Moduli of Elliptic Curves and Classical Diophantine Problems

Lead Research Organisation: University of Warwick
Department Name: Mathematics

Abstract

An equation is called Diophantine if we seek solutions that are either whole or fractional numbers. These equations are named after Diophantus of Alexandria who probably lived around the third century AD. However, the subject is far older; for example a Babylonian clay tablet around 4000 years old lists small whole number solutions of what is now known as the Pythagorean equation.

The subject of Diophantine equations was revived and popularised by 17th century French jurist and amateur mathematician Pierre de Fermat. In particular, Fermat's Last Theorem was an open Diophantine problem that captured public imagination for over 250 years and was finally settled by Andrew Wiles in 1994. Whilst the statement of Fermat's Last Theorem and many other Diophantine problems can be understood by any educated person, the discipline is one of the deepest in contemporary mathematics, and builds on profound connections with other mathematical disciplines such as algebraic geometry, analysis and representations theory.

The proposed research comprises of two themes. The first is concerned with modular curves, which in essence are Diophantine equations whose solutions classify certain other kinds of Diophantine objects called elliptic curves. Modular curves play a crucial role in modern number theory, and are the key to several difficult unresolved problems. In this project we develop theoretical and computational tools for studying the arithmetic of modular curves.

The second theme is concerned with certain families of classical Diophantine problems where Baker's theory provides bounds for the solutions but the search regions are so enormous that they are beyond the computational capabilities of even the most powerful computer clusters. We will develop new techniques for sifting search regions using the theory of lattices.

Planned Impact

The proposed research will lead to powerful new methods and techniques that will become invaluable to researchers working on Galois representations of elliptic curves, arithmetic of modular curves, or more broadly the arithmetic of curves, and also to computational number theorists, and the Diophantine approximation community. These will be the primary beneficiaries.

Diophantine equations have always been a tool for popularising mathematics, and the particularly results arising from the envisaged work on classical Diophantine problems will have the potential to capture public imagination, and raise awareness of front-line mathematical research.

Publications

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Derickx M (2020) Elliptic curves over totally real cubic fields are modular in Algebra & Number Theory

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Freitas N (2020) On asymptotic Fermat over Zp-extensions of Q in Algebra & Number Theory

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Freitas N (2021) The unit equation over cyclic number fields of prime degree in Algebra & Number Theory

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Freitas N (2021) Local criteria for the unit equation and the asymptotic Fermat's Last Theorem. in Proceedings of the National Academy of Sciences of the United States of America

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Bennett M (2021) Odd values of the Ramanujan tau function in Mathematische Annalen

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Siksek S (2021) Sums of integer cubes. in Proceedings of the National Academy of Sciences of the United States of America

 
Description The work towards the objectives of this award has resulted in a radically new research direction, which concerns studying rational points on a curve over a family of number fields. The idea is that extending the base field almost never results in new points.
Exploitation Route The research has initiated new ideas on the Galois theory of algebraic points and their relation to the geometry of curves. In particular, it is now clear that there is a relationship between the Galois theory of a divisor on a curve and its complete linear series. This is a largely new subject with much potential for further exploration and results.
Sectors Other

URL https://homepages.warwick.ac.uk/staff/S.Siksek/
 
Description The research from this grant has been featured in an article in following popular articles: ¿Todo número entero es la suma de, como mucho, nueve cubos?, El Pais, 12 June 2020. Needles in an infinite haystack, interview with Dana Mackenzie in What's new in the mathematical sciences, AMS, 2020.
First Year Of Impact 2020
Sector Other
Impact Types Cultural

 
Description National Academy consultation
Geographic Reach National 
Policy Influence Type Contribution to a national consultation/review
 
Title Thue-Mahler Solver 
Description This is a freely available implementation of the algorithm of Gherga (PDRA) and Siksek (PI) for solving Thue-Mahler equations. It is described in the paper, "Efficient Resolution of Thue-Mahler Equations" by Gherga and Siksek, which is submitted for publication (but available at https://doi.org/10.48550/arXiv.2207.14492) 
Type Of Material Improvements to research infrastructure 
Year Produced 2022 
Provided To Others? Yes  
Impact This program, in combination with an algorithm of Bennett, Gherga and Rechnitzer, allows for the efficient enumeration of elliptic curves of a given conductor. 
URL https://github.com/adelagherga/ThueMahler/tree/master/Code/TMSolver
 
