Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

TL;DR

This paper investigates the zeros of divisor polynomials of rational elliptic curves of prime conductor modulo p, revealing their connection to supersingular elliptic curves and quaternionic modular forms, with implications for elliptic curve ranks.

Contribution

It establishes a bijection between supersingular zeros and zeros of quaternionic modular forms, and proves a conjecture relating coefficients of these forms to elliptic curve rank.

Findings

Supersingular zeros correspond to zeros of quaternionic modular forms.

If the root number is -1, all supersingular j-invariants are zeros of divisor polynomials.

When the root number is 1, higher rank elliptic curves show more supersingular zeros.

Abstract

For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials of rational elliptic curves $E$ of conductor $p$. Ono made the observation that these zeros of 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.