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Supersingular zeros of divisor polynomials of elliptic curves of prime conductor. (English) Zbl 1433.11074

Summary: For a prime number \(p\), we study the zeros modulo \(p\) of divisor polynomials of rational elliptic curves \(E\) of conductor \(p\). K. Ono [The web of modularity: arithmetic of the coefficients of modular forms and \(q\)-series. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1119.11026)] made the observation that these zeros 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.

MSC:

11G05 Elliptic curves over global fields
11F32 Modular correspondences, etc.
11F37 Forms of half-integer weight; nonholomorphic modular forms

Citations:

Zbl 1119.11026
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References:

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