Description Galois groups of points on curves 
Organisation University of Sheffield
Country United Kingdom 
Sector Academic/University 
PI Contribution A conjecture was made by PI Siksek regarding the distribution of algebraic points with large Galois groups on hyperbolic curves. This shifts the emphasis from studying algebraic points by degree to studying algebraic points by Galois group.
Collaborator Contribution Khawaja (Sheffield) supplied examples which confirm the conjecture in particular cases. The PI and Khawaja collaborated to generalise these examples and formulated the notion of primitive points (points whose Galois group is primitive as a permutation group). They showed that the geometry of the curve, via the Castelnouvo-Severi inequality presents an obstruction to such points, and moreover that the Riemann-Roch spaces of divisors corresponding to such points have relatively small dimension. This work connected the arithmetic and geometry of curves to new research on fixed point ratios of groups by Burness and Guralnick.
Impact (Khawaja and Siksek) A single source theorem for primitive points on curves, 2023, submitted for publication. (Khawaja and Siksek) Primitive algebraic points on curves, 2024, submitted for publication.
Start Year 2022
 
Description Ramanujan tau-function 
Organisation University of British Columbia
Country Canada 
Sector Academic/University 
PI Contribution PI Siksek and PDRA Gherga developed new techniques for solving Thue-Mahler equations of high degree. In the ternary setting, these can make use of information coming from Galois representations of Frey curves.
Collaborator Contribution Bennett (UBC) and Patel (Manchester) related arithmetic properties of the Ramanujan-tau function to ternary Thue-Mahler problems. Working with Siksek and Gherga this has resulted in new theorems for perfect powers from values of the Ramanujan-tau function. Follow up work by Bennett and Siksek modified the techniques to obtain new results for multiparameter Ramanujan-Nagell equations with the help of a multi-Frey approach.
Impact (Bennett, Michaud-Jacobs and Siksek) Q-curves and the Lebesgue-Nagell equation, Journal de Théorie des Nombres de Bordeaux 35 (2023), 335-370. (Bennett and Siksek) Differences between perfect powers : prime power gaps , Algebra & Number Theory 17 (2023), 1789-1846. (Bennett and Siksek) Differences between perfect powers : the Lebesgue-Nagell Equation , Transactions of the AMS 376 (2023), 225-370. (Bennett, Gherga, Patel and Siksek) Odd values of the Ramanujan tau function, Math. Ann. 382 (2022), 203-238.
Start Year 2020
 
Description Ramanujan tau-function 
Organisation University of Manchester
Country United Kingdom 
Sector Academic/University 
PI Contribution PI Siksek and PDRA Gherga developed new techniques for solving Thue-Mahler equations of high degree. In the ternary setting, these can make use of information coming from Galois representations of Frey curves.
Collaborator Contribution Bennett (UBC) and Patel (Manchester) related arithmetic properties of the Ramanujan-tau function to ternary Thue-Mahler problems. Working with Siksek and Gherga this has resulted in new theorems for perfect powers from values of the Ramanujan-tau function. Follow up work by Bennett and Siksek modified the techniques to obtain new results for multiparameter Ramanujan-Nagell equations with the help of a multi-Frey approach.
Impact (Bennett, Michaud-Jacobs and Siksek) Q-curves and the Lebesgue-Nagell equation, Journal de Théorie des Nombres de Bordeaux 35 (2023), 335-370. (Bennett and Siksek) Differences between perfect powers : prime power gaps , Algebra & Number Theory 17 (2023), 1789-1846. (Bennett and Siksek) Differences between perfect powers : the Lebesgue-Nagell Equation , Transactions of the AMS 376 (2023), 225-370. (Bennett, Gherga, Patel and Siksek) Odd values of the Ramanujan tau function, Math. Ann. 382 (2022), 203-238.
Start Year 2020
 
