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Twisted Hasse-Weil L-Functions and the Rank of Mordell-Weil Groups

Published online by Cambridge University Press:  20 November 2018

Lawrence Howe
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Abstract

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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 PQn a PGL2(Z/pnZ)–extension of Q and E an elliptic curve over Q, define the motive E ⊗ ρ, where ρ is any irreducible representation of Gal(PQn /Q). Under some restrictions, the root number in the conjectural functional equation for the L-function of E ⊗ ρ is easily computable, and a ‘–1’ implies, by the Birch and Swinnerton–Dyer conjecture, that ρ is found in E(PQn) ⊗ C. Summing the dimensions of such ρ gives a conjectural lower bound of

p2n–p2n–1–p–1

for the rank of E(PQn).

Keywords

11G0514G10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 4 , 01 August 1997 , pp. 749 - 771
Copyright
Copyright © Canadian Mathematical Society 1997

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