Michael Bennett

Professor

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2021)
Computing elliptic curves over Q via Thue-Mahler equations and related problems (2020)

We present a practical and efficient algorithm for solving an arbitrary Thue-Mahler equation. This algorithm uses explicit height bounds with refined sieves, combining Diophantine approximation techniques of Tzanakis-de Weger with new geometric ideas. We begin by using methods of algebraic number theory to reduce the problem of solving the Thue-Mahler equation to the problem of solving a finite collection of related Diophantine equations. In the first part of this thesis, we establish the key results which allow us to drastically reduce the number of such Diophantine equations and subsequently reduce the running time. In the second part of this thesis, we show that, by fixing one exponent, there exists an effectively computable constant bounding the solutions of a Goormaghtigh equation under certain conditions. For small values of this fixed exponent, we solve the equation completely. For one such small exponent, we modify and specialize our Thue-Mahler algorithm to the resulting equation in order to fully resolve this case. In the third part, we discuss an algorithm for finding all elliptic curves over ℚ with a given conductor. Though based on classical ideas derived from reducing the problem to one of solving associated Thue-Mahler equations, our approach, in many cases at least, appears to be reasonably efficient computationally. We provide details of the output derived from running the algorithm, concentrating on the cases of conductor p or p², for p prime, with comparisons to existing data. Finally, we specialize the Thue-Mahler algorithm to degree 3, applying an analogue of Matshke-von Kanel’s elliptic logarithm sieve to construct a global sieve, leading to reduced search spaces. The algorithm is implemented in the Magma computer algebra system, and is part of an ongoing collaborative project.

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Twisted Extensions of Fermat's Last Theorem (2015)

Let x, y, z, n, α ∈ ℤ with α ≥ 1, p and n ≥ 5 primes. In 2011, Michael Bennett, Florian Luca and Jamie Mulholland showed that the equation involving a twisted sum of cubes [equation omitted] has no pairwise coprime nonzero integer solutions p ≥ 5,n ≥ p²p and p ∉ S where S is the set of primes q for which there exists an elliptic curve of conductor NE ∈ {18q,36q,72q} with at least one nontrivial rational 2-torsion point. In this dissertation, I present a solution that extends the result to include a subset of the primes in S; those q ∈ S for which all curves with conductor NE ∈ {18q,36q,72q} with nontrivial rational 2-torsion have discriminants not of the form ℓ² or -3m² with ℓ,m ∈ ℤ. Using a similar approach, I will classify certain integer solutions to the equation of a twisted sum of fifth powers [equation omitted] which in part generalizes work done from Billerey and Dieulefait in 2009. I will also discuss limitations of the methods for these equations and as they extendto further prime exponents.

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Thue equations (2014)

In this dissertation, we are mainly interested in effective methods to solveparametrized Thue equations. After briefly talking about the different effective methods, two parametrized families of cubic Thue equations are completelysolved by using Pad é approximation and linear forms in logarithms.The Thue inequality ∣x³ + pxy² + qy³ ∣≤ k,is studied by using Bombieri's method. We find all solutions under some conditions on k, p and q. As an application of Thue equations, we find theintegral points on the Mordell curves Y² = X³ + k for all nonzero integersk with ∣k∣ ≤ 10⁷ . Our approach uses a classical connection between theseequations and cubic Thue equations.

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Thue Equations and Related Topics (2008)

No abstract available.

Master's Student Supervision (2010 - 2020)
On a generalization of Waring's Conjecture (2013)

Pupyrev's paper "Effectivization of a Lower Bound for

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Implementation of a Thu-Mahler equation solver (2011)

A practical algorithm for solving an arbitrary Thue-Mahler equation is presented, and its correctness is proved. Methods of algebraic number theory are used to reduce the problem of solving the Thue-Mahler equation to the problem of solving a finite collection of related Diophatine equations having parameters in an algebraic number field. Bounds on the solutions of these equations are computed by employing the theory of linear forms in logarithms of algebraic numbers. Computational Diophantine approximation techniques based on lattice basis reduction are used to reduce the upper bounds to the point where a direct enumerative search of the solution space becomes possible. Such an enumerative search is carried out with the aid of a sieving procedure to finally determine the complete set of solutions of the Thue-Mahler equation. The algorithm is implemented in full generality as a function in the Magma computer algebra system. This is the first time a completely general algorithm for solving Thue-Mahler equations has been implemented as a computer program.

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