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Galois Representations and Diophantine Problems

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Removing restrictions of only whole-number solutions to Fermat’s Last Theorem

For centuries, mathematicians tried to solve Fermat’s Last Theorem using only natural numbers. In a first, EU-funded scientists expanded this number system to include bigger number systems with exotic values.

Fundamental Research

Natural numbers (positive integers) are not always enough to solve a problem. Over centuries, mathematicians realised this when seeking to solve Fermat’s Last Theorem which states that no three positive integers x, y and z can satisfy the equation xn + yn = zn for any integer value n > 2. This simple statement became the most famous open problem in mathematics. It tormented swarms of mathematicians for over 350 years since the lawyer and amateur mathematician Pierre de Fermat scribbled it in the margin of a copy of Diophantus’ ‘Arithmetica’. Diophantine equations, name after Diophantus of Alexandria, are combinations of variables, exponents and coefficients, such as 3x + 7y = 1 or x3 + y3 = z3. Since ancient times, mathematicians have known how to work out whole-number combinations to solve Diophantine equations with two variables and no exponents larger than two. The oldest known record comes from Plimpton 322, a Babylonian clay tablet believed to have been written about 1800 BC. The tablet which was discovered in 1920 contains 15 rows of Pythagorian triples. “A Pythagorean triple is a triple of whole numbers (x,y,z) that form the sides of a right-angled triangle. The corresponding Diophantine equation is x2 + y2 = z2,” notes Samir Siksek, coordinator of the GalRepsDiophantine project that was funded under the Marie Skłodowska-Curie programme. “Fermat’s conjecture implies that if you push the exponent value above 2 then the equation is fundamentally different from Pythagorean triples.”

Wiles’ monumental breakthrough moment

The only case of his theorem that Fermat actually proved and has survived intact is the case n = 4. Leonhard Euler found a proof for n = 3, and Sophie Germain proved Fermat's Last Theorem for a very large set of prime exponents n. The complete proof was found by the British mathematician Andrew Wiles in 1995. It relied on three concepts in number theory: elliptic curves, modular forms and Galois representations. “In the 1980s, Gerhard Frey proposed an astonishing link between the Fermat’s conjecture and a deep idea called the modularity conjecture for elliptic curves. Frey strongly suspected that the elliptic curves over the field of rational numbers are not modular. Frey’s non-modularity was proven a few years later. Wiles proved Fermat's Last Theorem by proving the semistable case of the modularity conjecture,” elaborates Siksek. Wiles’ proof predicts that the residual Galois representation of that elliptic curve comes from a finite computable set of modular Galois representations.

Wiles solution is a piece of a much larger puzzle

Whilst Wiles succeeded in resolving Fermat’s conjecture over the rationals, the proof strategy for many other Diophantine problems (including the Fermat equation over number fields) is insufficient. “Modern studies focus on Diophantine equations over other number systems as well. One can even think about towers of number systems, where the numbers are becoming increasingly abundant. It is natural to wonder whether the ideas of Frey, Wiles and others that led to the stunning proof of Fermat's Last Theorem can reveal clues to the Fermat equation over an infinite family of number fields of arbitrarily large dimensions,” explains Siksek. “Our study has been the first to consider the Fermat equation for towers containing infinite number systems. In particular, we succeeded in proving the asymptotic Fermat's Last Theorem over the layers of the Z2 extension of the rationals,” concludes Siksek who jointly worked with Marie Skłodowska-Curie fellow Nuno Freitas. Project results are published here.

Keywords

GalRepsDiophantine, Wiles, Diophantine equations, elliptic curves, Fermat’s Last Theorem, Galois representations, rationals, modularity conjecture, whole number

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