Supersingular J-invariants Research Articles

Supersingular loci from traces of Hecke operators

A classical observation of Deligne shows that, for any prime p ≥ 5 p \geq 5 , the divisor polynomial of the Eisenstein series E p − 1 ( z ) E_{p-1}(z) mod p p is closely related to the supersingular polynomial at p p , S p ( x ) ≔ ∏ E / F ¯ p supersingular ( x − j ( E ) ) ∈ F p [ x ] . \begin{align*} S_p(x) ≔\prod _{E/\overline {\mathbb {F}}_p \text { supersingular}}(x-j(E))\,\, \in \mathbb {F}_p[x]. \end{align*} Deuring, Hasse, and Kaneko and Zagier [Supersingular j-invariants, hypergeometric series, and Atkin’s orthogonal polynomials, Providence, RI, 1998, pp. 97–126] found other families of modular forms which also give the supersingular polynomial at p p . In a new approach, we prove an analogue of Deligne’s result for the Hecke trace forms T k ( z ) T_k(z) defined by the Hecke action on the space of cusp forms S k S_k . We use the Eichler-Selberg trace formula to identify congruences between trace forms of different weights mod p p , and then relate their divisor polynomials to S p ( x ) S_p(x) using Deligne’s observation.

The number of linear factors of supersingular polynomials and sporadic simple groups

The set of prime numbers p such that the supersingular j-invariants in characteristic p are all contained in the prime field is finite. And it is well known that this set of primes coincides with the set of prime divisors of the order of the Monster simple group. In this paper, we will present analogous coincidences of supersingular invariants in level 2 and 3 and the orders of the Baby monster group and the Fischer's group. The proof uses a connection between the number of supersingular invariants and class numbers of imaginary quadratic fields.

Congruence formulas for Legendre modular polynomials

Let p≥5 be a prime number. We generalize the results of E. de Shalit [4] about supersingular j-invariants in characteristic p. We consider supersingular elliptic curves with a basis of 2-torsion over F‾p, or equivalently supersingular Legendre λ-invariants. Let Fp(X,Y)∈Z[X,Y] be the p-th modular polynomial for λ-invariants. A simple generalization of Kronecker's classical congruence shows that R(X):=Fp(X,Xp)p is in Z[X]. We give a formula for R(λ) if λ is supersingular. This formula is related to the Manin–Drinfeld pairing used in the p-adic uniformization of the modular curve X(Γ0(p)∩Γ(2)). This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if λ is supersingular and is in Fp, then we also express R(λ) in terms of a CM lift (which is shown to exist) of the Legendre elliptic curve associated to λ.

Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

For a prime number p, we study the zeros modulo p of divisor polynomials of rational elliptic curves E of conductor p. Ono (CBMS regional conference series in mathematics, 2003, vol 102, p. 118) made the observation that these zeros are often j-invariants of supersingular elliptic curves over {overline{{mathbb {F}}_{p}}}. We show that these supersingular zeros are in bijection with zeros modulo p of an associated quaternionic modular form v_E. This allows us to prove that if the root number of E is -1 then all supersingular j-invariants of elliptic curves defined over {mathbb {F}}_{p} are zeros of the corresponding divisor polynomial. If the root number is 1, we study the discrepancy between rank 0 and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in {mathbb {F}}_p seems to be larger. In order to partially explain this phenomenon, we conjecture that when E has positive rank the values of the coefficients of v_E corresponding to supersingular elliptic curves defined over {mathbb {F}}_p are even. We prove this conjecture in the case when the discriminant of E is positive, and obtain several other results that are of independent interest.

On the Newton polygons of Kaneko-Zagier lifts of supersingular polynomials

Let p be a prime number. The supersingular polynomial $$\mathfrak {S}_p(x)$$ S p ( x ) in characteristic p is the polynomial over $$\mathbf {F}_p$$ F p whose roots in $$\overline{\mathbf {F}_p}$$ F p ¯ are the supersingular j-invariants of elliptic curves over $$\mathbf {F}_p$$ F p . In this paper we study the Newton polygons of certain rational lifts of $$\mathfrak {S}_p(x)$$ S p ( x ) connected to the theory of Jacobi polynomials. As a corollary to our results on the Newton polygons, we obtain new cases of irreducibility for these supersingular lifts, providing more evidence for the general irreducibility conjecture of Mahlburg and Ono.

