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arXiv · 1804.10993

On the Iwasawa main conjectures for modular forms at non-ordinary primes

Francesc Castella·Mirela Çiperiani·Christopher Skinner·Florian Sprung
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Abstract

In this paper, we prove under mild hypotheses the Iwasawa main conjectures of Lei--Loeffler--Zerbes for modular forms of weight $2$ at non-ordinary primes. Our proof is based on the study of the two-variable analogues of these conjectures formulated by B\"uy\"ukboduk--Lei for imaginary quadratic fields in which $p$ splits, and on anticyclotomic Iwasawa theory. As application of our results, we deduce the $p$-part of the Birch and Swinnerton-Dyer formula in analytic ranks $0$ or $1$ for abelian varieties over $\mathbb{Q}$ of ${\rm GL}_2$-type for non-ordinary primes $p>2$.

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Francesc Castella, Mirela Çiperiani, Christopher Skinner, Florian Sprung. 2018-04-29. On the Iwasawa main conjectures for modular forms at non-ordinary primes. https://arxiv.org/abs/1804.10993

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