Related papers: On the Bloch-Kato conjecture for Hilbert modular f…

In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove a one sided divisibility result toward the Iwasawa main conjecture. The proof relies on the first and second…

Let F be a totally real field of degree d and let p be an odd prime which is totally split in F. We define and study one-dimensional partial eigenvarieties interpolating Hilbert modular forms over F with weight varying only at a single…

We prove an explicit reciprocity law for the Euler system attached to the spin motive of a genus 2 Siegel modular form. As consequences, we obtain one inclusion of the Iwasawa Main Conjecture for such motives, and the Bloch--Kato conjecture…

We construct Stickelberger elements for Hilbert modular cusp forms of parallel weight 2 and use recent results of Dasgupta and Spiess to bound their order of vanishing from below. As a special case the vanishing part of Mazur and Tate's…

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…

We provide a simple and efficient numerical criterion to verify the Iwasawa main conjecture and the indivisibility of derived Kato's Euler systems for modular forms of weight two at any good prime under mild assumptions. In the ordinary…

We construct an Euler system -- a compatible family of global cohomology classes -- for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps…

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction…

We prove one inclusion of the Iwasawa Main Conjecture, and the Bloch-Kato conjecture in analytic rank 0, for the symmetric cube of a level 1 modular form.

We prove the Bloch--Kato conjecture for certain critical values of degree 8 $L$-functions associated to cusp forms on $\mathrm{GSp}_4 \times \mathrm{GL}_2$. We also construct a $p$-adic Eichler--Shimura isomorphism in Hida families for…

We prove the Bloch-Kato conjecture for critical values of Asai L-functions of p-ordinary Hilbert modular forms over quadratic fields (with p split); and one inclusion in the Iwasawa main conjecture for these L-functions (up to a power of…

We prove the existence of Euler systems for adjoint modular Galois representations using deformations of Galois representations coming from Hilbert modular forms and relate them to $p$-adic $L$-functions under a conjectural formula for the…

Let k be a positive integer divisible by 4, l>k a prime, and f an elliptic cuspidal eigenform of weight k-1, level 4, and non-trivial character. Let \rho_f be the l-adic Galois representation attached to f. In this paper we provide evidence…

We study the \'etale cohomology of Hilbert modular varieties, building on the methods introduced for unitary Shimura varieties in [CS17, CS19]. We obtain the analogous vanishing theorem: in the "generic" case, the cohomology with torsion…

It it shown that the Bloch-Kato conjecture on the norm residue homomorphism $K^M(F)/l \to H^*(G_F,Z/l)$ follows from its (partially known) low-degree part under the assumption that the Milnor K-theory algebra $K^M(F)/l$ modulo $l$ is…

We discuss the relationship between different notions of "integrality" in motivic cohomology/K-theory which arise in the Beilinson and Bloch-Kato conjectures, and prove their equivalence in some cases for products of curves (used in the…

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of…

We study the Beilinson-Bloch-Kato conjecture for polarized motives. In the conjugate self-dual case, we show that if the central $L$-value does not vanish, then the associated Bloch-Kato Selmer group with coefficients in a suitable local…

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 $\mathit{extra \ reducibility}$: the Eisenstein ideal is strictly larger than the ideal cutting out…

In this article, we study the Beilinson-Bloch-Kato conjecture for motives corresponding to the Rankin-Selberg product of conjugate self-dual automorphic representations, within the framework of the Gan-Gross-Prasad conjecture. We show that…