The Eisenstein ideal for weight k and a Bloch-Kato conjecture for tame families

2020/02/06 by Preston Wake, Wake, Preston
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models

paper · pdf · doi:10.48550/arxiv.2002.02442

Abstract

We study the Eisenstein ideal for modular forms of even weight k>2 and prime level N. We pay special attention to the phenomenon of extra reducibility: the Eisenstein ideal is strictly larger than the ideal cutting out reducible Galois representations. We prove a modularity theorem for these extra reducible representations. As consequences, we relate the derivative of a Mazur-Tate L-function to the rank of the Hecke algebra, generalizing a theorem of Merel, and give a new proof of a special case of an equivariant main conjecture of Kato. In the second half of the paper, we recall Kato's formulation of this main conjecture in the case of a family of motives given by twists by characters of conductor N and p-power order and its relation to other formulations of the equivariant main conjecture.

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