Seminar of Algebra

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From triangles to the research edge of arithmetic

Harald Grobner (University of Vienna, Austria)
Departamento de Álgebra
Thu, 5 feb 2015 13:00
My main mathematical interests are located within the realm of cohomology of arithmetic groups and its applications to number theory. In particular, I am interested in the arithmetic and analytic theory of automorphic forms and their associated L-functions.

In his book Liber Quadratorum (Book of Squares) from 1225, Fibonacci studied 3-term arithmetic progressions of integral squares, i.e., triples of the form $(x^2-n, x^2, x^2+n)$ with both $x^2 \pm n$ being squares. More generally, allowing triples of squares of rational numbers, the question of which triples of that form (Fibonacci's “congrua”) exist, is equivalent to asking which integers $n$ can be written as the area of a right triangle with rational sides. As we will see, this at first sight maybe simple looking question, leads us directly to consider elliptic curves and their rational points. This, in turn, is intimately linked to Fermat's Last Theorem and its proof by Wiles and Taylor.

The passage from elliptic curves to Fermat's Last Theorem is provided by cuspidal modular forms. These are certain functions on the Siegel upper half-space, which might be viewed as the most classical instance of what one would nowadays call an automorphic form. If time permits, we will shortly sketch the eminent role of automorphic forms in the Langlands Program, a program, which, from some perspective, generalizes some ideas behind the proof of FLT.