconstruction. It has order
O’N = 2 · · · · · · , and it has 30 conjugacy classes.It contains the first Janko group J , also not involved in the monster [95], as a subgroup. Theorem 1.1.
There is an infinite-dimensional virtual graded
O’N -module W := M In other prominent examples of moonshine (e.g. monstrous [29] and umbral [26, 27]moonshine) the McKay–Thompson series of a group element g is a modular form (essentially)of level o ( g ) , but in this work the McKay–Thompson series F [ g ] have level o ( g ) . This anomalycan be resolved by repackaging the F [ g ] as Jacobi forms as follows. For g ∈ O’N set ϕ [ g ] ( τ, z ) := F [ g ] , ( τ ) θ , ( τ, z ) + F [ g ] , ( τ ) θ , ( τ, z ) (1.1)where F [ g ] ,r ( τ ) := P m ≡ r mod tr( g | W m ) q m and θ ,r ( τ, z ) := P n ≡ r mod e πinz q n . Then ϕ [ g ] isa weakly holomorphic Jacobi form of weight and index on Γ ( o ( g )) , with a non-trivialcharacter in case o ( g ) = 16 . For the sake of simplicity we have chosen to formulate ourresults in terms of the scalar-valued modular forms F [ g ] in this work. However, we notethat one advantage of the Jacobi form formulation is that it illuminates an analogue of theHauptmodul property of monstrous moonshine. Namely, each ϕ [ g ] has the property that itis uniquely determined, up to a cusp form, by the condition that it has growth of a certainform (independent of g ) near the infinite cusp of Γ ( o ( g )) , and vanishes at all other cusps.This follows from the proof of Theorem 3.1. It may be compared to monstrous moonshine,in which the McKay–Thompson series are uniquely determined up to constant functions byan analogous condition, and to umbral and Thompson moonshine, in which the McKay–Thompson series are uniquely determined by such a condition up to theta series (althoughin the umbral case almost all the relevant spaces of theta series vanish; cf. [28]). It isthe appearance of cusp forms that allows us to connect the O’Nan group to elliptic curvearithmetic. Remark. The module W is virtual in the sense that some irreducible representations of O’N occur with negative multiplicity in W m for some m . The proof of Theorem 1.1 will show thatonly non-negative multiplicities appear for m / ∈ { , , , } . So in fact we can replace W with a non-virtual module for a small cost, by adding suitable multiples of weight / unarytheta functions (i.e. sums of the form P n ∈ Z nǫ ( n ) q λn where ǫ is an odd periodic functionand λ is a positive rational) to the McKay–Thompson series F [ g ] . This changes the modulestructure of W m when m = vd for v ∈ { , , , } , for certain integers d , but it does noteffect the validity of our other three main results, Theorems 1.2, 1.3, and 1.4, for − D < − .The price for such an adjustment to W is the property that the McKay–Thompson seriesattached to [ g ] have level o ( g ) . It is this property which motivates us to focus on theparticular module W that appears in Theorem 1.1.The F [ g ] will turn out to be expressible in terms of traces of singular moduli for Haupt-moduln (cf. Section 5), class numbers, and central critical L -values of quadratic twists ofweight 2 modular forms (cf. Section 4.2.2). The Hauptmoduln which arise are for the genus0 modular curves(1.2) { X ( N ) : N = 1 , . . . , , , , } ∪ { X +0 ( N ) : N = 11 , , , , , , , , } , where X +0 ( N ) is the modular curve corresponding to the extension of Γ ( N ) by all the level N Atkin–Lehner involutions. JOHN F. R. DUNCAN, MICHAEL H. MERTENS, AND KEN ONO Remark. Purely for the sake of curiosity we mention that it follows from the descriptionof the dimensions of the graded components W m in terms of traces of singular moduli (cf.Appendix D) that dim W = 12 ( α + α − , where α = l e π √ m = ⌈ . ... ⌉ denotes the Ramanujan constant. (This number was actually already discovered and studiedby Hermite in 1859 [59].) Remark. From Tables B.1 to B.3 we see that W is an irreducible O’N -module of dimension , and W has three irreducible constituents, with dimensions , and .On the other hand the specialization ϕ A ( τ, of Equation (1.1) is the derivative of the J function, up to a scalar factor. This leads to the identity · · , where the summands on the right are dimensions of irreducible representations of O’N . In-spired by the moonshine module vertex operator algebra [46] of Frenkel–Lepowsky–Meurmanwe may ask: is there a holomorphic vertex operator algebra with an action by O’N thatexplains this coincidence? (See the second remark in Section 3 for some further relatedcomments.)Armed with Theorem 1.1 and the explicit identities expressing the F [ g ] in terms of sin-gular moduli, class numbers and critical L -values, it is natural to ask whether the infinite-dimensional O’N -module W reveals arithmetic information about the modular curves theyorganize, which include the positive genus curves { X (11) , X (14) , X (15) , X (19) , X (20) , X (28) , X (31) } related to the X +0 ( N ) in (1.2). For example, are there interesting congruences modulo primes p | O’N which relate the graded components W m to classical objects in number theory andarithmetic geometry? This is indeed the case, and we now describe surprising congruenceswhich relate graded dimensions and traces of W to class numbers and Selmer groups andTate–Shafarevich groups of elliptic curves. Remark. Suppose that p is prime and g n (resp. g np ) are elements of O’N with order n (resp. np ). Then by Theorem 1.1, we have that tr( g n | W m ) ≡ tr( g np | W m ) (mod p ) for all m . Inparticular, if o ( g ) = p , then for all m we have dim W m ≡ tr( g | W m ) (mod p ) . The following theorem concerns congruences modulo small primes p and ideal class groupsof imaginary quadratic fields. Here and in the following, we denote by H ( D ) the Hurwitz classnumber of positive definite binary quadratic forms of discriminant − D < (cf. Section 5). Theorem 1.2. Suppose that − D < is a fundamental discriminant. Then the followingare true: ’NAN MOONSHINE AND ARITHMETIC 7 (1) If − D < − is even and g ∈ O’N has order 2, then dim W D ≡ tr( g | W D ) ≡ − H ( D ) ≡ ) . (2) If p ∈ { , , } , (cid:16) − Dp (cid:17) = − and g p ∈ O’N has order p , then dim W D ≡ tr( g p | W D ) ≡ ( − H ( D ) (mod 3 ) if p = 3 , − H ( D ) (mod p ) if p = 5 , . Remark. Systematic congruences which assert for (cid:16) − Dp (cid:17) = − that dim W D ≡ − H ( D ) (mod p ) do not seem to hold for p ≥ . However, this congruence holds for p = 13 , a bonus because ∤ O’N . Remark. As the proof of Theorem 1.2 will reveal, it holds true that if − D < − is an evenfundamental discriminant, then H ( D ) is even, and dim W D ≡ ) .In view of Theorem 1.2, it is natural to consider the primes p = 11 , and whichalso divide O’N . For these primes, a refinement of the congruences above is necessary. Inparticular, for the primes 11 and 19 we obtain congruences which relate dim W D to Selmergroups and Tate–Shafarevich groups of elliptic curves (cf. [84, Chapter X]).Let E/ Q be an elliptic curve given by E : y + a xy + a y = x + a x + a x + a where a , a , a , a , a ∈ Z . For a fundamental discriminant D , let E ( D ) denote its D -quadratic twist, and let rk( E ( D )) denote the Mordell–Weil rank of E ( D ) over Q . The O’N -module W encodes deep information about the Selmer and Tate–Shafarevich groupsof the quadratic twists of elliptic curves with conductor 11, 14, 15, and 19. To make thisprecise, suppose that ℓ is an odd prime. Then for each curve E ( D ) we have the short exactsequence → E ( D ) /ℓE ( D ) → Sel( E ( D ))[ ℓ ] → X ( E ( D ))[ ℓ ] → , where Sel( E ( D ))[ ℓ ] is the ℓ -Selmer group of E ( D ) , and X ( E ( D ))[ ℓ ] denotes the elements ofthe Tate–Shafarevich group X ( E ( D )) with order dividing ℓ .For p = 11 and , we let E p / Q be the Γ ( p ) -optimal elliptic curves given by the Weier-strass models E : y + y = x − x − x − ,E : y + y = x + x − x − (cf. [68, Elliptic Curve 11.a2, Elliptic Curve 19.a2]). We obtain the following congruencerelating the graded dimension dim W D to class numbers, and Selmer groups and Tate–Shafarevich groups of such twists. Theorem 1.3. Assume the Birch and Swinnerton-Dyer Conjecture. If p = 11 or and − D < is a fundamental discriminant for which (cid:16) − Dp (cid:17) = − , and g p ∈ O’N has order p ,then the following are true. JOHN F. R. DUNCAN, MICHAEL H. MERTENS, AND KEN ONO (1) We have that Sel( E p ( − D ))[ p ] = { } if and only if dim W D ≡ tr( g p | W D ) ≡ − H ( D ) (mod p ) . (2) Suppose that L ( E p ( − D ) , = 0 . Then we have that rk( E ( − D )) = 0 . Moreover, wehave p | X ( E p ( − D )) if and only if dim W D ≡ tr( g p | W D ) ≡ − H ( D ) (mod p ) . Remark. The claim about