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

Abstract

AbstractFollowing 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 ofp2n–p2n–1–p–1for the rank of E(PQn).