Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

Abstract

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 Fp¯. We show that these supersingular zeros are in bijection with zeros modulo p of an associated quaternionic modular form vE. This allows us to prove that if the root number of E is - 1 then all supersingular j-invariants of elliptic curves defined over Fp 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 Fp 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 vE corresponding to supersingular elliptic curves defined over Fp 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.