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Twisted Hasse-Weil $L$-functions and the rank of Mordell-Weil groups

  Published:1997-08-01
 Printed: Aug 1997
  • Lawrence Howe
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Abstract

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 $E(\PQ_{n})$.
MSC Classifications: 11G05, 14G10 show english descriptions Elliptic curves over global fields [See also 14H52]
Zeta-functions and related questions [See also 11G40] (Birch-Swinnerton-Dyer conjecture)
11G05 - Elliptic curves over global fields [See also 14H52]
14G10 - Zeta-functions and related questions [See also 11G40] (Birch-Swinnerton-Dyer conjecture)
 

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