ranks in Theorem 1.3 (2) is unconditional thanks to the work ofKolyvagin [67]. Remark. By Goldfeld’s famous conjecture on ranks of quadratic twists of elliptic curves [51],it turns out that the hypothesis in Theorem 1.3 (2) is expected to hold for of the − D for which (cid:16) − Dp (cid:17) = − . Therefore, for almost all such − D , we should have a test fordetermining the presence of order p elements in these Tate–Shafarevich groups. Remark. There is a more complicated congruence for the prime p = 31 . For fundamentaldiscriminants − D < satisfying (cid:0) − D (cid:1) = − , we have that dim W D ≡ tr( g | W D ) (mod 31) are related to the central critical values of the − D twists of the L -function for the genus 2curve C : y + ( x + x + 1) y = x + x + x − x − (cf. [68, Genus Curve 961.a.961.3]). Its L -function arises from the two newforms in S (Γ (31)) which are Galois conjugates. Namely, if φ := √ then the two newformsare f σ and f ( τ ) := ∞ X n =1 a ( n ) q n = q + φq − φq + ( φ − q + q − (2 φ + 2) q + O ( q ) , where σ ( √ 5) = −√ . If p ∤ is prime, then the local L -factor L p ( T ) at p is L p ( T ) := (1 − a ( p ) T + pT )(1 − σ ( a ( p )) T + pT ) . Remark. Apart from the claims about tr( g | W D ) (there are no elements of order 17 in O’N ),Theorem 1.3 holds for p = 17 as well. Namely, the congruences hold for E , the optimal Γ (17) elliptic curve over Q (cf. [68, Elliptic Curve 17.a3]) given by E : y + xy + y = x − x − x − . The two theorems on congruences above only pertain to the dimensions of the gradedcomponents of the O’N -module W . We now turn to congruences for graded traces forelements of order 2 and 3. To this end, we let E and E be the corresponding optimalelliptic curves over Q (cf. see [68, Elliptic Curve 14.a6, Elliptic Curve 15.a5]) given by E : y + xy + y = x + 4 x − ,E : y + xy + y = x + x − x − . Using work of Skinner and Skinner–Urban [85, 86] related to the Iwasawa main conjecturesfor GL , we obtain the following unconditional result. ’NAN MOONSHINE AND ARITHMETIC 9 Theorem 1.4. Assume the notation above, and suppose that N ∈ { , } . If p is the uniqueprime ≥ dividing N , then let δ p := p − and let p ′ := N/p . If − D < is a fundamentaldiscriminant for which (cid:16) − Dp (cid:17) = − and (cid:16) − Dp ′ (cid:17) = 1 , then the following are true.(1) We have that Sel( E N ( − D ))[ p ] = { } if and only if tr( g p ′ | W D ) ≡ tr( g N | W D ) ≡ δ p · ( H ( D ) − δ p H ( p ′ ) ( D )) (mod p ) . (2) Suppose that L ( E N ( − D ) , = 0 . Then we have that rk( E ( − D )) = 0 . Moreover, wehave p | X ( E N ( − D )) if and only if tr( g p ′ | W D ) ≡ tr( g N | W D ) ≡ δ p · ( H ( D ) − δ p H ( p ′ ) ( D )) (mod p ) . Remark. We note that Theorem 1.4 does not apply for p = 2 (resp. p = 3 ) when N = 14 (resp. N = 15 ). In the case of p = 2 the work of Skinner–Urban does not apply. For p = 3 the connection between graded traces and central values of Hasse-Weil L -functions does nothold. Namely, a critical hypothesis due to Kohnen in terms of eigenvalues of Atkin–Lehnerinvolutions fails (cf. Proposition 4.4). Remark. In view of the new results presented here, it is natural to wonder where one shouldlook for further moonshine. It seems likely that other sporadic groups will fall within thescope of weight 3/2 moonshine. In another direction, one can ask about other half-integralweights. Also, it is natural to wonder if there are extensions of moonshine to Shimura curvesand varieties. Are there infinite-dimensional G -modules which organize the arithmetic oftheir divisors?1.3. Methods. To prove Theorem 1.1, we employ the theory of Rademacher sums, harmonicMaass forms, and standard facts about the representation theory of finite groups. Namely, wemake use of the character table of O’N (cf. Table A.1), and the Schur orthogonality relationsfor group characters. In Section 2, we first recall essential facts about harmonic Maass formsand Rademacher sums. In Section 3, we prove a theorem which, using harmonic Maass forms,explicitly constructs weakly holomorphic weight 3/2 modular forms, one for each conjugacyclass of O’N . Furthermore, we establish that these modular forms have integer Fouriercoefficients. To complete the proof, we apply the Schur orthogonality relations to thesefunctions to construct weight 3/2 modular forms whose coefficients encode the multiplicitiesof the irreducibles of the graded components of the alleged module W . The proof is completeonce it is established that these multiplicities are integral. Since the obstruction to integralityis bounded by group theoretical considerations, the proof of integrality follows by confirmingsufficiently many congruence relations among these forms. These calculations confirm that W is a virtual module. However, as mentioned earlier, it turns out that the multiplicities ofeach irreducible are non-negative in W m once m > . This claim follows from an analyticargument which involves bounding sums of Kloosterman sums. These statements are provedin Section 4. In Section 5 we recall properties of singular moduli, and we interpret themodular forms number theoretically in terms of singular moduli and class numbers and cuspforms. We prove Theorems 1.2, 1.3 and 1.4 in Section 6. These proofs require the explicitformulas for the F [ g ] , the results in Section 5, and the work of Skinner–Urban on the Birch andSwinnerton-Dyer Conjecture. We conclude the paper in Section 7 with numerical examplesof some of these results. Acknowledgements The authors thank Kathrin Bringmann, Michael Griffin, Dick Gross, Maryam Khaqan, Win-fried Kohnen, Martin Raum, Jeremy Rouse, Jean-Pierre Serre and anonymous referees forhelpful comments and corrections. The authors thank Theo Johnson-Freyd for communica-tion regarding his joint work with David Treumann on the cohomology of the O’Nan group.The authors thank Drew Sutherland for computing the elliptic curve invariants in Tables 7.5and 7.6. 2. Rademacher Sums and Harmonic Maass Forms Harmonic Maass forms are now a central topic in number theory. Their study originatesfrom the work of Bruinier–Funke [18] on geometric theta lifts and Zwegers’ seminal work[101] on Ramanujan’s mock theta functions. These realizations played a central role in thework of Bringmann and one of the authors on the Andrews–Dragonette Conjecture andDyson’s partition ranks [15, 17]. For an overview on the subject of harmonic Maass formsand its applications in number theory and various other fields of mathematics, includingmathematical physics, we refer the reader to [14, 31, 74, 100].Here, we briefly recall the essential facts about harmonic Maass forms that are requiredin this paper. Namely, we recall Rademacher sums, and we describe their projection toKohnen’s plus space.2.1. Rademacher Sums. Here and throughout, we let τ = u + iv , u, v ∈ R , denote avariable in the upper half-plane H and we use the shorthand e ( α ) := e πiα . Definition 2.1. We call a smooth function f : H → C a harmonic Maass form of weight k ∈ Z and level N if the following conditions are satisfied:(1) We have f | k γ ( τ ) = f ( τ ) for all γ ∈ Γ ( N ) and τ ∈ H , where we define f | k γ ( τ ) := ( ( cτ + d ) − k f (cid:0) aτ + bcτ + d (cid:1) if k ∈ Z (cid:0)(cid:0) cd (cid:1) ε d (cid:1) k (cid:0) √ cτ + d (cid:1) − k f (cid:0) aτ + bcτ + d (cid:1) if k ∈ + Z . with ε d := ( if d ≡ ,i if d ≡ . and where we assume | N if k / ∈ Z .