Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

KOHEN, DANIEL; KAZALICKI, MATIJA

Research in the Mathematical Sciences

Springer

Año: 2017 vol. 4

For a prime number $p$ we study the zeros modulo $p$ of divisor polynomials of rational elliptic curves $E$ of conductor $p$. Ono cite[p.~118]{Ono} made the observation that these zeros of are often $j$-invariants of supersingular elliptic curves over $overline{F}$. 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.