Event Detail

Event Type: 
Number Theory Seminar
Tuesday, October 18, 2022 - 10:00 to 10:50

Speaker Info

University of Connecticut

Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$. Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where $K$ is an $\ell$-power extension of $k$. We have previously shown that, under mild rationality hypotheses, the case of interest is when $\ell=2$ and $K$ is the unique quadratic extension of $k$.
In this talk we determine the likelihood of such an occurrence by fixing a pair of 2-isogenous elliptic curves $E$, $E'$ over $\mathbf{Q}$ and asking for what proportion of primes $p$ do we have $E(\mathbf{F}_p) \simeq E'(\mathbf{F}_p)$ and $E(\mathbf{F}_{p^2}) \not \simeq E'(\mathbf{F}_{p^2})$.