(2) The function f is annihilated by the weight k hyperbolic Laplacian , ∆ k f := (cid:20) − v (cid:18) ∂ ∂u + ∂ ∂v (cid:19) + ikv (cid:18) ∂∂u + i ∂∂v (cid:19)(cid:21) f ≡ . (3) There is a polynomial P ( q − ) such that f ( τ ) − P ( e − πiτ ) = O ( v c ) for some c ∈ R as v → ∞ . Analogous conditions are required at all cusps of Γ ( N ) .We denote the space of harmonic Maass forms of weight k and level N by H k (Γ ( N )) .Remark. We note that condition (3) in the definition above differs from other definitionswhich occur commonly in the literature. For example, harmonic Maass forms with principalparts are those forms for which the O ( v c ) bound is replaced by O ( e − cv ) for c > . Namely, ’NAN MOONSHINE AND ARITHMETIC 11 the harmonic Maass forms we consider here are permitted to have th Fourier coefficientswhich are essentially powers of v .For the basic properties of these functions, we again refer to the literature mentionedabove. We mention however the following lemmas. Lemma 2.2. Let f ∈ H k (Γ ( N )) be a harmonic Maass form of weight k = 1 . Then there isa canonical splitting (2.1) f ( τ ) = f + ( τ ) + f − ( τ ) , where for some m ∈ Z we have the Fourier expansions f + ( τ ) := ∞ X n = m c + f ( n ) q n , and f − ( τ ) := c − f (0) (4 πv ) − k k − ∞ X n =1 c − f ( n ) n k − Γ(1 − k ; 4 πnv ) q − n , where Γ( α ; x ) := Z ∞ x t α e − t dtt denotes the usual incomplete gamma function. The q -series f + in (2.1) is called the holomorphic part of the harmonic Maass form f .An important differential operator in the theory of harmonic Maass forms is the ξ -operator,a variation of the Maass lowering operator. Proposition 2.3. The operator ξ k : H k (Γ ( N )) → M − k (Γ ( N )) , f ξ k ( f ) := 2 iy k ∂f∂τ is a well-defined and surjective anti-linear map with kernel M ! k (Γ ( N )) .Mock modular forms are the holomorphic parts of harmonic Maass forms. Any mockmodular form has an associated modular form, called its shadow , which is the image ofits corresponding harmonic Maass form under the ξ -operator. A mock modular form withvanishing shadow is a (weakly holomorphic) modular form.The next lemma seems to have been missed by the literature. Lemma 2.4. A harmonic Maass form whose holomorphic part vanishes at all cusps is a(holomorphic) cusp form.Proof. This is a direct consequence of the properties of the Bruinier–Funke pairing (cf.Proposition 3.5 in [18]). (cid:3) A convenient way to construct mock modular forms, which are holomorphic parts ofharmonic Maass forms, is through Rademacher sums . These were introduced by Rademacherin his work on coefficients of the J -function [79], and further developed in the context ofmoonshine mainly by Cheng, Frenkel and one of the authors [22, 23, 36]. Rademacher sums can be thought of as low weight analogues of Poincaré series. For afixed level N and some K > , we define the set Γ K,K ( N ) := (cid:8) ( a bc d ) ∈ Γ ( N ) : | c | < K and | d | < K (cid:9) . Given an integer µ we can use this to formally define the Rademacher sum R [ µ ] k,N ( τ ) := lim K →∞ X γ ∈ Γ ∞ \ Γ K,K ( N ) q µ | k γ where as usual Γ ∞ := {± ( n ) : n ∈ Z } denotes the stabilizer of ∞ in Γ ( N ) . If conver-gent, these sums define mock modular forms of the indicated weight, level and character.Convergence for these series however is in general a delicate matter when the weight k isbetween and . We will be interested in these series when the weight is k = in whichcase it has been established in [22, Section 5] that they do converge (possibly using a certainregularization explained in loc. cit.) and define holomorphic functions on H .By construction, Rademacher sums are -periodic and therefore have a Fourier expansion.It is given in terms of infinite sums of Kloosterman sums (2.2) K k ( m, n, c ) := X ∗ d (mod c ) (cid:16) cd (cid:17) ε kd e (cid:18) md + ndc (cid:19) weighted by Bessel functions. Here we have that k ∈ + Z , c is divisible by , the ∗ atthe sum indicates that it runs over primitive residue classes modulo c , and d denotes themultiplicative inverse of d modulo c . Computing the Fourier expansion of a Rademachersum is a standard computation, see for instance [23, Section 3.1] and [74, Section 8.3]. Theorem 2.5. Assuming locally uniform convergence, for µ ≤ and k ∈ + N and | N , theRademacher sum R [ µ ] k,N defines a mock modular form of weight k for Γ ( N ) whose shadow isgiven by a constant multiple of the Rademacher sum R [ − µ ]2 − k,N . Its Fourier expansion is givenby R [ µ ] k,N ( τ ) = q µ + ∞ X n =1 c [ µ ] k,N ( n ) q n , where (2.3) c [ µ ] k,N ( n ) = − πi k (cid:12)(cid:12)(cid:12)(cid:12) nµ (cid:12)(cid:12)(cid:12)(cid:12) k − X c> c ≡ N ) K k ( µ, n, c ) c · I k − π p | µn | c ! for µ < and (2.4) c [0] k,N ( n ) = ( − πi ) k n k − Γ( k ) X c> c ≡ N ) K k (0 , n, c ) c k . The completion [ R [ µ ] k,N of R [ µ ] k,N to a harmonic Maass form has a pole of order µ at the cusp ∞ and vanishes at all other cusps. ’NAN MOONSHINE AND ARITHMETIC 13 Remark. One can also consider Rademacher sums of weights ≤ / , which are the mainsubject of [22] and play a crucial rule in both umbral and Thompson moonshine. Theformulas look very similar in those cases, but since they are not needed, we omit them here.2.2. Kohnen’s Plus Space. In [66], Kohnen introduced the notion of the so-called plusspace, a natural subspace of weight k + cusp forms for Γ (4 N ) which is isomorphic viathe Shimura correspondence to the space of weight k cusp forms of level N as a Heckemodule, provided that N is odd and square-free. This space is easily characterized viaFourier expansions. Namely, it consists of all forms in S k + (Γ (4 N )) (or, by extension, M ! k + (Γ (4 N )) and also H k + (Γ (4 N )) ) whose Fourier coefficients are supported on expo-nents n with n ≡ , ( − k (mod 4) . There is a natural projection operator | pr : S k + (Γ (4 N )) → S + k + (Γ (4 N )) for N odd given in terms of slash operators (see loc. cit.), which extends to spaces ofweakly holomorphic modular forms and harmonic Maass forms. The action of this projectionoperator on principal parts of harmonic Maass forms is described in the following lemma (cf.Lemma 2.9 in [54]). Lemma 2.6. Let N be odd and let f ∈ H k + (Γ (4 N )) for some k ∈ N . Suppose that f + ( τ ) = q − m + ∞ X n =0 a n q n for some m > with − m ≡ , ( − k (mod 4) , and suppose also that f has a non-vanishingprincipal part only at the cusp ∞ and is bounded at the other cusps of Γ (4 N ) . Then theprojection f | pr of f to the plus space has a pole of order m at ∞ and has a pole of order m either at the cusp N if m ≡ , or at the cusp N if − m ≡ ( − k (mod 4) , and isbounded at all other cusps. For the purpose of this paper, we are particularly interested in the Fourier expansionsof weight / Rademacher sums projected to the plus space (see the following section).Convergence for these follows along the same lines as in [22, Section 5]. The followingproposition gives their Fourier expansion explicitly. Proposition 2.7. Consider the Rademacher sum R [ µ ] , N for µ ≤ such that µ ≡ , and N odd. Then we have that R [ µ ] , + , N ( τ ) := (cid:16) R [ µ ] , N | pr (cid:17) ( τ ) = q µ + X n> n ≡ , c [ µ ] , + , N ( n ) q n , where we have (2.5) c [ µ ] , + , N ( n ) = κ ( µ, n ) ∞ X c =1 (1 + δ odd ( N c )) K ( µ, n, N c ) · I ( µ, n, N c ) , with (2.6) κ ( µ, n ) := ( πe (cid:0) − (cid:1) if µ = 0 , πe (cid:0) − (cid:1) ( n/ | µ | ) otherwise, (2.7) δ odd ( n ) := ( if n is odd, otherwise,and (2.8) I ( µ, n, c ) := (2 πn ) c Γ(3 / if µ = 0 ,I (cid:18) π √ | µn | c (cid:19) c otherwise. The following proposition shows that the vanishing of Kloosterman sums automaticallyforces certain even level Rademacher sums to be in the plus space. Proposition 2.8. The Rademacher sum R [ µ ] , N is automatically in the plus space if N iseven and µ ≡ , . Moreover, if N, µ ≡ , then the Fourier coefficients of R [ µ ] , N are supported on exponents divisible by .Proof. We begin by noting that if c is divisible by , then the Kloosterman sum K ( m, n, c ) in (2.2) vanishes unless m − n ≡ , . If c is divisible by , the same sum vanishesunless m ≡ n (mod 4) . Therefore, the claim follows from Theorem 2.5. (cid:3) Remark. This is an easy restatement (and slight correction) of [54, Lemma 2.10]. Remark. Proposition 2.8 actually follows from the splitting properties of the Weil represen-tation, which is a stronger statement than the vanishing of Kloosterman sums we employedin the proof. However, since we don’t use the language of vector-valued modular forms inthis paper we use the above more elementary argument. Remark. We note that the formulas in Proposition 2.7 also hold for even N if one defines theprojection operator pr for even levels as a suitable sieving operator, which one easily sees bya comparison to Theorem 2.5.3. The Relevant Modular Forms Here we use the results from the previous section to realize the McKay–Thompson seriesfor the O’N -module W whose existence shall be proved later. The main result here is thefollowing theorem. To state it we define a character ρ [ g ] : Γ (4 o ( g )) → C ∗ for each conjugacyclass [ g ] of O’N by setting ρ [ g ] ( ∗ ∗ c ∗ ) := ( − c when o ( g ) = 16 , and letting ρ [ g ] be trivialotherwise. Theorem 3.1. Assuming the notation above, the following are true. ’NAN MOONSHINE AND ARITHMETIC 15 (1) For every conjugacy class [ g ] of O’N there is a unique weakly holomorphic modularform (3.1) F [ g ] ( τ ) = − q − + 2 + X n =1 a [ g ] ( n ) q n of weight / for the group Γ (4 o ( g )) , with character ρ [ g ] , satisfying the followingconditions:(a) F [ g ] ( τ ) lies in the Kohnen plus space, i.e., a [ g ] ( n ) = 0 if n ≡ , .(b) F [ g ] ( τ ) has a pole of order at the cusp ∞ , a pole of order at the cusp o ( g ) if o ( g ) is odd (as forced by the projection to the plus space, see Lemma 2.6), andvanishes at all other cusps.