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38
Integrality of a ratio of Petersson norms and LevelLowering Congruences
 ANN. OF MATH
, 2003
"... We prove integrality of the ratio f#/#g, g# (outside an explicit finite set of primes), where g is an arithmetically normalized holomorphic newform on a Shimura curve, f is a normalized Hecke eigenform on GL(2) with the same Hecke eigenvalues as g and denotes the Petersson inner product. The p ..."
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Cited by 15 (7 self)
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We prove integrality of the ratio f#/#g, g# (outside an explicit finite set of primes), where g is an arithmetically normalized holomorphic newform on a Shimura curve, f is a normalized Hecke eigenform on GL(2) with the same Hecke eigenvalues as g and denotes the Petersson inner product. The primes dividing this ratio are shown to be closely related to certain levellowering congruences satisfied by f and to the central values of a family of RankinSelberg Lfunctions. Finally we give two applications, the first to proving the integrality of a certain triple product Lvalue and the second to the computation of the Faltings height of Jacobians of Shimura curves.
SaitoKurokawa lifts and applications to the BlochKato conjecture
 REDUCIBLE REPRESENTATIONS 7
, 1994
"... Let f be a newform of weight 2k −2 and level 1. In this paper we provide evidence for the BlochKato conjecture for modular forms. We demonstrate an implication that under suitable hypotheses if ̟  Lalg(k,f) then p  # Hf(Q,Wf(1 − k)) where p is a suitably chosen prime and ̟ a uniformizer of a fi ..."
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Cited by 15 (9 self)
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Let f be a newform of weight 2k −2 and level 1. In this paper we provide evidence for the BlochKato conjecture for modular forms. We demonstrate an implication that under suitable hypotheses if ̟  Lalg(k,f) then p  # Hf(Q,Wf(1 − k)) where p is a suitably chosen prime and ̟ a uniformizer of a finite extension K/Qp. We demonstrate this by establishing a congruence between the SaitoKurokawa lift Ff of f and a cuspidal Siegel eigenform G that is not a SaitoKurokawa lift. We then examine what this congruence says in terms of Galois representations to produce a nontrivial ptorsion element in H 1 f (Q,Wf(1 − k)). 1.
RIGID LOCAL SYSTEMS, HILBERT MODULAR FORMS, AND FERMAT’S LAST THEOREM
 VOL. 102, NO. 3 DUKE MATHEMATICAL JOURNAL
, 2000
"... ..."
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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Cited by 9 (0 self)
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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.
An Eisenstein ideal for imaginary quadratic fields
, 2005
"... For certain algebraic Hecke characters χ of an imaginary quadratic field F we define an Eisenstein ideal in a padic Hecke algebra acting on cuspidal automorphic forms of GL2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a l ..."
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Cited by 9 (5 self)
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For certain algebraic Hecke characters χ of an imaginary quadratic field F we define an Eisenstein ideal in a padic Hecke algebra acting on cuspidal automorphic forms of GL2/F. By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special Lvalue L(0,χ). We further prove that its index is bounded from above by the order of the Selmer group of the padic Galois character associated to χ −1. This uses the work of R. Taylor et al. on attaching Galois representations to cuspforms of GL2/F. Together these results imply a lower bound for the size of the Selmer group in terms of L(0,χ), coinciding with the value given by the BlochKato conjecture.
The Iwasawa main conjectures for GL2
, 2010
"... In this paper we prove the IwasawaGreenberg Main Conjecture for a large class of elliptic curves and modular forms. 1.1. The IwasawaGreenberg Main Conjecture. Let p be an odd prime. Let Q ⊂ C be the algebraic closure of Q in C. We fix an embedding Q ↩ → Q p. For simplicity we ..."
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Cited by 8 (0 self)
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In this paper we prove the IwasawaGreenberg Main Conjecture for a large class of elliptic curves and modular forms. 1.1. The IwasawaGreenberg Main Conjecture. Let p be an odd prime. Let Q ⊂ C be the algebraic closure of Q in C. We fix an embedding Q ↩ → Q p. For simplicity we
Companion forms and the structure of padic Hecke algebras
 J. reine angew. Math
"... Abstract. We study the structure of the Eiesenstein component of Hida’s ordinary padic Hecke algebra attached to modular forms, in connection with the companion forms in the space of modular forms (mod p). We show that such an algebra is a Gorenstein ring if certain space of modular forms (mod p) h ..."
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Cited by 6 (0 self)
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Abstract. We study the structure of the Eiesenstein component of Hida’s ordinary padic Hecke algebra attached to modular forms, in connection with the companion forms in the space of modular forms (mod p). We show that such an algebra is a Gorenstein ring if certain space of modular forms (mod p) having companions is onedimensional; and also give a numerical criterion for this onedimensionality. This in part overlaps with a work of Skinner and Wiles; but our method, based on a work of Ulmer, is totally different. We then consider consequences of the above mentioned Gorenstein property. We especially discuss the connection with the Iwasawa theory.
Yoshida lifts and the BlochKato conjecture for the convolution Lfunction, preprint
, 2012
"... Abstract. Let f1 (resp. f2) denote two (elliptic) newforms of prime level N, trivial character and weight 2 (resp. k + 2, where k ∈ {8, 12}). We provide evidence for the BlochKato conjecture for the motive M = ρf1 ⊗ρf2 (−k/2−1) by proving that under some assumptions the pvaluation of the order of ..."
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Cited by 5 (4 self)
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Abstract. Let f1 (resp. f2) denote two (elliptic) newforms of prime level N, trivial character and weight 2 (resp. k + 2, where k ∈ {8, 12}). We provide evidence for the BlochKato conjecture for the motive M = ρf1 ⊗ρf2 (−k/2−1) by proving that under some assumptions the pvaluation of the order of the BlochKato Selmer group of M is bounded from below by the pvaluation of the relevant Lvalue (a special value of the convolution Lfunction of f1 and f2). We achieve this by constructing congruences between the Yoshida lift Y (f1 ⊗ f2) of f1 and f2 and Siegel modular forms whose padic Galois representations are irreducible. Our result is conditional upon a conjectural formula for the Petersson norm of Y (f1 ⊗ f2). 1.
Eisenstein deformation rings
 Compositio Math
"... We prove R = T theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin on whether certain Hecke algebras T are discrete valuation rings. In order to prove these results we determine (using the theory of Breuil modules) when two finite f ..."
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Cited by 3 (0 self)
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We prove R = T theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin on whether certain Hecke algebras T are discrete valuation rings. In order to prove these results we determine (using the theory of Breuil modules) when two finite flat group schemes G and H of order p over an arbitrarily tamely ramified discrete valuation ring admit an extension not killed by p. 1