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Dr Oscar Rivero Salgado, Newton International Fellowship

Euler Systems and Congruences Among Modular Forms

Number theory is the branch of mathematics which studies natural numbers and their relations. Its biggest goal is solving equations involving these numbers, which may seem simple at first sight, but which can be extremely complicated.

For instance, one example already known by the Greeks is the problem of finding all the Pythagorean triples, that is, natural numbers x, y and z such that x2+y2=z2. An ostensibly harder challenge is Fermat’s Last Theorem, which asks for triples of natural numbers, such that xn+yn=zn, where n is now an integer number greater than 2.

Its resolution took more than 300 years, introduced many interesting concepts and raised other questions. One of them is the Birch and Swinnerton-Dyer conjecture, one of the six unsolved millennium problems posed by the Clay Mathematics Institute, with very deep applications and which draws tantalizing connections among different mathematical objects.

Roughly speaking, the conjecture tries to predict how many rational points an elliptic curve has. Elliptic curves are a central actor in arithmetic, even with applications to cryptography!

Examples of elliptic curves.

Euler systems have relevance to different areas of mathematics, including algebraic geometry, complex analysis and representation theory.

The quest for proof of the conjecture

Not many results towards the proof of this conjecture have been done in the last thirty years. In the 90s, the mathematicians Kolyvagin and Rubin envisaged the notion of Euler system, which was rather powerful to successfully attack several cases of the conjecture. In recent years, there has been an enormous growth in that direction. This has helped create a better understanding of the concept and its relations with other problems. This also draws a fascinating theory which combines elements from different areas of mathematics: algebraic geometry, complex analysis and representation theory.

Dr Oscar Rivero Salgado's Fellowship is devoted to achieving a better understanding of the relations between different Euler systems and to derive some applications to problems like the Birch and Swinnerton-Dyer, but also to other intriguing questions like the celebrated Gross-Stark conjectures or the Iwasawa main conjecture.

Find out more about Dr Oscar Rivero Salgado's research