Description The Unit Equation and the Fermat Equation 
Organisation Institute of Mathematical Sciences
Country Spain 
Sector Charity/Non Profit 
PI Contribution Siksek (PI) introduced new Mordell-Weil sieve ideas into the context of unit equations. These new sieving techniques relate local obstructions to global obstructions to the solutions of unit equations. Using these techniques it is now possible to rule out solutions to the unit equation for many families of number fields where the only restriction is a local restriction on the behaviour of one prime.
Collaborator Contribution Partners Nuno Freitas (ICMAT, Madrid), and Alain Kraus (Sorbonne Universite) worked with the PI to show that the sieving technique to provide obstructions to the Fermat equation over number fields. This has resulted in a proof of the asymptotic Fermat's Last Theorem for several infinite families of number fields where the restriction is a local one at 2 and one other odd prime.
Impact (Freitas, Kraus, Siksek) Local criteria for the unit equation and the asymptotic Fermat's Last Theorem , Proceedings of the National Academy of Sciences 118 (2021), No. 12, on arXiv (Freitas, Kraus, Siksek) The unit equation over cyclic number fields of prime degree , Algebra & Number Theory 15 (2021), 2647-2653. on arXiv (Freitas, Kraus, Siksek) Chevalley's class number formula, unit equations and the asymptotic Fermat's Last Theorem on arXiv (Freitas, Kraus, Siksek) On Asymptotic Fermat over Zp-extensions of Q, Algebra & Number Theory 14 (2020), 2571-2574. on arXiv
Start Year 2020
 
Description The Unit Equation and the Fermat Equation 
Organisation Sorbonne University
Country France 
Sector Academic/University 
PI Contribution Siksek (PI) introduced new Mordell-Weil sieve ideas into the context of unit equations. These new sieving techniques relate local obstructions to global obstructions to the solutions of unit equations. Using these techniques it is now possible to rule out solutions to the unit equation for many families of number fields where the only restriction is a local restriction on the behaviour of one prime.
Collaborator Contribution Partners Nuno Freitas (ICMAT, Madrid), and Alain Kraus (Sorbonne Universite) worked with the PI to show that the sieving technique to provide obstructions to the Fermat equation over number fields. This has resulted in a proof of the asymptotic Fermat's Last Theorem for several infinite families of number fields where the restriction is a local one at 2 and one other odd prime.
Impact (Freitas, Kraus, Siksek) Local criteria for the unit equation and the asymptotic Fermat's Last Theorem , Proceedings of the National Academy of Sciences 118 (2021), No. 12, on arXiv (Freitas, Kraus, Siksek) The unit equation over cyclic number fields of prime degree , Algebra & Number Theory 15 (2021), 2647-2653. on arXiv (Freitas, Kraus, Siksek) Chevalley's class number formula, unit equations and the asymptotic Fermat's Last Theorem on arXiv (Freitas, Kraus, Siksek) On Asymptotic Fermat over Zp-extensions of Q, Algebra & Number Theory 14 (2020), 2571-2574. on arXiv
Start Year 2020
 
Description Interview with El Pais 
Form Of Engagement Activity A press release, press conference or response to a media enquiry/interview
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Media (as a channel to the public)
Results and Impact Article in El Pais newspaper which is the Spanish language newspaper with the largest circulation (75 million unique browsers per month).

The article was about the PI's work on Jacobi's the sums of seven cubes problem and intended for the general public.
Year(s) Of Engagement Activity 2020
URL https://elpais.com/ciencia/2020-06-12/todo-numero-entero-es-la-suma-de-como-mucho-nueve-cubos.html#
 
Description Needles in an Infinite Haystack 
Form Of Engagement Activity A press release, press conference or response to a media enquiry/interview
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Schools
Results and Impact This is an interview with journalist Dana Mackenzie concerning the collaboration between Siksek (PI) and Freitas (Madrid) and Kraus (Paris). It focused on the links uncovered between the Fermat equation and the unit equation. The article is intended for a broad audience.
Year(s) Of Engagement Activity 2020
URL http://www.ams.org/publicoutreach/math-history/happening-series#vol9