Abstract

Let $E$ be an elliptic curve over a number field $K$ defined by a monic irreducible cubic polynomial $F(x)$. When $E$ is nice at all finite primes of $K$, we bound its $2$-Selmer rank in terms of the $2$-rank of a modified ideal class group of the field $L=K[x]∕(F(x))$, which we call the seminarrow class group of $L$. We then provide several sufficient conditions for $E$ being nice at a finite prime.

As an application, when $K$ is a real quadratic field, $E∕K$ is semistable and the discriminant of $F$ is totally negative, we frequently determine the $2$-Selmer rank of $E$ by computing the root number of $E$ and the $2$-rank of the narrow class group of $L$.

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Mathematical Subject Classification

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