Ternary Diophantine Equations via Galois Representations and Modular Forms

By Michael A. Bennett, Chris M. Skinner. Canadian Journal of Mathematics - Journal Canadien de Mathematiques, 2004

229 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f11da190634ee2fb39f15c3b6df419b3ec0114b2

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Citations in the corpus

The Modular Approach to Diophantine Equations (2007, methodology): “• p ≥ 17 and j(E) 6∈ Z[ 1 2 ], • or p ≥ 11 and E is a semi-stable elliptic curve, • or p ≥ 5, #E(Q)[2] = 4, and E is a semi-stable elliptic curve, Then E does not have any p-isogenies.”
Fibonacci numbers at most one away from a perfect power (2008, background): “Several alternative strategies do apply in special cases (see for example [3], [13], [16]), though there does not seem to be a general strategy that is guaranteed to succeed.”
The asymptotic Fermat’s Last Theorem for five-sixths of real quadratic fields (2013, background): “in the wider Diophantine context by mentioning related results on Fermat-type equations. Many results of the FLT or asymptotic FLT kind (but over Q) are established by Kraus [30], Bennett and Skinner [3], and Bennett, Vatsal and Yazdani [4] for equations of the form Ax p+By p+Cz = 0, Axp +By = Cz2, ”
Generalized Fermat equations: A miscellany (2015, background): “(n, n, n), n ≥ 3 Wiles [74], Taylor–Wiles [72] (n, n, 2), n ≥ 4 Darmon–Merel [35], Poonen [63] (n, n, 3), n ≥ 3 Darmon–Merel [35], Poonen [63] (2n, 2n, 5), n ≥ 2 Bennett [1] (2, 4, n), n ≥ 4 Ellenberg [41], Bennett–Ellenberg–Ng [5], Bruin [16] (2, 6, n), n ≥ 3 Bennett–Chen [2], Bruin [16] (2, n, 4),”
Powers from Products of Consecutive Terms in Arithmetic Progression (2006, methodology): “The cases of equation (21) with p = 3 or 5, and β 1, while not all explicitly treated in [1], follow immediately from the arguments of that paper, upon noting that the modular curves X0(N) have genus 0 for all N dividing 6 or 10.”
Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation (2004, methodology): “In [6], Bennett and Skinner give an exhaustive recipe for Frey curves and level-lowering for equations of the form Ax + By = Cz under the assumption that the three terms in the equation are coprime.”