(c) We have a [ g ] (3) = χ ( g ) , and a [ g ] (4) = χ ( g ) + χ ( g ) + χ ( g ) , and a [ g ] (7) asgiven in Tables B.1 to B.3, where χ j , for j = 1 , ..., , denotes the j th irreduciblecharacter of O’N as given in Table A.1.(2) The function F [ g ] ( τ ) above has integer Fourier coefficients.Remark. One can also give a more intrinsic description of the conditions in part (c) above.The proof of the theorem will show that F [ g ] is already determined by conditions (a) and (b)in part (1), for the conjugacy classes [ g ] such that o ( g ) / ∈ { , , , , , , } . Forthe remaining conjugacy classes we remark that whenever a prime p divides o ( g ) , we needthe congruence(3.2) a [ g ] ( n ) ≡ a [ g ′ ] ( n ) (mod p ) where o ( g ′ ) = o ( g ) /p in order for these to be generalized characters for O’N . Wheneverone can choose the coefficient a [ g ] ( n ) for the function F [ g ] —which turns out to be the casefor n = 3 and o ( g ) ∈ { , , } , for n = 4 for o ( g ) ∈ { , } and for n ∈ { , } for o ( g ) ∈ { , } —we pick the least integer in absolute value satisfying (3.2) for all primes p | o ( g ) . Remark. The mod cohomology of the O’Nan group was computed by Adem–Milgram [2].Using this, Johnson-Freyd–Treumann determined [65] that H ( O’N , Z ) is cyclic of order .Furthermore, they have explained to us [64] that there is an element whose image underthe restriction map H ( O’N , Z ) → H ( h g i , Z ) is zero unless o ( g ) = 16 , in which case itis the element of order in H ( Z / Z , Z ) ≃ Z / Z . The significance of this is that if V is a holomorphic vertex operator algebra with an action by a finite group G then it isconjectured that the ( G -twisted) representation theory of V , including the modularity ofits associated trace functions, is controlled by an element of H ( G, Z ) ≃ H ( G, U (1)) . Inparticular, an element of H ( G, Z ) associates a multiplier system on Γ ( o ( g )) to each g ∈ G .The above statements about H ( O’N , Z ) imply that there is an element that associates thetrivial multiplier to Γ ( o ( g )) for all g ∈ O’N except those with o ( g ) = 16 , and in the lattercase the multiplier arising is the order two character on Γ (16) with kernel Γ (32) . That is,the characters determined by this element of H ( O’N , Z ) are exactly those that are satisfiedby the Jacobi forms ϕ [ g ] of Equation (1.1), and this is compatible with the existence of aholomorphic vertex operator algebra that realizes these functions, and hence also the F [ g ] ,and the O’N -module of Theorem 1.1. We refer to §2 of [49], and references therein, for more on the relationship between H ( G, U (1)) , modular forms and vertex operator algebra. Werefer to §3.2 of [42] for a recent account of the aforementioned conjecture on vertex operatoralgebras. Remark. The O’Nan group has a non-split extension · GL (2) as a subgroup, whilst thesporadic simple Higman–Sims group contains a splitting extension : GL (2) . In bothcases the mentioned subgroups contain Sylow -subgroups, so can be used to detect theexistence, or not, of -power elements in the cohomology of the corresponding simple groups.There may be something to be gained from a comparison of these groups, especially inlight of the fact that the Higman–Sims group inherits a natural counterpart to monstrousmoonshine by virtue of generalized moonshine, and the fact that it appears in the centralizersof suitable elements of order in the monster. We thank an anonymous referee for offeringthis observation. We also thank the referee for pointing out that the extensions of GL (2) by were studied by Alperin [5] and Griess [53]. Results on the cohomology of the Higman–Sims group can be found in [1] and [65]. Proof of Theorem 3.1. Let g ∈ O’N be any element with o ( g ) = 16 . Then the difference ofRademacher sums − R [ − , o ( g ) ( τ ) + 2 R [0] , o ( g ) ( τ ) of level o ( g ) is a mock modular form with the correct principal part at infinity and vanishesat all other cusps by Theorem 2.5. If o ( g ) is even, then we know from Proposition 2.8 thatthis function is in the plus space. If on the other hand, o ( g ) is odd, then we use the projectionoperator | pr to map it into the plus space, which by Lemma 2.6 introduces an additionalpole of order at the cusp o ( g ) . This establishes the existence of a function e F [ g ] ( τ ) := − R [ − , + , o ( g ) ( τ ) + 2 R [0] , + , o ( g ) ( τ ) satisfying properties (a) and (b) in Theorem 3.1 (1) for o ( g ) = 16 . To achieve this much for o ( g ) = 16 we use e F [ g ] ( τ ) := − (2 R [ − , + , ( τ ) − R [ − , + , ( τ )) + 2(2 R [0] , + , ( τ ) − R [0] , + , ( τ )) since ρ [ g ] in this case is trivial on Γ (128) , and − on Γ (64) \ Γ (128) .By Lemma 2.4 we see that the above properties determine a mock modular form uniquelyup to cusp forms. Unless o ( g ) ∈ { , , , , , , } there are no cusp forms of weight / in the plus spaces with the required characters, so one checks directly that in all thosecases condition (c) is satisfied. In the remaining cases, condition (c) uniquely determines thecontribution from cusp forms, because, as one can check using standard computer algebrasystems (the authors used Magma [13] and PARI [77]), any weight / cusp form of one ofthe given levels in the plus space with the relevant character is uniquely determined by thecoefficients of q , q , and q .We now show that the functions F [ g ] are actually all weakly holomorphic instead of justmock modular. First suppose that o ( g ) is odd or || o ( g ) . Then, because in those cases o ( g ) is square-free, the shadow of F [ g ] ( τ ) must be a multiple of ϑ ( τ ) := X n ∈ Z q n , ’NAN MOONSHINE AND ARITHMETIC 17 which follows from the Serre-Stark basis theorem [83]. We compute Bruinier–Funke pairings(see Proposition 3.5 in [18]) and find that { \ R [ − , + , o ( g ) ( τ ) , ϑ ( τ ) } = 2 c and { \ R [0] , + , o ( g ) ( τ ) , ϑ ( τ ) } = c, where c is some constant. This shows that the shadow of the mock modular form F [ g ] ( τ ) = − R [ − , + , o ( g ) ( τ ) + 2 R [0] , + , o ( g ) ( τ ) is , whence it is indeed a weakly holomorphic modular form.If o ( g ) is divisible by or , but not , the space of possible shadows is a priori -dimensional, generated by ϑ ( τ ) and ϑ (4 τ ) , but Proposition 2.8 and the fact that the shadowof a Rademacher sum is again a Rademacher sum show that the shadow’s Fourier coefficientsmust be supported on exponents divisible by . So in fact, only multiplies of ϑ (4 τ ) canoccur as shadows and the same computation as above shows the claim in these cases. For o ( g ) = 16 the space of possible shadows is a priori one-dimensional, spanned by ϑ ( τ ) − ϑ (4 τ ) ,but this function is supported on odd exponents so is also ruled out by the Rademacher sumconstruction.It remains to show that the coefficients of the F [ g ] are all rational integers. This follows bychecking finitely many coefficients and applying Sturm’s Theorem [87], in a manner directlysimilar to the proof of Proposition 3.2 in [54], for example. As we will also see in Section 4.1,a possible bound up to which coefficients need to be checked to verify the claim is 225. (cid:3) Proof of Theorem 1.1 Here we prove that the weakly holomorphic modular forms given in Theorem 3.1 areMcKay–Thompson series for the infinite-dimensional O’N -module W . We begin by statinga refined form of Theorem 1.1. Theorem 4.1. There is an infinite-dimensional graded virtual O’N -module W = ∞ M m =3 m ≡ , W m such that we have tr( g | W m ) = a [ g ] ( m ) , for all m . Moreover, W m is an honest O’N -module for m / ∈ { , , , } (see Tables B.1to B.3). We break down the proof of this theorem into separate pieces. Using the Schur orthogo-nality relations on the irreducible representations of O’N we construct weakly holomorphicmodular forms of weight 3/2 whose coefficients are the multiplicities of the irreducible compo-nents if and only if W exists. Then the proof of Theorem 4.1 boils down to proving that thesemultiplicities are integral for all m , and non-negative for m 6∈ { , , , } . In Section 4.1we establish integrality, and in Section 4.2 we establish the claim on non-negativity. Integrality of Multiplicities. For every prime p | O’N we find linear congruencesamong the alleged McKay–Thompson series. Here, we prove these, but first we note thattheir truth implies the following systematic congruences. Theorem 4.2. Let g j ∈ O’N of order d j , j = 1 , , with d = p c · d for some prime number p and c ≥ . Then we have the congruence F [ g ] ≡ F [ g ] (mod p ) . In Appendix C, we list these congruences, which sometimes hold with higher prime powermoduli than stated in Theorem 4.2. Assuming their correctness for the moment, we canshow integrality