aa r X i v : . [ m a t h . N T ] S e p A supersingular coincidence
G.K. SankaranSeptember 25, 2020
The list of fifteen primes S = { , , , , , , , , , , , , , , } known as the supersingular primes ( https://oeis.org/A002267 ) appearsin several different contexts. Here are five of them.1. p ∈ S if and only if p divides the order of the Monster sporadic simplegroup.2. p ∈ S if and only if g ( X ( p ) + ) = 0, where X ( p ) + = H / Γ ( p ) + is themodular curve associated with the group Γ ( p ) + < GL(2 , Q )3. p ∈ S if and only if the supersingular values of the j -invariant all liein F p .4. p ∈ S if and only if the space J cusp2 ,p of Jacobi cusp forms of weight 2and index p is of dimension 0.5. If p
6∈ S then the moduli space A p of complex abelian surfaces with apolarisation of type (1 , p ) is of general type.Of these, (1)–(3) are described in [Ogg], where the equivalence of theconditions in (2) and (3) is proved. In [Ogg] a prize (a bottle of Jack Daniels)is offered for an explanation of why the condition in (1) is equivalent to thosein (2) and (3): it is still unclaimed.This note is primarily about (5). The proof that A p is of general typefor p
6∈ S is due to Erdenberger [Er], and specialists in moduli of abeliansurfaces are occasionally asked to explain the apparent coincidence [HMc].In fact the answer consists of a series of well-known facts, but because theyare not all well known to the same people, the question continues to recur.The purpose of this note is to set the answer out clearly.1
Moduli of abelian surfaces
An abelian surface equipped with a polarisation of type (1 , d ) (for d ∈ N )may be thought of as a complex torus C / Λ, where Λ ⊂ C is the subgroup(lattice) generated by the columns of Ω = ( I d , τ ) for I d := (cid:18) d (cid:19) , τ = (cid:18) τ τ τ τ (cid:19) ∈ H = { τ = t τ ∈ M × ( C ) | Im τ > } . The paramodular group Γ d = (cid:26) γ ∈ GL(4 , Q ) | t γ (cid:18) I d − I d (cid:19) γ = (cid:18) I d − I d (cid:19)(cid:27) acts on the Siegel upper half-plane H by fractional linear transformations (cid:18) A BC D (cid:19) : τ −→ ( Aτ + B )( Cτ + D ) − . This group action is properly discontinuous and the quotient A d := H / Γ d is a coarse moduli space for (1 , d )-polarised abelian varieties. It is a quasi-projective variety, and one may ask for its Kodaira dimension, or more pre-cisely, for the Kodaira dimension κ ( Y d ) of a desingularisation Y d of a projec-tive compactification A d . For more on this and related spaces, see [HKW].In practice one expects that A d is of general type, i.e. κ ( Y d ) = 3, exceptfor some small values of d . Very loosely, this is because k -fold differentialforms on Y d correspond to suitable modular forms of weight 3 k for Γ d , andthese become abundant as d grows at least for k sufficiently divisible. How-ever, not every modular form of weight 3 k will do: obstructions come fromthe boundary A d \ A d and from the branching of H → A d .The obstructions at the boundary may be overcome by using the low-weight cusp form trick [Gr2]: if we can find a cusp form f of weight 2 forΓ d then we may consider modular forms f of weight 3 k of the form f = f k f k ,where f k is a modular form of weight k . These are also abundant, if d and k are large enough, and because they vanish to high order at the boundary,the associated differential forms extend.The branching behaviour has to be analysed separately, and it dependson the factorisation of d . For that reason much work in this direction hasconcentrated, for simplicity, on the case d = p prime. The case d = p hassome simplifying features and was treated in [OG] and [GS].By this method it was shown in [Sa] that A p is of general type for p > p came from the branching, so all that was necessary was to verify thata weight 2 cusp form exists for all p > p according to Gritsenko2[Gr2, Theorem 3], [Gr1]). The dimension of the space of Jacobi cusp formsis computed in [EZ, SZ] and in this case it takes the form [Gr2, Sa]dim J cusp2 ,p = p X j =1 (cid:22) j (cid:23) − δ ( j ) − (cid:22) j p (cid:23) where δ ( j ) = 1 if 6 | j and 0 otherwise. This is positive for all p > p >
173 to p ≥
37, so thatthe existence of the Jacobi form becomes the effective constraint. Then it iseasy to compute from the formula above that p ∈ S exactly when no Jacobicusp form of weight 2 and index p exists, i.e. when the condition in (4) holds.It is not necessarily to be expected that A p is of general type exactlywhen p
6∈ S . The method of proof of [Er] fails for p ∈ S , as we shall see, andit is known that A d is unirational (so in particular not of general type) forsome small values of d , including all primes p ≤
11: see [GP]. However, if p ≥
