About me
I am a postdoctoral researcher in the group of Steffen Müller at the University of Groningen, Netherlands.
Research interests
I am interested in arithmetic geometry, specifically the use of fundamental groups to study rational and integral points of schemes. The Chabauty–Kim method, while rooted in deep considerations about the motivic fundamental group and still being largely conjectural, has nevertheless proved very successful in explicitly solving polynomial equations in recent years. I am exploring both the theoretical and practical aspects of the method, in particular in the setting of S-integral points on affine hyperbolic curves. What I find exciting about this line of research is that deep ideas from geometry can be used to solve very concrete problems from number theory.
I am moreover interested in Grothendieck’s conjectures on anabelian geometry which he formulated in his letter to Faltings. There, the Section Conjecture predicts a description of rational points purely in terms of étale fundamental groups. Recently, with A. Betts and T. Kumpitsch, we could prove a precise relation between the Section Conjecture and the Chabauty–Kim method which brings the conjecture closer to the realm of explicit calculations.
Keywords
Chabauty–Kim method, section conjecture, arithmetic fundamental group, rational points
Recent research
When applying the Chabauty–Kim method to study S-integral points on affine hyperbolic curves, a refined version of the method due to A. Betts and N. Dogra often gives stronger results. The first explicit calculations using this method were carried out by our project group at the 2020 Arizona Winter School. We could solve the S-unit equation in cases where S contains 2 primes (see here for the paper and here for the Sage code). However, these calculations in depth 2 were insufficient to verify Kim’s conjecture for S = {2,3} for any choice of auxiliary prime p. Recently, I derived new Kim functions for the Chabauty–Kim locus in depth 4 and used them to verify Kim’s conjecture for S = {2,3} for all p < 10,000. The preprint is here and the accompanying Sage code is here.
With D. Corwin and I. Dan-Cohen, we are developing and applying the mixed Tate motivic Chabauty–Kim method for the thrice-punctured line beyond the polylogarithmic quotient of the fundamental group. We obtain new constraints on Chabauty–Kim loci by using functions involving p-adic multiple polylogarithms (rather than just single polylogarithms). The preprint will be available soon.
Kim’s conjecture predicts that the set of p-adic points produced by the Chabauty–Kim method agrees precisely with the set of S-integral points of the curve. With A. Betts and T. Kumpitsch we showed that one can prove the “locally geometric” version of Grothendieck’s Section Conjecture by verifying Kim’s conjecture for almost all choices of p. We successfully carried this out for Z[1/2]-integral points on the thrice-punctured line, thereby obtaining the first example of a curve where Kim’s conjecture is proved for infinitely many choices of p. The preprint is here.
For a smooth projective hyperbolic curve the classical Chabauty method applies whenever the rank-genus inequality g – r > 0 holds; the quadratic Chabauty method applies whenever g – r + ρ – 1 > 0 holds. With J. S. Müller and M. Leonhardt we determined the analogous inequalities for S-integral points on affine hyperbolic curves. We also obtain bounds on the number of S-integral points whenever these inequalities are satisfied. The paper is here.
The p-adic section conjecture for a smooth projective curve X over a finite extension K of Qp predicts that every section of the fundamental exact sequence is induced by a K-rational point. Previously, only the birational variant, where X is replaced by its generic point, was known. In my dissertation I extended this result to larger localisations of X. The dissertation is here.