Bloch-Kato Conjecture for the Adjoint of H¹(X₀(N)) with Integral Hecke Algebra

• Main Author: Lin, Qiang
• Format: Others
• Language: en
• Published: 2004
• Online Access: https://thesis.library.caltech.edu/4595/1/BurnsFlachConjectureForIntegralHeckeAlgebra.pdf; Lin, Qiang (2004) Bloch-Kato Conjecture for the Adjoint of H¹(X₀(N)) with Integral Hecke Algebra. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/QD4X-J291. https://resolver.caltech.edu/CaltechETD:etd-11182003-084742 <https://resolver.caltech.edu/CaltechETD:etd-11182003-084742>

<p>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 H¹(X₀(N)) for prime N and where T is the (commutative) integral Hecke algebra for cusp forms of weight 2 and the congruence group Γ₀(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.</p> <p>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. </p>