Bloch-Kato Conjecture for the Adjoint of H1(Xo(N)) with Integral Hecke Algebra

Abstract

Let M be a motive that is defined over a number field and admits an action of a finite dimensional semisimple Q-algebra T. David Burns and Matthias Flach formulated a conjecture, which depends on a choice of Z-order T in T, for the leading coefficient of the Taylor expansion at 0 of the T-equivariant L-function of M. For primes l outside a finite set we prove the l-primary part of this conjecture for the specific case where M is the trace zero part of the adjoint of H^1(X_0(N)) for prime N and where T is the (commutative) integral Hecke algebra for cusp forms of weight 2 and the congruence group [Gamma_0](N), thus providing one of the first nontrivial supporting examples for the conjecture in a geometric situation where T is not the maximal order of T. We also compare two Selmer groups, one of which appears in Bloch-Kato conjecture and the other a slight variant of what is defined by A. Wiles. A result on the Fontaine-Laffaille modules with coefficients in a local ring finite free over Z_l is obtained.