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Ternary Diophantine equations via Galois representations and modular forms
 CANAD J. MATH
, 2004
"... In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B and C ..."
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Cited by 26 (3 self)
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In this paper, we develop techniques for solving ternary Diophantine equations of the shape Ax n + By n = Cz 2, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters A, B and C. We conclude with an application of our results to certain classical polynomialexponential equations, such as those of Ramanujan–Nagell type.
On the surjectivity of the Galois representations associated to nonCM elliptic curves
 Canadian Math. Bulletin
"... 1 Let E be an elliptic curve defined over Q, of conductor N and without complex multiplication. For any positive integer k, let φk be the Galois representation associated to the kdivision points of E. From a celebrated 1972 result of Serre we know that φl is surjective for any sufficiently large pr ..."
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Cited by 15 (5 self)
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1 Let E be an elliptic curve defined over Q, of conductor N and without complex multiplication. For any positive integer k, let φk be the Galois representation associated to the kdivision points of E. From a celebrated 1972 result of Serre we know that φl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which φl is not surjective. 1
Uniform results for Serre’s theorem for elliptic curves
 MR 2189500 ↑1.5
"... A celebrated theorem of Serre from 1972 asserts that if E is an elliptic curve defined over Q and without complex multiplication, then its associated mod ℓ representation is surjective for all sufficiently large primes ℓ. In this paper we address the question of what sufficiently large means in Serr ..."
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Cited by 13 (3 self)
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A celebrated theorem of Serre from 1972 asserts that if E is an elliptic curve defined over Q and without complex multiplication, then its associated mod ℓ representation is surjective for all sufficiently large primes ℓ. In this paper we address the question of what sufficiently large means in Serre’s theorem. More precisely, we obtain a uniform version of Serre’s theorem for nonconstant elliptic curves defined over function fields, and a uniform version of Serre’s theorem for oneparameter families of elliptic curves defined over Q.
Toric modular forms and nonvanishing of Lfunctions
 J. Reine Angew. Math
"... Abstract. In a previous paper [1], we defined the space of toric forms T (l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two ..."
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Cited by 6 (4 self)
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Abstract. In a previous paper [1], we defined the space of toric forms T (l), and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group Γ1(l). In this article we prove the following theorem: modulo Eisenstein series, the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that L(f, 1) ̸ = 0. The proof uses work of Merel, and involves an explicit computation of the intersection pairing on Manin symbols. 1.
Normalizers of split Cartan subgroups and supersingular elliptic curves
"... Let us recall the following theorem of JP. Serre ([20], result (7)). Theorem 1 (Serre [20]). — Let E be an elliptic curve without complex multiplication over a number field K. There exists a number B(E, K) such that for any prime number p> B(E, K), the image GE,p of Gal ( ¯ K/K) in the group Aut(E ..."
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Cited by 5 (0 self)
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Let us recall the following theorem of JP. Serre ([20], result (7)). Theorem 1 (Serre [20]). — Let E be an elliptic curve without complex multiplication over a number field K. There exists a number B(E, K) such that for any prime number p> B(E, K), the image GE,p of Gal ( ¯ K/K) in the group Aut(E[p]) ≃ GL2(Fp) of
unknown title
"... 1.1. Background. Lfunctions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathemati ..."
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1.1. Background. Lfunctions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. The fundamental importance of these functions in mathematics is supported by the fact that two of the seven Clay Mathematics Million Dollar Millennium Problems [20] deal with properties of these functions, namely the