MIT Number Theory Seminar
Tuesday, November 19, 2019 at 4:30pm to 5:30pm
MIT Room 2-143 -- MIT - Dept. of Mathematics, 77 Mass. Ave., Cambridge, MA
SPEAKER: Michael Stoll (Universität Bayreuth)
TITLE: An application of "Selmer group Chabauty" to arithmetic dynamics
The irreducibility or otherwise of iterates of polynomials is an important question in arithmetic dynamics. For example, it is conjectured that whenever the second iterate of x2 + c (with c a rational number) is irreducible over Q, then so are all iterates.
A sufficient criterion for the iterates to be irreducible can be expressed in terms of rational points on certain hyperelliptic curves. We will show how to use the "Selmer group Chabauty" method developed by the speaker to determine the set of rational points on a hyperelliptic curve of genus 7. This leads to a proof that the seventh iterate of x2 + c must be irreducible if the second iterate is. Assuming GRH, we can extend this to the tenth iterate.