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- W66838809 abstract "In this thesis we use Rankin's method to evaluate the central critical value of the L-series attached to an elliptic curve E over Q and certain odd irreducible 2-dimensional Artin representation tau: G_Q --> GL_2(C). The motivation for this study is the twisted BSD conjecture which states that:For any continuous irreducible finite-dimensional complex representation tau of Gal(Qbar}/Q), we have:ord_{s=1}L(rho_E tensor tau,s)= multiplicity of tau in C tensor E(K)where E is an elliptic curve over Q, K is the fixed field of Ker(tau) and rho_E is the 2-dimensional l-adic Galois representation attached to the elliptic curve E, namely, the p-adic Tate module of E.Deligne and Serre proved that there is a correspondence between the modular forms of weight 1 and certain two-dimensional Galois representations. In the twisted BSD conjecture, we put tau=rho_g where rho_g:Gal(K/Q)-->GL_2(C) is an odd, continuous and irreducible Galois representation arising from a cusp form g belonging to S_1(Gamma_0(N),chi). Using the Rankin method, we compute the constant term of L(rho_{g}otimes rho_{E},s) at s=1. Thus, if we assume the BSD, we deduce:L(rho_E tensor rho_g,1)= 0 ---> Hom_{G_{Q}} (rho_g, C tensor E(K)) 0.In this case, one can say that r_{K}(E) >= 2 .We present some tables of computations we did with Sage for different modular forms g and elliptic curves E." @default.
- W66838809 created "2016-06-24" @default.
- W66838809 creator A5018810169 @default.
- W66838809 date "2012-09-14" @default.
- W66838809 modified "2023-09-27" @default.
- W66838809 title "Rankin L-functions and the Birch and Swinnerton-Dyer Conjecture" @default.
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