Naslov
Supersingular zeros of divisor polynomials of elliptic curves of prime conductor

Autori
Kazalicki, Matija ; Kohen, Daniel

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
Journées Algophantiennes Bordelaises 2017 / - , 2017, 1-1

Skup
Journées Algophantiennes Bordelaises 2017

Mjesto i datum
Bordeaux, Francuska, 07.06.2017. - 09.06.2017

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Podatak o recenziji nije dostupan

Ključne riječi
supersingular elliptic curves

Sažetak
For a prime number p we study the zeros modulo p of divisor polynomials of elliptic curves E/Q of conductor p. Ono made the observation that these zeros of 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 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 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 v_E corresponding to supersingular elliptic curves defined over Fp are even. We prove this conjecture in the case when the discriminant of E is positive. This is joint work with Daniel Kohen.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA