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Colloquium Mathematics, Professor P. Stevenhagen

24 June 2014

Join us for coffee and tea at 15.30 p.m.


Tuesday, June 24th 2014


Prof. Peter Stevenhagen,
University Leiden


5161.0293 (Bernoulliborg)



Title:   From Artin’s conjecture to Serre curves


In 1927, the German mathematician Emil Artin conjectured that for any non-square integer x different from -1, the powers of x will fill up all (non-zero) residue classes modulo a prime number p for infinitely many p. For instance, the powers of 2 fill up the residue classes modulo 3, 5, and 11, but not modulo 7 or 17. The conjecture also predicts a density for the set of such primes p, which for x=2 it is about 37%. I will explain why the conjecture is still unproved for all values of x, but can be proved if one is willing to assume the generalized Riemann hypothesis. Then, I will focus on the computation of the associated densities, for which Artin made a mistake that was only discovered after computer calculations in the 1950’s(!). The computation of such densities now be realized in a more conceptual way that allows various generalizations, of which I will discuss the example of Serre curves. These are elliptic curves over the rationals for which the associated Galois representation on their torsion points is `maximal’ in a sense that I will define.

Colloquium coordinators are Prof.dr. A.C.D. van Enter (e-mail : and
Dr. A.V. Kiselev (e-mail: )

Last modified:06 June 2018 2.05 p.m.

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