This week's seminars
Pure Maths Colloquium: Is there a prime number theorem in algebraic dynamics?
10 December 2018 16:00 in CM219
There is an (easy) analogue of the prime number theorem (PNT) for polynomials over a finite field k with q elements: there are approximately q^d/d monic irreducible polynomials of degree d. You might also have counted the number of invertible matrices of fixed size n over k by “picking a basis”: (q^n-1)(q^n-q)…. In the context of this talk, both of these results are related to the study of the dynamics of the Frobenius operator (“raising to the power q”) on, respectively, the affine line and the algebraic group GL(n) over the algebraic closure K of k. For the first result, we count “prime” orbits; for the second, we count fixed points.
The general framework of our work is that of counting orbits and fixed points of endomorphisms of algebraic groups over K. We show there is a sharp dichotomy: either the associated dynamical zeta function is a rational function (like the Weil zeta function, e.g. in the above examples), and an analogue of PNT holds; or the zeta function is transcendental, and the set of limit points in PNT is uncountable. In the latter case, the number of fixed points of the k-th iteratre involves p-adic properties of k. The distinction is very similar to dichotomies observed in measurable dynamics (mixing/non-mixing). Sometimes, the dichotomy has a clear geometric interpretation (e.g., on abelian varieties, and on reductive groups, in relation to a famous formula of Steinberg generalizing the count for GL(n)). [Joint work with Jakub Byszewski and Marc Houben.]
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