Conjectures of Langlands, Fontaine and Mazur [22] predict that certain Galois representations ρ: Gal(Q/Q) → GL2(Qℓ) (where ℓ denotes a fixed prime) should arise from modular forms. This applies in particular to representations defined by the action of Gal(Q/Q) on the ℓ-adic Tate

This paper concerns the Tamagawa number conjecture of Bloch and Kato [B-K] for adjoint motives of modular forms of weight k ≥ 2. The conjecture relates the value at 0 of the associated L-function to arithmetic invariants of the motive. We prove that it holds up to powers of certain “bad primes. ” The strategy for achieving

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In this paper, we prove many cases of the anticyclotomic main conjecture for general CM fields with p-ordinary CM type.

1.1. Let G be a connected reductive group over Q. Diamond [16] and Fujiwara [27] (independently) have axiomatized the Taylor-Wiles method which allows

Abstract. In 1987 Serre conjectured that any mod ℓ two-dimensional irreducible odd representation of the absolute Galois group of the rationals came from a modular form in a precise way. We present a generalisation of this conjecture to 2-dimensional representations of the absolute Galois group of a totally real field where ℓ is unramified. The hard work is in formulating an analogue of the “weight ” part of Serre’s conjecture. Serre furthermore asked whether his conjecture could be rephrased in terms of a “mod ℓ Langlands philosophy”. Using ideas of Emerton and Vigneras, we formulate a mod ℓ local-global principle for the group D ∗ , where D is a quaternion algebra over a totally real field, split above ℓ and at 0 or 1 infinite places, and show how it implies the conjecture. 1.

Abstract. Fix a residual ordinary representation ¯ρ: GF → GLn(k) of the absolute Galois group of a number field F. Generalizing work of Greenberg– Vatsal and Emerton–Pollack–Weston, we show that the Iwasawa invariants of Selmer groups of deformations of ¯ρ depends only on ¯ρ and the ramification of the deformation. Let p be an odd prime and let K be a finite extension of Qp with residue field k. Consider a continuous representation ¯ρ: GF → GLn(k) of the absolute Galois group of a number field F. Assume that ¯ρ is nearly ordinary in the sense that for any place v dividing p the image of a decomposition group at v lies in some Borel subgroup Bv of GLn. In this paper we show, under appropriate hypotheses, that the Iwasawa invariants of the Selmer group of a nearly ordinary deformation of ¯ρ depend only on ¯ρ and the tame ramification of the deformation. For simplicity, we specialize now to the case F = Q. Assume that ¯ρ satisfies the conditions of [11, Section 7] which guarantee that it has a reasonable deformation

this article was written. This research was funded by grants from NSERC and by an Alfred P. Sloan research fellowship

§1.1. Selmer groups §1.2. Greenberg’s L-invariant §1.3. Factorization of L-invariants