Colloquium Mathematics - dr. J.S. Müller (Bernoulli Instituut, RUG)
|When:||Tu 23-10-2018 16:00 - 17:00|
|Where:||5161.0293 (Bernoulliborg, Zernike)|
Solving hyperelliptic diophantine equations
A hyperelliptic diophantine equation is an equation of the form $y^2 = f(x)$, where $f$ is a polynomial with integral coefficients without repeated roots, and one is interested in integral or rational solutions. These equations have been studied for thousands of years, but it is still only possible to describe all integral or rational solutions in specific cases. Nowadays such equations are often studied from a geometric point of view. In this setting, one is interested in integral or rational points on the curve defined by the equation.
I will discuss joint work with Jennifer Balakrishnan and Amnon Besser which allows us to find the integral points on a hyperelliptic curve if a certain geometric condition is satisfied. For this, we use a combination of geometric, algebraic, (p-adic) analytic and algorithmic techniques. If time permits, I will also talk about a generalization which sometimes lets us compute the rational points.