Abstract
The aim of this paper is to prove inequalities towards instances of the Bloch–Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the L-function at the central point is zero or one. We achieve this implementing an inductive Euler system argument which relies on explicit reciprocity laws for cohomology classes constructed using congruences of automorphic forms and special points on several Shimura curves.
Highlights
In this case we prove, under suitable assumptions, inequalities towards the special value formulas predicted by Bloch–Kato
It may be worth pointing out that such a criterion is somewhat optimal: the equality in the relevant special value formula in analytic rank zero in particular implies the existence of non-trivial Selmer classes if Lalg( fK, 1) is not a p-adic unit
Such a statement follows from the Skinner–Urban divisibility in the Iwasawa main conjecture [43,47] but is a priori much weaker; it would be interesting to know whether the implication at the end of our theorem can be established by other means. 1.9
Summary
We denote by M(U ) the space of Hilbert modular forms of parallel weight two and level U with trivial central character and by S(U ) the subspace of cusp forms. They are equipped with an action of the Hecke algebra H(U \G(A f )/U ) of compactly supported, left and right U -invariant functions G(A f ) → C. Let f ∈ S(n) be a newform and O the ring generated by the eigenvalues λ f (Tv), λ f (Uv) of the Hecke operators acting on f It is an order in the ring of integers of a number field E ⊂ C which is totally real (since f has trivial central character). We define S B×/Z (U , A) by requiring F ×-invariance in addition
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