Open Access Publications from the University of California

Strong Selmer Companion Elliptic Curves

• Author(s): Chiu, Ching-Heng
Let $E_1$ and $E_2$ be elliptic curves defined over a number field $K$. Suppose that for all but finitely many primes $\ell$, and for all finite extension fields $L/K$,
$$\dim_{\mathbb{F}_\ell}\mathrm{Sel}_{\ell}(L,E_1)=\dim_{\mathbb{F}_\ell}\mathrm{Sel}_{\ell}(L,E_2).$$ We prove that $E_1$ and $E_2$ are isogenous over $K$.