Parity of ranks of elliptic curves with equivalent mod 𝑝 Galois representations

Shekhar, Sudhanshu

doi:10.1090/proc/12993

Parity of ranks of elliptic curves with equivalent mod $p$ Galois representations
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Proc. Amer. Math. Soc. 144 (2016), 3255-3266 Request permission

Abstract:

Given two elliptic curves $E_1$ and $E_2$ defined over the field of rational numbers $\mathbb {Q}$ that have good and ordinary reduction at an odd prime $p$, and have equivalent, irreducible mod $p$ Galois representations, we study the variation of the parity of Selmer ranks and analytic ranks of $E_1$ and $E_2$ over certain number fields.

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