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

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 Büyükboduk--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 ${\mathrm{GL}}_2$-type for non-ordinary primes $p>2$.

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Cited by in corpus (5)

• Ranks of elliptic curves over Z_p^2-extensions
• On fine Selmer groups and signed Selmer groups of elliptic modular forms
• Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction
• The vanishing of anticyclotomic $\mu$-invariants for non-ordinary modular forms
• A proof of Kolyvagin's Conjecture via the BDP main conjecture