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63
On Serre’s conjecture for 2dimensional mod p representations of Gal(Q̄/Q)
"... We prove the existence in many cases of minimally ramified padic lifts of 2dimensional continuous, odd, absolutely irreducible, mod p representations ¯ρ of the absolute Galois group of Q. It is predicted by Serre’s conjecture that such representations arise from newforms of optimal level and weig ..."
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We prove the existence in many cases of minimally ramified padic lifts of 2dimensional continuous, odd, absolutely irreducible, mod p representations ¯ρ of the absolute Galois group of Q. It is predicted by Serre’s conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre’s conjectures in low levels and weights. For instance we prove that there are no irreducible (p, p) type group schemes over Z. We prove that a ¯ρ as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Deltafunction. In the last part of the paper we present arguments that reduce Serre’s conjecture to proving generalisations of modularity lifting theorems of the type pioneered by Wiles.
NEARLY ORDINARY GALOIS DEFORMATIONS OVER ARBITRARY NUMBER FIELDS
, 2009
"... Let K be an arbitrary number field, and let ρ: Gal ( ¯ K/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation ..."
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Let K be an arbitrary number field, and let ρ: Gal ( ¯ K/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic ’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM ’ forms, and (3) ‘even ’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many ’ positivedimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘padic functoriality’, as well as some remarks on how our methods should apply to ndimensional representations of Gal ( ¯ Q/Q) when n>2.
Anticyclotomic Main Conjectures
 DOCUMENTA MATH.
, 2006
"... In this paper, we prove many cases of the anticyclotomic main conjecture for general CM fields with pordinary CM type. ..."
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Cited by 17 (10 self)
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In this paper, we prove many cases of the anticyclotomic main conjecture for general CM fields with pordinary CM type.
Diagonal cycles and Euler systems I: A padic GrossZagier formula
"... Abstract. This article is the first in a series devoted to studying generalised GrossKudlaSchoen diagonal cycles in the product of three KugaSato varieties and the Euler system properties of the associated Selmer classes, with special emphasis on their application to the Birch–SwinnertonDyer con ..."
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Cited by 17 (8 self)
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Abstract. This article is the first in a series devoted to studying generalised GrossKudlaSchoen diagonal cycles in the product of three KugaSato varieties and the Euler system properties of the associated Selmer classes, with special emphasis on their application to the Birch–SwinnertonDyer conjecture and the theory of StarkHeegner points. The basis for the entire study is a padic formula of GrossZagier type which relates the images of these diagonal cycles under the padic AbelJacobi map to special values of certain padic Lfunctions attached to the GarrettRankin triple convolution of three Hida families of modular forms. The main goal of this article is to describe and prove this formula. Contents
Mazur’s Principle for totally real fields of odd degree
, 1998
"... Abstract. In this paper, we prove an analogue of the result known as Mazur’s Principle concerning optimal levels of mod ` Galois representations. The paper is divided into two parts. We begin with the study (following Katz–Mazur) of the integral model for certain Shimura curves and the structure of ..."
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Abstract. In this paper, we prove an analogue of the result known as Mazur’s Principle concerning optimal levels of mod ` Galois representations. The paper is divided into two parts. We begin with the study (following Katz–Mazur) of the integral model for certain Shimura curves and the structure of the special fibre. It is this study which allows us to generalise, in the second part of this paper, Mazur’s result to totally real fields of odd degree.
Nonvanishing modulo p of Hecke L–values and application
 LFUNCTIONS AND GALOIS REPRESENTATIONS EDITED BY DAVID BURNS , KEVIN BUZZARD , JAN NEKOVÁR
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COMPANION FORMS OVER TOTALLY REAL FIELDS
"... Abstract. We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2 < k < p, which is ordinary at all primes dividing p and has tamely ramified Galois representation at all primes dividing p, then there is a “companion f ..."
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Abstract. We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2 < k < p, which is ordinary at all primes dividing p and has tamely ramified Galois representation at all primes dividing p, then there is a “companion form ” of parallel weight k ′: = p + 1 − k. This work generalises results of Gross and Coleman–Voloch for modular forms over Q. 1.
lowering for modular mod ` representations over totally real
, 1999
"... In this paper, we continue the study of part of the analogue of Serre’s conjecture for mod ` Galois representations for totally real fields. More precisely, one knows, through results of Carayol and Taylor, that to any Hilbert cuspidal eigenform over a totally real field F, one can attach a compatib ..."
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In this paper, we continue the study of part of the analogue of Serre’s conjecture for mod ` Galois representations for totally real fields. More precisely, one knows, through results of Carayol and Taylor, that to any Hilbert cuspidal eigenform over a totally real field F, one can attach a compatible system of λadic representations of the corresponding absolute Galois group. One may ask if a given λadic or modulo ` representation is attached by this process to a Hilbert modular form, and, if so, what weights and levels this form can have. We prove some analogues of results known in the case F = Q.