13, nothing currently excludes the possibility that A p is of general type.On the other hand, A p being unirational, other than in the known cases p ≤ is excluded. Gritsenko [Gr1] showed that A d has non-negative Ko-daira dimension, so is not uniruled, for all d ≥
13, prime or not, except pos-sibly for d = 14 , , , , , , ,
36. Of these, the cases d = 14 , , , A d are in fact unirational) and onlyfor d = 15 , , ,
36 is nothing known about the Kodaira dimension of A d . Since we have now established a connection between (4) and (5), to achievea moderately satisfactory explanation of the apparently coincidental appear-ance of S in (5) we should show, without direct computation, that the con-ditions in (2) and (4) are equivalent. (A fully satisfactory explanation wouldalso involve (1): this we are not able to give.) This is well known amongspecialists in Jacobi forms, and follows easily from a small part of [SZ].It is shown in [SZ] that the space J k,d of Jacobi forms of weight k and in-dex d is isomorphic (even as a Hecke module) to a certain subspace M − k − ( d )of the space M k − ( d ) of modular forms of weight 2 k − ( d ). Thissubspace is defined by M − k − ( d ) = M − k − ( d ) ∩ M k − ( d ), where M − k − ( d )is the space of weight 2 k − ( d ) that satisfy an extracondition on the behaviour under the Fricke involution w : τ − dτ , namely f ( − dτ ) = ( − k d k − τ k − f ( τ ) . In our case ( k = 2 and d = p ) this is equivalent to saying that f is a modularform of weight 2 for the group Γ ( p ) + < GL(2 , Q ) generated by Γ ( p ) and3 = (cid:18) − d (cid:19) . See, for example, the definition of automorphic form in [Sh,Chapter 2]. So if there is a weight 2, index p Jacobi form, then Γ ( p ) + hasa weight 2 modular form.Conversely, inspecting the definition of M in [SZ] we find that thereare no other conditions for p prime: simply M k − ( p ) = M k − ( p ). Thiscan be seen at once, for instance, from [SZ, Equation (4), p. 116], since M (1) ⊂ M (1) = 0. In other words, the space of Jacobi forms in this caseis isomorphic exactly to the space of weight 2 modular forms for Γ ( p ) + .Moreover, the isomorphism respects cusp forms: see [SZ, Theorem 5].We remark that for k = 2 and p square-free (in particular for p prime)there are no Eisenstein series, so the condition (3) is equivalent to the samestatement but with J cusp2 ,p replaced by J ,p .However, Ogg [Ogg] shows that the modular curve X ( p ) + correspondingto Γ ( p ) + is of genus 0 precisely for p ∈ S , i.e. he shows (2). One can computethe dimension of the space of weight 2 forms from the formulae given in [Sh,Theorem 2.23]: it is g + m −
1, where g is the genus of X ( p ) + and m isthe number of cusps. Because p is prime, the curve X ( p ) has two cusps,which are interchanged by the Fricke involution; so m = 1, and so the spaceof modular forms for Γ ( p ) + has dimension g (i.e. they are all cusp forms,as one should also expect from the remark above). So for p ∈ S , there canbe no weight 2, index p Jacobi forms; so we definitely cannot prove that A p is of general type result by the methods of [Sa] and [Er] for any p ∈ S . References [Er] C. Erdenberger, The Kodaira dimension of certain moduli spaces ofAbelian surfaces.
Math. Nachr. , 32–39 (2004).[Gr1] V. Gritsenko, Irrationality of the moduli spaces of polarized Abeliansurfaces.
Int. Math. Research Notices , 235–243 (1994).[Gr2] V. Gritsenko, Irrationality of the moduli spaces of polarized abeliansurfaces. Abelian varieties (Egloffstein, 1993) , 63–84, de Gruyter,Berlin, 1995.[GS] V. Gritsenko & G.K. Sankaran, Moduli of Abelian surfaces witha (1 , p ) polarisation. Izv. Ross. Akad. Nauk. Ser. Mat. , 19–26(1997).[GP] M. Gross & S. Popescu, The moduli space of (1,11)-polarized abeliansurfaces is unirational. Compositio Math. (2001), 1–23.[HMc] Y.-H. He & J. McKay, Sporadic and exceptional. Preprint arXiv:1505.06742 , 2015. 4HKW] K. Hulek, C. Kahn & S. Weintraub, Moduli Spaces of Abelian Sur-faces: Compactification, Degenerations, and Theta Functions. Ex-positions in Mathematics (de Gruyter, Berlin, 1993).[Ogg] A. Ogg, Automorphismes de courbes modulaires. S´eminaireDelange-Pisot Poitou (16e ann´ee: 1974/75) , Th´eorie des nombres,Fasc. , Exp. No. 7, 8 pp. Secr´etariat Math´ematique, Paris, 1975.[OG] K. O’Grady, On the Kodaira dimension of moduli spaces of Abeliansurfaces. Compositio Math. , 121–163 (1989).[Sa] G.K. Sankaran, Moduli of polarised Abelian surfaces. Math.Nachr. , 321–340 (1997).[EZ] M. Eichler & D. Zagier, The theory of Jacobi forms.
Progress inMathematics . Birkh¨auser Boston, Inc., Boston, MA, 1985.[Sh] G. Shimura, Introduction to the arithmetic theory of automor-phic functions. Kanˆo Memorial Lectures, No. 1.