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Classical and modular approaches to exponential Diophantine equations I. Fibonacci and Lucas perfect powers
 Annals of Math
"... Abstract. This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we use a general ..."
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Cited by 37 (12 self)
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Abstract. This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat’s Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical, elementary and substantially improved modular methods to solve completely the LebesgueNagell equation for D in the range 1 ≤ D ≤ 100. x 2 + D = y n, x, y integers, n ≥ 3, 1.
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 33 (6 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.
Faltings plus epsilon, Wiles plus epsilon, and the generalized Fermat equation
 C. R. Math. Rep. Acad. Sci. Canada
, 1997
"... Wiles ’ proof of Fermat’s Last Theorem puts to rest one of the most famous unsolved problems in mathematics, a question that has been a wellspring for much of modern algebraic number theory. While celebrating Wiles ’ achievement, one also feels a twinge of regret at Fermat’s demise. Is the Holy Grai ..."
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Cited by 18 (3 self)
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Wiles ’ proof of Fermat’s Last Theorem puts to rest one of the most famous unsolved problems in mathematics, a question that has been a wellspring for much of modern algebraic number theory. While celebrating Wiles ’ achievement, one also feels a twinge of regret at Fermat’s demise. Is the Holy Grail of number theorists to become a mere footnote in the history books? Hoping to keep some of the spirit of Fermat alive, I would like to discuss the generalized Fermat equation x p + y q = z r, (1) where p, q and r are fixed exponents. As in the case of Fermat’s Last Theorem, one is interested in integer solutions (x, y, z), which are nontrivial in the sense that xyz ̸ = 0. One might expect the equation above to have no such solutions if the exponents p, q, and r are large enough. But observe that, if p = q is odd, and r = 2, then any solution to a p +b p = c (of which there is an abundant supply!) yields the solution (ac, bc, c p+1 2) to the equation xp + yp = z2. A similar construction works whenever the exponents p, q, and r are pairwise coprime. However, the solutions produced in this way are not very interesting: the integers x, y and z have a large common factor. ∗This is a transcription of the author’s Aisenstadt prize lecture given at the CRM in March 1997. It is a pleasure to thank Andrew Granville and Loïc Merel for stimulating collaborations related to the topics of this essay, as well as Dan Abramovich for many helpful conversations over the years. This research was supported by CICMA and by grants from the Sloan Foundation, NSERC and FCAR. 1 Accordingly, one calls a solution (x, y, z) to the generalized Fermat equation primitive if gcd(x, y, z) = 1. Main Question: What are the nontrivial primitive solutions to the generalized Fermat equation? In [DG], Andrew Granville and I made the following conjecture:
The rational function analogue of a question of Schur and exceptionality of permutation representations
, 2008
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Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik’s problem
, 2001
"... 1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are ..."
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Cited by 16 (3 self)
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1 Let E be an elliptic curve defined over Q and of conductor N. For a prime p ∤ N, we denote by E the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which E(Fp) is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J.P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = pE for which the group E(Fp) is cyclic and we show that, under the generalized Riemann hypothesis, pE = O � (log N) 4+ε � if E is without complex multiplication, and pE = O � (log N) 2+ε � if E is with complex multiplication, for any 0 < ε < 1. 1
J.,Galois representations attached to Qcurves and the generalized Fermat equation A 4 + B 2 = C p , preprint 7
 Duke Math. J
"... We prove that the equation A 4 +B 2 = C p has no solutions in coprime positive integers when p≥211. The main step is to show that, for all sufficiently large primes p, every Qcurve over an imaginary quadratic field K with a prime of bad reduction greater than 6 has a surjective mod p Galois represe ..."
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Cited by 15 (2 self)
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We prove that the equation A 4 +B 2 = C p has no solutions in coprime positive integers when p≥211. The main step is to show that, for all sufficiently large primes p, every Qcurve over an imaginary quadratic field K with a prime of bad reduction greater than 6 has a surjective mod p Galois representation. The bound on p depends on K and the degree of the isogeny between E and its Galois conjugate, but is independent of the choice of E. The proof of this theorem combines geometric arguments due to Mazur, Momose, Darmon, and Merel with an analytic estimate of the average special values of certain Lfunctions. 1
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 11 (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.
Ternary Diophantine equations of signature (p, p, 3
 Compos. Math
"... In this paper, we develop machinery to solve ternary Diophantine equations of the shape Axn+Byn = Cz3 for various choices of coefficients (A,B,C). As a byproduct of this, we show, if p is prime, that the equation xn + yn = pz3 has no solutions in coprime integers x and y with xy > 1 and prime n ..."
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Cited by 11 (3 self)
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In this paper, we develop machinery to solve ternary Diophantine equations of the shape Axn+Byn = Cz3 for various choices of coefficients (A,B,C). As a byproduct of this, we show, if p is prime, that the equation xn + yn = pz3 has no solutions in coprime integers x and y with xy > 1 and prime n> p4p2. The techniques employed enable us to classify all elliptic curves over Q with a rational 3torsion point and good reduction outside the set {3, p}, for a fixed prime p. 1.