just as described in [54]. For the convenience of the reader, we recall themethod briefly.Let C ∈ Z ×∞ denote the matrix formed by the coefficients of the functions F [ g ] ( τ ) foreach of the conjugacy classes of O’N (in practice one uses a × B matrix for somelarge B ). Further denote by X ∈ Q × the matrix whose rows are indexed by irreduciblecharacters and whose columns are indexed by conjugacy classes of O’N , with X χ, [ g ] := χ ( g ) C ( g ) , where C ( g ) denotes the centralizer of g ∈ O’N . By the first Schur orthogonality relation wesee that the matrix m := XC gives the multiplicities of each irreducible representation in the alleged virtual representationin Theorem 4.1. Since there are repetitions among the rows of C , because the functions F [ g ] ( τ ) depend only on the order of elements in [ g ] , it does not have full rank, but by justdeleting the repetitions it does turn out to have full rank, which is . Let N ∗ ∈ Z × denote the matrix performing this operation and let N ∈ Z × be the matrix that undoesit, so that m = XNN ∗ C . Now for each prime p | O’N , we can reduce the matrix N ∗ C according to the aforementionedcongruences as in [54] by left-multiplying by a matrix M p ∈ Q × , which may be seen tohave full rank. Hence we get m = ( XNM − p ) · ( M p N ∗ C ) . The congruences in Appendix C ensure that the matrix M p N ∗ C ∈ Q ×∞ has all integerentries and one can check directly that the matrix XNM − p ∈ Q × has p -integral (rational)entries for every p . This shows that m has p -integral entries as well for each p | O’N , henceits entries must be integers, as claimed.It remains to show the congruences. Since by Theorem 3.1 all the functions F [ g ] are weaklyholomorphic modular forms, we can prove all the congruences with standard techniques fromthe theory of modular forms. For example, we may multiply each of the congruences by theunique cusp form g in S + (Γ (4)) such that g ( τ ) = q + O ( q ) (which has integral coefficients),thereby reducing the problem to congruences among holomorphic modular forms of weight ’NAN MOONSHINE AND ARITHMETIC 19 . These can be checked in all cases using the Sturm bound [87], which is at most 225 inall cases.4.2. Positivity of Multiplicities. Denote by mult j ( n ) the multiplicity of the irreduciblecharacter χ j of O’N in the virtual module W n as in Theorem 4.1, whose associated generalizedcharacter is given by the coefficients a [ g ] ( n ) , cf. Theorem 3.1. Then the Schur orthogonalityrelations and the triangle inequality tell us that(4.1) mult j ( n ) = X [ g ] ⊆ O’N C ( g ) a [ g ] ( n ) χ j ( g ) ≥ | a ( n ) | O’N χ j (1) − X [ g ] =1 A | a [ g ] ( n ) | C ( g ) | χ j ( g ) | , where the summations run over conjugacy classes of O’N . Hence in order to show the eventualpositivity of all mult j ( n ) , we want to establish explicit lower bounds on a A ( n ) , and upperbounds on a [ g ] ( n ) for g = 1 . Recall that F [ g ] ( τ ) = − q − + 2 + ∞ X n =1 a [ g ] ( n ) q n = − R [ − , + , o ( g ) ( τ ) + 2 R [0] , + , o ( g ) ( τ ) + cusp formfor o ( g ) = 16 , and F [ g ] ( τ ) = − (2 R [ − , + , ( τ ) − R [ − , + , ( τ )) + 2(2 R [0] , + , ( τ ) − R [0] , + , ( τ )) for o ( g ) = 16 . (For o ( g ) = 16 we have dim S + (Γ (4 o ( g )) , ρ [ g ] ) = 1 but condition (c) inTheorem 3.1 rules out any cuspidal contribution to F [ g ] . Cf. Appendix D.)We bound each of the components individually, following the strategy already employedin [37, 49, 54], which we sketch briefly for the convenience of the reader. Note however thatin the cited papers, only the coefficients of one Rademacher sum had to be considered, sincethe corrections there were known to come from weight modular forms, whose coefficientsare bounded. In our case the corrections can grow with n .Since the computations necessary to bound the contribution coming from the Rademachersum R [ − , N , which is obviously going to be the dominant part, have been carried out in detailin [49, 54], we omit them here. The idea is to use the known formula for the coefficients ofthe Rademacher sum in terms of infinite sums of Kloosterman sums weighted by I -Besselfunctions, see Section 2. One then splits this sum into three parts, a dominant part, anabsolutely convergent remainder term and a value of a Selberg–Kloosterman zeta function ,the first two of which may be bounded by elementary means, and for the third, one usesProposition 4.1 in [54] (which we note is directly applicable to our situation).4.2.1. Bounding Coefficients of Rademacher Sums. From Proposition 5.2 below, we see thatthe µ = 0 Rademacher sum can be explicitly given in terms of generating functions ofgeneralized Hurwitz class numbers H ( N ) ( n ) (see Section 5 for the definition). While strongbounds for class numbers are known (see for instance Chapter 23 in [63] and the referencestherein), they are usually not explicit. For our purposes, crude bounds on class numberssuffice. Proposition 4.3. For every N ∈ N , − D ≤ − a negative discriminant and ε > we have H ( N ) ( D ) ≤ [SL ( Z ) : Γ ( N )] c ε D ε · √ D π (cid:18) D (cid:19) , where we can choose c ε = Y p First we note that we trivially have the bound H ( N ) ( D ) ≤ [SL ( Z ) : Γ ( N )] H (1) ( D ) by definition. Now suppose for the moment that D is a fundamental discriminant. ThenDirichlet’s class number formula gives H ( D ) = √ D π · L (cid:18) , (cid:18) − D • (cid:19)(cid:19) . Theorem 13.3 of Chapter 12 in [61] tells us that for D ≥ we have the upper bound L (cid:18) , (cid:18) − D • (cid:19)(cid:19) < D. By [97, pp. 73f.], Dirichlet’s formula is also valid for non-fundamental discriminants if onlyprimitive forms are counted, so that we get the bound H ( D ) ≤ τ (cid:3) ( D ) √ D π (cid:18) D (cid:19) , where τ (cid:3) ( n ) denotes the number of square divisors of n . Considering the prime factorisationof D , it is elementary to see that τ (cid:3) ( D ) ≤ c ε D ε for any ε > and c ε as claimed. (cid:3) This result together with Proposition 5.2 gives a sufficient and explicit bound for thecoefficients of the Rademacher sum R [0] , + , N . For the actual computations we choose ε = ,which yields c ε ≈ . .4.2.2. Bounding Coefficients of Cusp Forms. For g ∈ O’N with o ( g ) ∈ { , , , , , } , there are non-trivial cusp forms in S + (Γ (4 o ( g ))) contributing to our modular forms F [ g ] ,see Appendix D. According to the Ramanujan–Petersson conjecture, the coefficients of thesecusp forms should grow like O ( n + ε ) (for n square-free). Unconditional bounds (again forsquare-free n ) have been obtained by Iwaniec [62] for weights ≥ / and Duke [32] for weight / (see also [33]). These bounds have one main disadvantage for our purposes, namely thatthe constants involved in them are not explicit or not computable. Here, we outline howto give completely explicit and computable, but very crude, estimates for the cusp formcoefficients in question.Let P [ m ]4 N denote the cuspidal Poincaré series of weight / characterized by the Peterssoncoefficient formula ,(4.2) h f, P [ m ]4 N i = b f ( m )[SL ( Z ) : Γ (4 N )] √ m ∀ f ( τ ) = ∞ X n =1 b f ( n ) q n ∈ S + (Γ (4 N )) , ’NAN MOONSHINE AND ARITHMETIC 21 where the Petersson inner product on S + (Γ (4 N )) is defined by the usual double integral h f , f i = 1[SL ( Z ) : Γ (4 N )] Z Γ (4 N ) \ H f ( τ ) f ( τ ) y du dvv . The Fourier coefficients of these Poincaré series are given in terms of infinite sums ofKloosterman sums times J -Bessel functions (see Proposition 4 in [66]), and essentially thesame computation used to bound the coefficients of the Rademacher sums R [ − , N can beused here as well. It is then only necessary to express the cusp forms G ( o ( g )) (see againAppendix D) in terms of these Poincaré series, which is particularly easy in the cases where o ( g ) = 31 is odd, since in those cases, the space S + (Γ (4 o ( g ))) is one-dimensional and G ( o ( g )) is a newform. Hence we have h G ( o ( g )) , P [ m ]4 N i = β h G ( o ( g )) , G ( o ( g )) i , where we choose m to be theorder of G ( o ( g )) at ∞ . It therefore remains to compute the Petersson norm of the newform G ( o ( g )) . This can be done by means of the following result due to Kohnen, which is an explicitversion of Waldspurger’s theorem (see Corollary 1 in [66]). Proposition 4.4. Let N ∈ N be odd and square-free, f ∈ S + k + (Γ (4 N )) be a newform and F ∈ S k (Γ ( N )) the image of f under the Shimura correspondence. For a prime ℓ | N , let w ℓ be the eigenvalue of F under the Atkin–Lehner involution W ℓ and choose a fundamentaldiscriminant D with ( − k D > and (cid:0) Dℓ (cid:1) = w ℓ for all ℓ . Then we have h f, f i = h F, F i π k ω ( N ) ( k − | D | k − L ( F, D ; k ) · | b f ( | D | ) | , where L ( F, D ; s ) denotes the twist of the newform F by the quadratic character (cid:0) D • (cid:1) and ω ( N ) denotes the number of distinct prime divisors of N . Since the twisted L -series has a functional equation of the usual type, there are efficientmethods to compute its values numerically. (The authors used the built-in intrinsics of Magma [13].) Computing the Petersson norm of F is also possible to high accuracy, e.g. byusing the well-known relationship (cf. [30, 98]) h F, F i = vol( E )4 π deg( ϕ E ) , where F is the newform associated to the elliptic curve E/ Q , we denote the covolume of theperiod lattice of E by vol( E ) , and use ϕ E for the modular parametrization of E . (Everyelliptic curve E we consider in this paper has deg( ϕ E ) = 1 .) Alternatively, the Peterssonnorm of F is computed for N prime by Theorem 2 in [98]. Remark. Kohnen’s result Proposition 4.4 has been extended to many situations, e.g. by Uedaand his collaborators [91, 92] to certain even levels and forms not in the plus-space (see inparticular Corollary 1 in [69]). So the above reasoning carries over to o ( g ) ∈ { , } , bynoting that G (14) and G (28) both arise from the unique normalized cusp form in S (Γ (28)) (not in the plus space). The former may be obtained by applying sieve operators, the latterby applying the V -operator. Remark. For o ( g ) = 31 , the above reasoning only needs to be modified to take into accountthat G (31) is not a Hecke eigenform, but its decomposition into newforms is given in Appen-dix D. Using the fact that these newforms are orthogonal, the only difference becomes thatone needs to take into account two Poincaré series instead of one.Putting the estimates for the Rademacher sums R [ − , N , R [0] , N , and the occuring cusp formstogether and plugging them all into (4.1), one finds that the multiplicities are nonnegativeas soon as n ≥ (the worst case occurs for the character χ ). Inspecting the remainingcoefficients by computer then completes the proof of Theorem 4.1.5. Traces of Singular Moduli In this section, we discuss and recall some basic notation and facts about traces of sin-gular moduli. Their study originates in seminal work by Zagier [99], and has since been animportant subject in number theory (cf. for instance [8, 16, 19, 70], just to name a few).They appeared in connection with moonshine for the Thompson group in [58].5.1. Genus Zero Levels. It is well-known that for N ∈ { , , , , , , , , , , } themodular curve X ( N ) has genus , so that in those cases, there is a Hauptmodul J ( N ) .These Hauptmoduln are given explicitly in Table 5.1 in terms of the Dedekind eta function η ( τ ) := q Q n> (1 − q n ) and the Eisenstein series E ( τ ) := 1 + 240 P n> n q n (1 − q n ) − . N J ( N ) ( τ ) E ( τ ) η ( τ ) − η ( τ ) η (2 τ ) + 24 η ( τ ) η (3 τ ) + 12 η ( τ ) η (4 τ ) + 8 η ( τ ) η (5 τ ) + 6 η ( τ ) η (3 τ ) η (2 τ ) η (6 τ ) + 5 N J ( N ) ( τ ) η ( τ ) η (7 τ ) + 4 η ( τ ) η (4 τ ) η (2 τ ) η (8 τ ) + 4 η ( τ ) η (5 τ ) η (2 τ ) η (10 τ ) + 3 η ( τ ) η (4 τ ) η (6 τ ) η (2 τ ) η (3 τ ) η (12 τ ) + 3 η ( τ ) η (8 τ ) η (2 τ ) η (16 τ ) + 2 Table 5.1. Hauptmoduln for some Γ ( N ) To make use of these Hauptmoduln we require some notation. Denote by Q ( N ) − D the setof positive definite quadratic forms Q = ax + bxy + cy =: [ a, b, c ] of discriminant − D = b − ac < such that N | a . It is well-known that Γ ( N ) acts on Q ( N ) − D with finitely manyorbits, which correspond to the so-called Heegner points on the modular curve X ( N ) . For Q = [ a, b, c ] ∈ Q ( N ) − D , we denote by τ Q := − b + i √ D a the unique root of Q ( x, in H . For afunction f : H → C invariant under the action of Γ ( N ) we then define the trace function(5.1) Tr ( N ) D ( f ) := X Q ∈Q ( N ) − D / Γ ( N ) f ( τ Q ) ω ( N ) ( Q ) , ’NAN MOONSHINE AND ARITHMETIC 23 where ω ( N ) ( Q ) = · Γ ( N ) ( Q ) . Further let H ( N ) ( τ ) := − [SL ( Z ) : Γ ( N )]12 + X D> D ≡ , H ( N ) ( D ) q D denote the generating function of the (generalized) Hurwitz class numbers of level N whichare defined as H ( N ) ( D ) := Tr ( N ) D (1) . The special case of N = 1 yields the classical Hurwitzclass numbers H (1) ( D ) := H ( D ) .It is a straightforward consequence of Theorem 1.2 in [70], analogous to Theorem 1.2 in[8], that we can describe the Fourier coefficients of the Rademacher sums R [ − , + , N as tracesof the Hauptmoduln in Table 5.1. Proposition 5.1. Let N ∈ N such that X ( N ) has genus and (5.2) Tr ( N )4 ( D ) := 12 (cid:16) Tr ( N ) D ( J ( N )2 ) − Tr ( N/d ) D ( J ( N/d ) ) (cid:17) , where J ( N )2 = q − + O ( q ) is the unique modular function for Γ ( N ) with this Fourier expansionat infinity and no poles anywhere else and d := gcd( N, . Then we have (5.3) T ( N ) ( τ ) := − q − + X D> D ≡ , Tr ( N )4 ( D ) q D = − R [ − , + , N ( τ ) − c H ( N ) ( τ )+ c H ( N/d ) ( τ ) for certain rational numbers c and c . In particular, the function T ( N ) has integer Fouriercoefficients.Remark. It should be pointed out that Theorem 1.2 in [70] is only stated for odd levels,although the proof goes through for even levels as well. Remark. The rational numbers c and c in Proposition 5.1 are the constant terms of theweight Rademacher sums R [ − ,N and R [ − ,N/d , respecitvely. For a proof of the rationality ofthese numbers see Lemma 3.2 in [8].For the Rademacher sum R [0] , + , N we get the following. Proposition 5.2. For N ∈ N we have R [0] , + , N = − ϕ ( N ) X d | N d [SL ( Z ) : Γ ( d )] µ (cid:18) Nd (cid:19) H ( d ) where µ and ϕ denote the Möbius function and Euler’s totient function, respectively.Proof. This follows from a straightforward modification of the proof of Theorem 1.2 in [70]. (cid:3) Note that the above Proposition 5.2 is indeed valid for all N , not just those such that X ( N ) has genus .Putting Propositions 5.1 and 5.2 together we obtain explicit descriptions of the functions F [ g ] in terms of singular moduli for o ( g ) ∈ { , , , , , , , , , } . These are given inAppendix D. Positive Genus Levels. In the remaining cases, i.e. where o ( g ) ∈ { , , , , , , , } , our F [ g ] involve (cf. the proof of Theorem 3.1) Rademacher sums R [ − , + , N where N is suchthat X ( N ) has positive genus. So there is no notion of a Hauptmodul there. However, it isknown that for all these levels, the modular curve X +0 ( N ) , being the quotient of X ( N ) by allAtkin–Lehner involutions, does have genus (see e.g. [43]). So there exists a Hauptmodul J ( N, +) ( τ ) for the corresponding group Γ +0 ( N ) . See Table 5.2 for these. There, E ( τ ) :=1 − P n> nq n (1 − q n ) − is the quasimodular Eisenstein series, f = q − q + O ( q ) denotes the weight newform associated to the elliptic curve E : y + y = x + x − x − ([68, Elliptic Curve 19.a2]), and f = q + √ q + O ( q ) denotes the unique newform in S (Γ (31)) up to Galois conjugation (which is denoted by an exponent σ ). N 11 14 J ( N, +) ( τ ) − E ( τ ) − E (11 τ )10 η ( τ ) η (11 τ ) − − E ( τ )+2 E (2 τ ) − E (7 τ ) − E (14 τ )18 η ( τ ) η (2 τ ) η (7 τ ) η (14 τ ) − N 15 16 J ( N, +) ( τ ) − E ( τ )+3 E (3 τ ) − E (5 τ ) − E (15 τ )16 η ( τ ) η (3 τ ) η (5 τ ) η (15 τ ) − η (2 τ ) η (8 τ ) η ( τ ) η (4 τ ) η (16 τ ) − N 19 20 28 J ( N, +) ( τ ) − E ( τ ) − E (19 τ )18 f ( τ ) − η (2 τ ) η (10 τ ) η ( τ ) η (4 τ ) η (5 τ ) η (20 τ ) − η (2 τ ) η (14 τ ) η ( τ ) η (4 τ ) η (7 τ ) η (28 τ ) − N 31 32 J ( N, +) ( τ ) √ f ( τ )+ f σ ( τ ))2( f ( τ ) − f σ ( τ )) − η (2 τ ) η (16 τ ) η ( τ ) η (4 τ ) η (8 τ ) η (32 τ ) − Table 5.2. Hauptmoduln for some Γ +0 ( N ) Armed with these Hauptmoduln we can now express the Fourier coefficients of all theremaining Rademacher sums R [ − , + , N in terms of singular moduli of holomorphic modularfunctions and class numbers. Proposition 5.3. Let N ∈ N such that X +0 ( N ) has genus and define (5.4) Tr ( N, +)4 ( D ) := 12 (cid:18) ω ( N ) Tr ( N ) D (cid:16) J ( N, +)2 (cid:17) − ω ( N/d ) Tr ( N/d ) D (cid:0) J ( N/d, +) (cid:1)(cid:19) ’NAN MOONSHINE AND ARITHMETIC 25 where J ( N, +)2 = q − + O ( q ) is the unique modular function for Γ +0 ( N ) with this Fourierexpansion at infinity and no poles anywhere else and d := gcd( N, . Then we have (5.5) T ( N, +) ( τ ) := − q − + X D> D ≡ , Tr ( N, +)4 ( D ) q D = − R [ − , + , N ( τ ) − c H ( N ) ( τ ) + c H ( N/d ) ( τ ) for some rational numbers c and c , where ω ( N ) denotes the number of distinct primefactors of N .Proof. Proposition 5.1 turns out to be valid for all N , if one replaces the Hauptmodul J ( N ) by the completed Rademacher sum [ R [ − ,N , normalized so that its constant term is and J ( N )2 by [ R [ − ,N with the same normalization, which is the original formulation in [70]. Note thatthese Rademacher sums coincide with the Hauptmoduln where applicable. Now we considerfor N ′ || N , i.e. gcd( N ′ , N/N ′ ) = 1 , the Atkin–Lehner involution W N ′ . These involutionsmap the set Q ( N ) − D / Γ ( N ) bijectively to itself (see e.g. Section 1 of [56]), so for any Γ ( N ) -invariant function f we have that Tr ( N ) D ( f ) = Tr ( N ) D ( f | W N ′ ) . Since there are exactly ω ( N ) Atkin–Lehner involutions of level N this means that Tr ( N ) D ( f ) = 12 ω ( N ) Tr ( N ) D ( ˜ f ) where ˜ f := P N ′ || N f | W N ′ . The function ˜ f is clearly Γ +0 ( N ) -invariant. By checking the polarparts we conclude that if f is [ R [ − ,N or [ R [ − ,N then ˜ f has to coincide with J ( N, +) or J ( N, +)2 ,respectively, up to a rational additive constant. This proves the result. (cid:3) We now put Propositions 5.2 and 5.3 together to obtain explicit descriptions of the func-tions F [ g ] in terms of singular moduli for o ( g ) ∈ { , , , , , } . For o ( g ) = 16 weuse Proposition 5.1 as well since F [ g ] involves Rademacher sums R [ µ ] , + , N for both N = 16 and N = 32 in this case. The resulting expressions are given in Appendix D.6. Number Theoretic Applications In this section we prove the arithmetic applications of O’Nan moonshine given in Theo-rems 1.2 to 1.4. All these proofs rely on the following easy observation. Lemma 6.1. Let N > be an integers and − D < a discriminant which is not a squarein Z /N Z . Then the set Q ( N ) − D is empty. In particular, we have that Tr ( N ) − D ( f ) = H ( N ) ( D ) = 0 for any Γ ( N ) -invariant function f .Proof. A quadratic form [ a, b, c ] ∈ Q ( N ) − D satisfies − D = b − ac and N | a , hence if − D is nota square modulo N , there cannot be any such forms. (cid:3) Proof of Theorem 1.2. Suppose first that p ∈ { , } and let D be as in Theorem 1.2.The congruences in Appendix C together with the identities in Appendix D imply the con-gruence dim( W D ) ≡ tr( g p | W D ) ≡ Tr ( p )4 ( D ) − H ( D ) + α p H ( p ) ( D ) (mod p ) for some integer α p . By Lemma 6.1, the terms Tr ( p )4 ( D ) and H ( p ) ( D ) vanish for D as required,proving the result. For p = 3 , one replaces the modulus above by , making the congruencenon-trivial.For p = 2 , we note that there is a congruence between dim W D and tr( g | W D ) modulo by Appendix C. As one easily sees through a Sturm bound argument, we also have dim W D ≡ tr( g | W D ) ≡ for D ≡ , , which is the case in particular when − D < is an even fundamentaldiscriminant. The fact that for these D the class number is even can be seen in variousways, for example by noting that by a famous theorem of Gauss and Hermite we have that H ( D ) = 2 r ( D/ , where r ( n ) is the number of representations of n as the sum of threesquares. Since − D is fundamental, it follows that D/ is square-free and hence is not thesum of three or just two equal squares. Through an easy case-by-case analysis one then findsthat r ( D/ is always divisible by . Alternatively, one could also show the modular formscongruence X n ≡ , r ( n ) q n ≡ ∞ X n =0 q (2 n +1) + 4 ∞ X n =0 q n +1) (mod 8) . This completes the proof.6.2. Preliminaries on Elliptic Curves. The proofs of Theorems 1.3 and 1.4 require alittle preparation which we provide in this section.One of the most important open problems in the theory of elliptic curves is the Birch andSwinnerton-Dyer Conjecture. Conjecture 6.2. Let E/ Q be an elliptic curve. Then we have that (6.1) L ( r ) ( E, r !Ω E = X ( E ) · Reg( E ) Q ℓ c ℓ ( E )( E ( Q ) tors ) , where r denotes the order of vanishing of L ( E, s ) at s = 1 , which equals the Mordell–Weil rank of E , Ω E is the real period of E , X ( E ) and Reg( E ) denote the order of theTate-Shafarevich group and the regulator of E , respectively, the c ℓ ( E ) for prime ℓ are theTamagawa numbers of E , and E ( Q ) tors signifies the order of the torsion subgroup of the Q -rational points of E . The weak Birch and Swinnerton-Dyer conjecture—that the order of vanishing of L ( E, s ) at s = 1 equals the rank of E —was established for curves of ranks and through workof Gross–Zagier [57] and Kolyvagin [67]. More recently, Bhargava–Shankar [9] proved, usingKolyvagin’s theorem and the proof of the Iwasawa main conjectures for GL by Skinner–Urban [86] (among other deep results), that a positive proportion of all elliptic curves satisfythe weak Birch and Swinnerton-Dyer Conjecture.It is known that the left-hand side of (6.1) is always a rational number, see for instance[3, Theorem 3.2]. The following result shows that in certain situations, a local version ofConjecture 6.2, which is going to be sufficient for our purposes, holds. ’NAN MOONSHINE AND ARITHMETIC 27 Theorem 6.3 ([85], Theorem C) . Let E/ Q be an elliptic curve and p ≥ a prime of goodordinary or multiplicative reduction. Further assume that the Gal( Q / Q ) -representation E [ p ] is irreducible and that there exists a prime p ′ = p at which E has multiplicative reductionand E [ p ] ramifies. If L ( E, = 0 , then we have that ord p (cid:18) L ( E, E (cid:19) = ord p X ( E ) Y ℓ c ℓ ( E ) ! . If L ( E, 1) = 0 , then we have Sel( E )[ p ] = { } . We are especially interested in quadratic twists of elliptic curves. In this context, thefollowing result by Agashe, giving the real period of such a twist, turns out to be very useful. Lemma 6.4 ([4], Lemma 2.1) . Let E/ Q be an elliptic curve of conductor N and let − D < be a fundamental discriminant coprime to N . Then we have that Ω E ( − D ) = c E · c ∞ ( E ( − D )) · ω − ( E ) / √ D, where c E denotes the Manin constant of E , c ∞ ( E ( − D )) denotes the number of componentsof E ( − D ) over R , and ω − ( E ) denotes the second period of the period lattice of E .Remark. The famous Manin Conjecture states that c E = 1 .Combining this with a theorem of Kohnen [66] (cf. Proposition 4.4), we obtain the follow-ing. Lemma 6.5. Let E/ Q be an elliptic curve of odd, square-free conductor N and let − D < be a fundamental discriminant satisfying (cid:0) − Dℓ (cid:1) = w ℓ , where w ℓ denotes the eigenvalue of thenewform F E ∈ S (Γ ( N )) associated to E and the Atkin–Lehner involution W ℓ , ℓ | N . Denoteby D the smallest such discriminant. Further let f E ( τ ) = P ∞ n =3 b E ( n ) q n ∈ S + (Γ (4 N )) bethe weight / cusp form associated to F E under the Shintani lift. For p ≥ prime we thenhave that ord p (cid:18) L ( E ( − D ) , E ( − D ) (cid:19) = ord p (cid:18) L ( E ( − D ) , E ( − D ) (cid:19) + ord p (cid:0) b E ( | D | ) (cid:1) . Proof. By combining Proposition 4.4 and Lemma 6.4, we find for the fundamental discrimi-nants − D < as in the lemma that(6.2) L ( E ( − D ) , E ( − D ) = π h F, F i c E · c ∞ ( E ( − D ))2 ω ( N ) h f, f i ω − ( E ) · | b E ( D ) | . We see that the only quantities in this formula depending on D are c ∞ ( E ( − D )) and b E ( D ) .Since the former is always either or and p is odd, it doesn’t affect the p -adic valuationat all, which proves the lemma. (cid:3) Remark. If the conductor N is even but still square-free, the same result still holds alongthe same lines, using the remark following Proposition 4.4. The exact formula in this caseonly differs from (6.2) by a power of , which doesn’t affect the p -adic valuation. Proofs. In this section, we prove Theorems 1.3 and 1.4. The proofs of both theoremsare very similar in their main steps, so we combine them here. Proof of Theorems 1.3 and 1.4. By applying the expressions for the relevant F [ g ] in terms ofthe traces of singular moduli, class numbers and weight / cusp forms in Appendix D, andusing the congruences in Appendix C, we find that dim( W D ) ≡ tr( g | W D ) ≡ Tr (11)4 ( D ) − H ( D ) + α H (11) ( D ) + γ b ( D ) (mod 11) , tr( g | W D ) ≡ tr( g | W D ) ≡ Tr (14)4 ( D ) + δ ( H ( D ) − δ H (2) ( D ))+ α H (7) ( D ) + β H (14) ( D ) + γ b ( D ) (mod 7) , tr( g | W D ) ≡ tr( g | W D ) ≡ Tr (15)4 ( D ) + δ ( H ( D ) − δ H (3) ( D ))+ α H (5) ( D ) + β H (15) ( D ) + γ b ( D ) (mod 5) , dim( W D ) ≡ tr( g | W D ) ≡ Tr (19)4 ( D ) − H ( D ) + α H (19) ( D ) + γ b ( D ) (mod 19) , where δ p = p − , α p , β p are some integers, γ p are p -adic units, and b N ( D ) denotes the D th coefficient of the weight / cusp form G ( N ) specified in Appendix D. If − D is a fundamentaldiscriminant as specified in Theorems 1.3 and 1.4 respectively, then by Lemma 6.1, the terms Tr ( N )4 ( D ) as well as H ( p ) ( D ) and H ( N ) ( D ) above disappear. This shows that the class numbercongruences in our theorems hold if and only if p divides the coefficient b N ( D ) , i.e. if andonly if