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Search: MSC category 14G10 ( Zeta-functions and related questions [See also 11G40] (Birch-Swinnerton-Dyer conjecture) )

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1. CJM 1997 (vol 49 pp. 749)

Howe, Lawrence
Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups
Following a method outlined by Greenberg, root number computations give a conjectural lower bound for the ranks of certain Mordell-Weil groups of elliptic curves. More specifically, for $\PQ_{n}$ a \pgl{{\bf Z}/p^{n}{\bf Z}}-extension of ${\bf Q}$ and $E$ an elliptic curve over {\bf Q}, define the motive $E \otimes \rho$, where $\rho$ is any irreducible representation of $\Gal (\PQ_{n}/{\bf Q})$. Under some restrictions, the root number in the conjectural functional equation for the $L$-function of $E \otimes \rho$ is easily computible, and a `$-1$' implies, by the Birch and Swinnerton-Dyer conjecture, that $\rho$ is found in $E(\PQ_{n}) \otimes {\bf C}$. Summing the dimensions of such $\rho$ gives a conjectural lower bound of $$ p^{2n} - p^{2n - 1} - p - 1 $$ for the rank of $E(\PQ_{n})$.

Categories:11G05, 14G10

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