p-Converse to a Theorem of Gross-Zagier, Kolyvagin, and Rubin for Small Primes

Author: Yu, Qiyao (Vivian)

Year: 2021

Degree: Senior thesis (Major)

Advisor: Burungale, Ashay

Committee Members: Burungale, Ashay; Mantovan, Elena; Ramakrishnan, Dinakar

Option: Mathematics

DOI: 10.7907/ccbk-ht28

Abstract

Let E be a CM elliptic curve over the rationals and p a good, ordinary prime. We show the implication: corank(Zp)Sel(p∞)(E/Q) = 1 ⟹ ord(s=1)L(s,E/Q) = 1 for the p∞-Selmer group Sel(p∞)(E/Q) and the complex L-function L(s,E/QQ) in the case of p = 2, 3.

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