Pure Maths Colloquium: A little bit of number theory: Langlands, quadratic forms, elliptic curves, and modular forms.
28 April 2008 16:15 in CM221
This talk will be very elementary and perfectly accessible for the whole departmental family (from pure to applied, from postgraduate to upper undergraduate).
We discuss two seemingly completely unrelated topics in number theory.
On one hand, we will consider certain positive definite quadratic forms in
four variables and study their representation numbers. These are
variants of the question in how many ways one can write a given integer as
the sum of 4 squares.
On the other hand, we will study a particular elliptic curve, that is, a
certain cubic equation in the xy-plane. In particular, we will investigate
the solutions of this equation over the finite field with p elements,
where p is a prime.
We will see that these two problems are in fact closely related. The
explanation of such a relationship leads naturally to questions in modern
number theory. In particular, we will outline the role of modular forms in
this context and give a glimpse of the so-called Langlands program, which
asserts far reaching generalizations of the examples presented in this
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