We prove modularity for a huge class of rigid Calabi-Yau threefolds over Q. In particular we prove that every rigid Calabi-Yau threefold with good reduction at 3 and 7 is modular. 1

We study Shafarevich–Tate groups of motives attached to modular forms on Γ0(N) of weight> 2. We deduce a criterion for the existence of nontrivial elements of these Shafarevich–Tate groups, and give 16 examples in which a strong form of the Beilinson–Bloch conjecture would imply the existence of such elements. We also use modular symbols and observations about Tamagawa numbers to compute nontrivial conjectural lower bounds on the orders of the Shafarevich–Tate groups of modular motives of low level and weight ≤ 12. Our methods build upon the idea of visibility due to Cremona and Mazur, but in the context of motives rather than abelian varieties. 1

Abstract. The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let’s mention: − the control of the image of the Galois representation modulo p [37][35], − Hida’s congruence criterion outside an explicit set of primes p [21], − the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras [30][16]. We study the arithmetic of the Hilbert modular forms by studying their modulo p Galois representations and our main tool is the action of the inertia groups at the primes above p. In order to determine this action, we compute the Hodge-Tate (resp. the Fontaine-Laffaille) weights of the p-adic (resp. the modulo p) étale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine [31, 33] on the cohomology of the Siegel modular varieties and builds upon the geometric constructions of [10, 11]. Contents

The Tamagawa number conjecture of Bloch and Kato describes the behavior at integers of the L-function associated to a motive over Q. Let f be a newform of weight k # 2, level N with coe#cients in a number field K. Let M be the motive associated to f and let A be the adjoint motive of M . Let # be a finite prime of K. We verify the #-part of the Bloch-Kato conjecture for L(A, 0) and L(A, 1) when # # Nk! and the mod # representation associated to f is absolutely irreducible when restricted to the Galois group over Q ## (-1) (#-1)/2 # # where # | #. 1.

Let f be a newform of weight k ≥ 2, level N, and character ω. Let K be the number field generated by the Fourier coefficients of f. For any prime λ of K Deligne has constructed a semisimple Galois representation ¯ρf,λ: G Q,S∪{ℓ} → GL2 kλ

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Abstract. Let f be a newform of weight k ≥ 3 with Fourier coefficients in a number field K. We show that the universal deformation ring of the mod λ Galois representation associated to f is unobstructed, and thus isomorphic to a power series ring in three variables over the Witt vectors, for all but finitely many primes λ of K. We give an explicit bound on such λ for the 6 known cusp forms of level 1, trivial character, and rational Fourier coefficients. We also prove a somewhat weaker result for weight 2. 1.

Abstract. We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms. 1.

Let f = ∑ anq n be a newform of weight k ≥ 2, level N, and character ω. Let K be the number field generated by the an. Let λ be a prime of K with residue field kλ of characteristic ℓ and let ¯ρf,λ: G Q,S∪{ℓ} → GL2 kλ be the semisimple Galois representation associated to f by Deligne and Serre; here S is any finite set of primes containing all primes dividing N and G Q,S∪{ℓ} is the Galois group of the maximal extension of Q unramified outside S ∪ {ℓ}. The representation ¯ρf,λ is absolutely irreducible for almost all λ; for such λ let R S f,λ denote the universal deformation ring parameterizing lifts of ¯ρf,λ (up to strict equivalence) to two dimensional representations of G Q,S∪{ℓ} over noetherian local rings with residue field kλ (see Section 1.1 for a precise definition). Using recent work of Diamond, Flach, and Guo [5] on the Bloch–Kato conjectures for adjoint motives of modular forms we prove the following theorem. Theorem 1. If k> 2, then (0.1) R S f,λ ∼ = W(kλ)[[T1, T2, T3]] for all but finitely many primes λ of K; here W(kλ) is the ring of Witt vectors of kλ. If k = 2, then (0.1) holds for all but finitely primes λ of K dividing rational primes ℓ such that (0.2) a 2 ℓ ̸ ≡ ω(ℓ) (mod λ). The special case of Theorem 1 for elliptic curves was proven by Mazur [15] using results of Flach [8] on symmetric square Selmer groups of elliptic curves. For weight k ≥ 3 Theorem 1 answers Mazur’s question of [15, Section 11] concerning the finiteness of the set of obstructed primes for modular forms; previously this finiteness was not known for a single modular form. We refer to [15] for a discussion of additional applications of Theorem 1. Our methods are in principle effective; that is, given enough information about the modular form f it is possible to determine a finite set of primes λ containing all those violating (0.1). We study in detail the cases of the six normalized cusp