Iwasawa theory for quadratic Hilbert modular forms

Abstract

We study the Iwasawa main conjecture for quadratic Hilbert modular forms over the p-cyclotomic tower. Using an Euler system in the cohomology of Siegel modular varieties, we prove the "Kato divisibility" of the Iwasawa main conjecture under certain technical hypotheses. By comparing this result with the opposite divisibility due to Wan, we obtain the full Main Conjecture over the cyclotomic Zp-extension. As a consequence, we prove new cases of the Bloch--Kato conjecture for quadratic Hilbert modular forms, and of the equivariant Birch--Swinnerton-Dyer conjecture in analytic rank 0 for elliptic curves over real quadratic fields twisted by Dirichlet characters. As a "by-product" of the theory developed here, we also present new results on Iwasawa theory for Rankin--Selberg convolutions of modular forms, relaxing hypotheses of $p$-distinction or $p$-regularity assumed in previous works. This gives new cases of the equivariant BSD conjecture for elliptic curves over $\mathbf{Q}$ twisted by 2-dimensional odd Artin representations, giving finiteness of the $p$-part of the Tate--Shafarevich group for all but finitely many ordinary primes.