Chebyshev’s bias for elliptic curves over function fields

Since Chebyshev's observation that there seems to be more primes of the form 4n+3 than of the form 4n+1, many other types of ‘arithmetical biases’ have been found. As was observed by Mazur and explained by Sarnak, such a bias appears in the count of points on reductions of a fixed elliptic curve E, and is created by the analytic rank. In this talk we will discuss the analogous question for elliptic curves over function fields. We will first discuss the occurrence of extreme biases, which originate from very different source than in the number field case. Secondly, we will discuss what happens to a ‘typical curve’, and discuss results of linear independence of the zeros of the associated L-functions. This is joint work with Byungchul Cha and Florent Jouve