The Jack Daniels problem

In 1975 Ogg offered a bottle of Jack Daniels for an explanation of the fact that the prime divisors of the order of the monster M are the primes p for which the characteristic p supersingular j-invariants are all defined over Fp. This coincidence is often suggested as the first hint of monstrous moonshine, the deep unexpected interplay between the monster and modular functions. We revisit Ogg's problem, and we point out (using existing tools) that the moonshine functions for order p elements give the set of characteristic p supersingular j-invariants (apart from 0 and 1728). Furthermore, we discuss this coincidence of the two seemingly unrelated sets of primes using the first principles of moonshine.

ON p-ADIC PROPERTIES OF TWISTED TRACES OF SINGULAR MODULI

We prove that logarithmic derivatives of certain twisted Hilbert class polynomials are holomorphic modular forms modulo p of filtration p + 1. We derive p-adic information about twisted Hecke traces and Hilbert class polynomials. In this framework, we formulate a precise criterion for p-divisibility of class numbers of imaginary quadratic fields in terms of the existence of certain cusp forms modulo p. We explain the existence of infinite classes of congruent twisted Hecke traces with fixed discriminant in terms of the factorization of the associated Hilbert class polynomial modulo p. Finally, we provide a new proof of a theorem of Ogg classifying those p for which all supersingular j-invariants modulo p lie in Fp.

P-Adic liftings of the supersingular j-invariants and j-zeros of certain Eisenstein series

Let p > 3 be a prime. We consider j-zeros of Eisenstein series E k of weights k = p − 1 + M p a ( p 2 − 1 ) with M , a ⩾ 0 as elements of Q ¯ p . If M = 0 , the j-zeros of E p − 1 belong to Q p ( ζ p 2 − 1 ) by Hensel's lemma. Call these j-zeros p-adic liftings of supersingular j-invariants. We show that for every such lifting u there is a j-zero r of E k such that ord p ( r − u ) > a . Applications of this result are considered. The proof is based on the techniques of formal groups.

SUPERSINGULAR RANK TWO DRINFEL'D MODULES AND ANALOGS OF ATKIN'S ORTHOGONAL POLYNOMIALS

The theory of elliptic curves and modular forms provides a precise relationship between the supersingular j-invariants and the congruences between modular forms. Kaneko and Zagier discuss a surprising generalization of these results in their paper on Atkin orthogonal polynomials. In this paper, we define an analog of the Atkin orthogonal polynomials for rank two Drinfel'd modules.

Minimal resolution of Atkin–Lehner quotients of [formula omitted

Let X 0 ( N ) be the classic modular curve of level N over Z . Let W M be the Atkin–Lehner involution of X 0 ( N ) associated to a divisor M with ( M , N / M ) = 1 . In this paper an explicit description is given for the minimal resolution over Z [ 1 / 6 ] of the Atkin–Lehner quotient X 0 ( N ) / W M . As an application a new proof of Deuring's formula on the number of supersingular j-invariants in F p is given. In certain cases it is also shown that the action of Hecke operators on the component group of the Jacobian of the Atkin–Lehner quotient is Eisenstein.

NUMBER THEORETIC PROPERTIES OF WRONSKIANS OF ANDREWS–GORDON SERIES

For positive integers 1 ≤ i ≤ k, we consider the arithmetic properties of quotients of Wronskians in certain normalizations of the Andrews–Gordon q-series [Formula: see text] This study is motivated by their appearance in conformal field theory, where these series are essentially the irreducible characters of [Formula: see text] Virasoro minimal models. We determine the vanishing of such Wronskians, a result whose proof reveals many partition identities. For example, if Pb(a;n) denotes the number of partitions of n into parts which are not congruent to 0, ±a ( mod b), then for every positive integer n, we have [Formula: see text] We also show that these quotients classify supersingular elliptic curves in characteristic p. More precisely, if 2k + 1 = p, where p ≥ 5 is prime, and the quotient is non-zero, then it is essentially the locus of supersingular j-invariants in characteristic p.

