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

Abstract

Using the theory of elliptic curves, we show that the class number hðÀpÞ of the field Qð ffiffiffiffiffiffi ffi Àp p Þ appears in the count of certain factors of the Legendre polynomials P m ðxÞ ðmod pÞ; where p is a prime 43 and m has the form ðp À eÞ=k; with k ¼ 2; 3 or 4 and p e ðmod kÞ: As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y 2 þ axy þ y ¼ x 3 and find an elementary expression for the supersingular polynomial ss p ðxÞ whose roots are the supersingular j-invariants of elliptic curves in characteristic p: As a corollary we show that the class number hðÀpÞ also shows up in the factorization ðmod pÞ of certain Jacobi polynomials.

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

“…With all notation as above, set r = 1 in Parts (1) and (3) of Theorem 1 and r = 2 in Part (2). Then for λ, μ ∈ {±1} we have…”

“…For parts (2) and (3) of Theorem 1, the ideas and proofs are nearly identical, so we give brief sketches of the arguments rather than detailed proofs. Both parts (2) and (3) can be proved via a similar sequence of Lemmas:…”

On the $j$-invariants of CM-elliptic curves defined over $\mathbb{Z}_p$

“…(3) Given an F p -rational supersingular j-invariant which admits CM by O, when does there exist an elliptic curve defined over Z p , which (after base extension) admits CM by O, which reduces to it. Though they are not necissarily framed in this way, related questions are treated in [Sta12], [Mor14] and [BM04] and some of our results can naturally be viewed as generalizations to the context of non-maximal orders. Furthermore, there are natural connections between some of our results and those presented in [LV15].…”

A formula for the supersingular polynomial

“…Several papers have dealt with the supersingular polynomial in the past, notably [5], [1], and [6]. In [1], J. Brillhart and P. Morton give an explicit formula for the supersingular polynomial, which depends on the Jacobi polynomials P (α,β) n .…”

“…In [1], J. Brillhart and P. Morton give an explicit formula for the supersingular polynomial, which depends on the Jacobi polynomials P (α,β) n . In [5], which is partially expository, a few different polynomials in Q[X] are given that reduce to the supersingular polynomial and, in particular, the Atkin's polynomials are quite explicit.…”