ord p (cid:16) L ( E N ( − D ) , EN ( − D ) (cid:17) > by Lemma 6.5. (A Magma computation reveals that the ratio L ( E N ( − D ) , EN ( − D for the smallest possible D is in each case a p -adic unit.)Suppose for simplicity that L ( E N ( − D ) , = 0 . According to the Birch and Swinnerton-Dyer Conjecture 6.2, this implies that ord p ( X ( E N ( − D )) Y ℓ c ℓ ( E ( − D ))) > , so our theorems follow, conditionally on Conjecture 6.2, if the Tamagawa numbers c ℓ ( E ( − D )) are never divisible by p in our cases. To establish this, we note (cf. [84, Appendix C, Table15.1]) that for an elliptic curve E/ Q we have that p | c ℓ ( E ) if and only if the reductiontype of E at ℓ is I n with p | n , which means that ord ℓ (∆( E )) = n , where ∆( E ) denotesthe (minimal) discriminant of E . An inspection of Tate’s algorithm for the computationof Tamagawa numbers and the well-known formulas for minimal discriminants from theKraus–Laska algorithm reveals that in our case, because we are considering twists of ellipticcurves by fundamental discriminants, all the Tamagawa numbers must be in { , , , } .The argumentation in the case L ( E ( − D ) , 1) = 0 is similar. This completes the proof ofTheorem 1.3 for N ∈ { , } .The truth of Theorem 1.4 does not depend on the Birch and Swinnerton-Dyer Conjecture,but rather on Skinner’s Theorem 6.3. A lemma of Serre [82, §2.8, Corollaire, p. 284]shows that the Galois representations E ( − D )[7] and E ( − D )[5] are surjective and henceirreducible. Furthermore, it is immediate to check that E ( − D ) (resp. E ( − D ) ) hasmultiplicative reduction modulo (resp. ) and that E ( − D )[7] (resp. E ( − D )[5] ) ramifiesthere, so the conditions of Theorem 6.3 are satisfied, completing the proof of Theorem 1.4. (cid:3) ’NAN MOONSHINE AND ARITHMETIC 29 Examples Here we offer some numerical examples which illustrate the congruences described in theintroduction.7.1. Class Number Congruences. Here we present some class number congruences thatarise from Theorem 1.2. Recall that this theorem offers congruences modulo 16, 9, 5, and7 for certain fundamental discriminants − D < which satisfy given congruence conditions.The three columns in Tables 7.1 to 7.4 are congruent, which illustrates the theorem. D dim W D tr( g | W D ) − H ( D ) ≡ ≡ − ≡ ... ≡ − ≡ − ≡ ... ≡ − ≡ − ≡ Table 7.1. p = 2 D dim W D tr( g | W D ) − H ( D ) ≡ ≡ − ≡ ≡ ≡ − ≡ ≡ ≡ − ≡ ≡ ≡ − ≡ Table 7.2. p = 3 D dim W D tr( g | W D ) − H ( D ) ≡ ≡ − ≡ ≡ ≡ − ≡ ≡ ≡ − ≡ ≡ ≡ − ≡ Table 7.3. p = 5 D dim W D tr( g | W D ) − H ( D ) ≡ ≡ − ≡ ≡ ≡ − ≡ ≡ ≡ − ≡ ≡ ≡ − ≡ ≡ ≡ − ≡ Table 7.4. p = 7 Selmer and Tate–Shafarevich Group Congruences. Theorems 1.3 and 1.4 offercriteria for detecting elements in p -Selmer groups and Tate–Shafarevich groups of quadratictwists of certain elliptic curves. Theorem 1.3 assumes the truth of the Birch and Swinnerton-Dyer Conjecture. Theorem 1.4 is unconditional thanks to results of Skinner–Urban.Here we offer data related to the curves E and E . In the notation of Theorem 1.4,we consider fundamental discriminants − D such that (cid:16) − Dp (cid:17) = − and (cid:16) − Dp ′ (cid:17) = 1 . Forconvenience let H ( D ) := δ ( H ( D ) − δ H (2) ( D )) , H ( D ) := δ ( H ( D ) − δ H (3) ( D )) , tr ( D ) := tr( g | W D ) , tr ( D ) := tr( g | W D ) , Diff ( D ) := H ( D ) − tr ( D ) , Diff ( D ) := H ( D ) − tr ( D ) . We have the following numerics. In Tables 7.5 and 7.6, the second and third columns offergraded traces and differences of class numbers. The fourth and fifth columns offer Mordell–Weil ranks and orders of Tate–Shafarevich groups assuming the Birch and Swinnerton-DyerConjecture. By Theorem 1.4, these columns are congruent if and only the corresponding p -Selmer group is nontrivial. First note that if these two columns are incongruent, then boththe Mordell–Weil rank over Q and the p -part of the Tate–Shafarevich groups are trivial.However, when these columns are congruent, notice that either the rank is positive or theTate–Shafarevich group is nontrivial at p . D tr ( D ) H ( D ) Diff ( D ) (mod 7) rk( E ( − D )) X an ( E ( − D )) 15 -96256 -30 3 0 123 -1746944 -45 0 2 139 -165767168 -60 4 0 171 -156822906880 -105 4 0 179 -669595144192 -75 3 0 1239 -6190369...040 -225 0 2 12671 -1630362...664 -345 0 0 49 Table 7.5. Examples for the curve E ’NAN MOONSHINE AND ARITHMETIC 31 D tr ( D ) H ( D ) Diff ( D ) (mod 5) rk( E ( − D )) X an ( E ( − D )) Table 7.6. Examples for the curve E The authors thank Drew Sutherland for computing the elliptic curve invariants in Ta-bles 7.5 and 7.6. Appendix A. The Character Table of O’N Here we give the character table of the O’Nan group O’N over the complex numbers. For n ∈ N we let ζ n := e πin and define A := 1 + 3 √ , B := √ ,C := − ζ − ζ − ζ − ζ − ζ − ζ ,D := − ζ − ζ − ζ − ζ − ζ − ζ ,E := − ζ − ζ − ζ − ζ − ζ − ζ ,F := i √ , G := √ , H := − i √ . We use A , B , &c. to denote images under the obvious Galois involutions. Note that C, D, and E are in one Galois orbit as well, since ( X − C )( X − D )( X − E ) = X − X − X + 7 . The character table is reproduced from Gap4 [50]. J O HN F . R . D UN C AN , M I C HA E L H . M E R T E N S , AN D K E N O N O A A A A B A A A B A B A A A A A B A B C D A B C A B A B A Bχ χ χ H Hχ H Hχ F − F -1 -1 0 0 χ − F F -1 -1 0 0 χ χ χ χ χ χ χ χ χ χ B − B − B B χ − B B B − B χ χ χ χ B − B B − B χ − B B − B B χ A A χ A A χ χ C E D χ D C E χ E D C χ G − G -1 -1 χ − G G -1 -1 Table A.1. Character table of O’N ’NAN MOONSHINE AND ARITHMETIC 33 Appendix B. Multiplicities of Irreducible Representations in W We denote by V j the O’N -module corresponding to the irreducible χ j in Table A.1.The following table gives the multiplicities of V j in the (virtual) modules W m in Theo-rem 4.1. Negative multiplicities are printed in bold. m V V V V V V V V V V -2 -2 -1 33 44 44 76 76 88 98 98 12215 0 406 581 581 1061 1061 1010 1252 1252 156816 -2 978 1193 1193 2316 2316 2386 2892 2892 336219 2 9484 11205 11205 21948 21948 23114 27766 27766 3189420 5 18951 23161 23161 44930 44930 46322 56156 56156 6527123 2 144238 177831 177831 343685 343685 352892 428308 428308 49990024 25 277191 338794 338794 656282 656282 677588 820362 820362 95478327 212 1795740 2189365 2189365 4245047 4245047 4388491 5310882 5310882 617447028 292 3264537 3989983 3989983 7730566 7730566 7979966 9663217 9663217 1124451031 1562 18513448 22644956 22644956 43863830 43863830 45258570 54815104 54815104 6380336032 2960 32416998 39620773 39620773 76765848 76765848 79241546 95957290 95957290 11165853435 15432 165271652 201946677 201946677 391304807 391304807 403986962 489174874 489174874 56916500636 25645 279985728 342204752 342204752 663020690 663020690 684409504 828775828 828775828 964395212 Table B.1. Multiplicities, part I. m V V V V V V V V V V Table B.2. Multiplicities, part II. m V V V V V V V V V V Table B.3. Multiplicities, part III. Appendix C. Congruences p = 31 : ≡ F A − F AB (mod 31) p = 19 : ≡ F A − F ABC (mod 19) p = 11 : ≡ F A − F A (mod 11) p = 7 : ≡ F A − F AB (mod 7 ) ≡ F A − F A (mod 7) ≡ F AB − F AB (mod 7) p = 5 : ≡ F A − F A (mod 5 ) ≡ F A − F A (mod 5) ≡ F A − F AB (mod 5) ≡ F AB − F AB (mod 5) ’NAN MOONSHINE AND ARITHMETIC 35 p = 3 : ≡ F A − F A (mod 3 ) ≡ F A − F A (mod 3 ) ≡ F AB − F A (mod 3 ) ≡ F A − F AB (mod 3 ) p = 2 : ≡ F A + 303 F A + 3024 F AB + 4864 F AB + 57344 F ABCD (mod 2 ) ≡ F A + 7 F AB + 8 F AB + 112 F ABCD (mod 2 ) ≡ F A + F A + 6 F A (mod 2 ) ≡ F AB + F AB + 14 F ABCD (mod 2 ) ≡ F A + F A + 6 F AB (mod 2 ) ≡ F A + F A (mod 2) ≡ F AB + F AB (mod 2 ) ≡ F AB + 7 F ABCD (mod 2 ) ≡ F A + F AB (mod 2) ≡ F A + F AB (mod 2) Appendix D. Traces of Singular Moduli We give the explicit descriptions of F [ g ] in terms of traces of singular moduli and classnumbers as described in Section 5. F A = T (1) ,F A = T (2) + 12 H (1) − H (2) ,F A = T (3) + 12 H (1) − H (3) ,F AB = T (4) + 12 H (2) − H (4) ,F A = T (5) + 6 H (1) − H (5) ,F A = T (6) − H (1) + 8 H (2) + 212 H (3) − H (6) ,F AB = T (7) + 4 H (1) − H (7) ,F AB = T (8) + 4 H (4) − H (8) ,F A = T (10) − H (1) + 4 H (2) + 112 H (5) − H (10) ,F A = T (11 , +) + 125 H (1) − H (11) − G (11) ,F A = T (12) − H (2) + 4 H (4) + 52 H (6) − H (12) ,F A = T (14 , +) − H (1) + 83 H (2) + 154 H (7) − H (14) + 83 G (14) ,F AB = T (15 , +) − H (1) + 94 H (3) + 52 H (5) − H (15) + 94 G (15) ,F ABCD = 2 T (32 , +) − T (16) − H (8) + 4 H (16) − H (32) ,F ABC = T (19 , +) + 43 H (1) − H (19) + 43 G (19) ,F AB = T (20 , +) − H (2) + 2 H (4) + 32 H (10) − H (20) ,F AB = T (28 , +) − H (2) + 43 H (4) + 2524 H (14) − H (28) + 83 G (28) ,F AB = T (31 , +) + 45 H (1) − H (31) + 35 G (31) . 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