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54
Serre's modularity conjecture (I)
, 2007
"... This paper is the first part of a work which proves Serre’s modularity conjecture. We first prove the cases p ̸ = 2 and odd conductor, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see [13]. We then reduce the general case to a modularity statement ..."
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Cited by 97 (0 self)
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This paper is the first part of a work which proves Serre’s modularity conjecture. We first prove the cases p ̸ = 2 and odd conductor, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see [13]. We then reduce the general case to a modularity statement for 2adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin [19].
Modularity of rigid CalabiYau threefolds over Q
 in “CalabiYau Varieties and Mirror Symmetry”, Fields Institute Communications, 38, AMS (2003
"... We prove modularity for a huge class of rigid CalabiYau threefolds over Q. In particular we prove that every rigid CalabiYau threefold with good reduction at 3 and 7 is modular. 1 ..."
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Cited by 33 (13 self)
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We prove modularity for a huge class of rigid CalabiYau threefolds over Q. In particular we prove that every rigid CalabiYau threefold with good reduction at 3 and 7 is modular. 1
The FontaineMazur conjecture for GL2
 Journal of the A.M.S
"... 1. BreuilMézard conjecture and the padic local Langlands 644 (1.1) The BreuilMézard conjecture 644 (1.2) Review of Colmez’s functor 647 ..."
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1. BreuilMézard conjecture and the padic local Langlands 644 (1.1) The BreuilMézard conjecture 644 (1.2) Review of Colmez’s functor 647
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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Cited by 27 (1 self)
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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.
On Serre’s conjecture for mod ℓ Galois representations over totally real fields
, 2009
"... In 1987 Serre conjectured that any mod ℓ twodimensional 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 2dimensional representations of the absolute Galois group of a totally ..."
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Cited by 19 (2 self)
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In 1987 Serre conjectured that any mod ℓ twodimensional 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 2dimensional 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 localglobal 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.
ON THE MODULARITY OF QCURVES
 DUKE MATHEMATICAL JOURNAL VOL. 109, NO. 1
, 2001
"... A Qcurve is an elliptic curve over a number field K which is geometrically isogenous to each of its Galois conjugates. K. Ribet [17] asked whether every Qcurve is modular, and he showed that a positive answer would follow from J.P. Serre’s conjecture on mod p Galois representations. We answer Rib ..."
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Cited by 17 (2 self)
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A Qcurve is an elliptic curve over a number field K which is geometrically isogenous to each of its Galois conjugates. K. Ribet [17] asked whether every Qcurve is modular, and he showed that a positive answer would follow from J.P. Serre’s conjecture on mod p Galois representations. We answer Ribet’s question in the affirmative, subject to certain local conditions at 3.
ON SERRE’S MODULARITY CONJECTURE FOR 2DIMENSIONAL MOD p REPRESENTATIONS OF ... Unramified Outside p
, 2005
"... We prove the level one case of Serre’s conjecture. Namely, we prove that any continuous, odd, irreducible representation ¯ρ: Gal ( ¯ Q/Q) → GL2(Fp) which is unramified outside p arises from a cuspidal eigenform in S k(¯ρ)(SL2(Z)). The proof relies on the methods introduced in an earlier joint wor ..."
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We prove the level one case of Serre’s conjecture. Namely, we prove that any continuous, odd, irreducible representation ¯ρ: Gal ( ¯ Q/Q) → GL2(Fp) which is unramified outside p arises from a cuspidal eigenform in S k(¯ρ)(SL2(Z)). The proof relies on the methods introduced in an earlier joint work with JP. Wintenberger, together with a new method of “weight reduction”.
Galois representations
 Proceedings of the International Congress of Mathematicians, Beijing, 2002, vol I. World Scientific
"... In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie groups. In the second part we briefly review some limited re ..."
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In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie groups. In the second part we briefly review some limited recent progress on these conjectures.
Root numbers and the parity problem
, 2003
"... Let E be a oneparameter family of elliptic curves over a number field K. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average roo ..."
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Cited by 10 (3 self)
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Let E be a oneparameter family of elliptic curves over a number field K. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any nondegenerate family E has average root number 0, provided that two classical arithmetical conjectures hold for two homogeneous polynomials with integral coefficients constructed explicitly in terms of E. The first such conjecture – commonly associated with Chowla – asserts the equidistribution of the parity of the number of primes dividing the integers represented by a polynomial. More precisely: given a homogeneous polynomial f ∈ Z[x, y], it is believed that µ(f(x, y)) averages to zero. This conjecture can be said to represent the parity problem in its pure form, while covering the same notional ground as the BunyakovskySchinzel and HardyLittlewood conjectures taken together. For deg f = 1 and deg f = 2, Chowla’s conjecture is essentially equivalent to the prime number theorem. For deg f> 2, the conjecture has been unproven up to now; the traditional approaches by means of analysis and sieve theory fail. We prove the conjecture for deg f = 3. There remains to state the second arithmetical conjecture referred to previously. It is believed that any nonconstant homogeneous polynomial f ∈ Z[x, y] yields to a squarefree sieve. We sharpen the existing bounds on the known cases by a sieve refinement and a new approach combining height functions, sphere packings and sieve methods. iii ��������������������� � ��������� � ����� � �����