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

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 ex...

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Veröffentlicht in:Journal of the European Mathematical Society : JEMS 2023-01, Vol.25 (7), p.2815-2861
1. Verfasser: Wake, Preston
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:1435-9855
1435-9863
DOI:10.4171/jems/1251