Reduction of CM elliptic curves and modular function congruences

Here we investigate congruences for the coefficients of modular functions on SL2(ℤ). We obtain a simple criterion, in terms of the locus of supersingular j-invariants, which guarantees certain congruences. We implement this criterion in the case of modular functions obtained from Hilbert class polynomials.

WEIERSTRASS POINTS ON $X_0(\lowercase{p}M)$ AND SUPERSINGULAR $\lowercase{j}$-INVARIANTS

Weierstrass points are special points on a Riemann surface that carry a lot of information. Ogg studied such points on X0(pM) (for M such that X0(M) has genus zero and p prime with p∤ M), and he proved that if Q is a Q-rational Weierstrass point on X0(pM), then its reduction modulo p is supersingular. The paper shows that, for square-free M on the list, all supersingular j-invariants are obtained in this way. Furthermore, for most cases where M is prime, the explicit correspondence between Weierstrass points and supersingular j-invariants in characteristic p is described. Along the way, a useful formula of Rohrlich for computing a certain Wronskian of modular forms modulo p is generalized.

Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial

Using the theory of elliptic curves, we show that the class number h(− p) of the field Q ( −p ) appears in the count of certain factors of the Legendre polynomials P m(x) ( mod p) , where p is a prime >3 and m has the form ( p− e)/ k, with k=2,3 or 4 and p≡e ( mod k) . As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y 2+ αxy+ y= x 3 and find an elementary expression for the supersingular polynomial ss p ( x) whose roots are the supersingular j-invariants of elliptic curves in characteristic p. As a corollary we show that the class number h(− p) also shows up in the factorization ( mod p) of certain Jacobi polynomials.

A remark on Hecke operators and a theorem of Dwork and Koike

Let $p\geq 5$ be prime, $\mathfrak{S}_p$ the set of all characteristic $p$ supersingular j-invariants in $\mathbb{F}_p-\{0,1728\}$, and $\mathfrak{M}_p$ the set of all monic irreducible quadratic polynomials in $\mathbb{F}_p[x]$ whose roots are supersingular j-invariants. A theorem of Dwork and Koike asserts that there are integers $A_p(\alpha),B_p(g),C_p(g)$, and a polynomial $D_p(x)\in\mathbb{F}_p[x]$ of degree $p-1$, for which \begin{multline*} j(pz) \equiv j(z)^p + pD_p(j(z)) + p\sum_{\alpha\in\mathfrak{S}_p}\frac{A_p(\alpha)}{j(z)-\alpha} + p \sum_{ g(x) \in \mathfrak{M}_p} \frac{B_p(g)j(z)+C_p(g)}{g(j(z))} \pmod{p^2}. \end{multline*} It is natural to seek a description of the polynomials $D_p(x)$. Here we provide such a description in terms of certain Hecke polynomials.

Arithmetic of certain hypergeometric modular forms

In a recent paper, Kaneko and Zagier studied a sequence of modular forms Fk(z) which are solutions of a certain second order differential equation. They studied the polynomials e Fk(j) = Y τ∈H/Γ−{i,ω} (j − j(τ))τ k, where ω = e2πi/3 and H/Γ is the usual fundamental domain of the action of SL2(Z) on the upper half of the complex plane. If p ≥ 5 is prime, they proved that e Fp−1(j) (mod p) is the nontrivial factor of the locus of supersingular j-invariants in characteristic p. Here we consider the irreducibility of these polynomials, and consider their Galois groups.

Weierstrass points on X 0 (p) and supersingular j-invariants

Weierstrass points on X 0 (p) and supersingular j-invariants

On Modular Forms Arising from a Differential Equation of Hypergeometric Type

Modular and quasimodular solutions of a specific second order differential equation in the upper-half plane, which originates from a study of supersingular j-invariants in the first author's work with Don Zagier, are given explicitly. Positivity of Fourier coefficients of some of the solutions as well as a characterization of the differential equation are also discussed.

Supersingular j-invariants as singular moduli mod p

Supersingular j-invariants as singular moduli mod p