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 (F-p) over bar. 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 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 vE corresponding to supersingular elliptic curves defined over 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.