# Jeroen Sijsling

The common denominator of my research is arithmetic geometry. Current subjects of focus include endomorphisms of Jacobians and Galois descent of curves and maps.

## Publications

This is a list of my papers in reverse chronological order, with the arXiv identifier when available. Most of these have some calculation or implementation that goes with them. All these related programs can be found at my GitHub page or at Algorithms.

 17 [1807.02605] Numerical computation of endomorphism rings (with Nils Bruin and Sasha Zotine), accepted by the Proceedings of ANTS-XIII, 16 pp. 16 [1805.07751] A database of Belyi maps (with Michael Musty, Sam Schiavone, and John Voight), accepted by the Proceedings of ANTS-XIII and winner of the Selfridge Prize, 16 pp. 15 [1707.01158] Canonical models of arithmetic $(1; \infty)$-curves, accepted by the Proceedings of AGC2T 2017, 19 pp. 14 [1705.09248] Rigorous computation of the endomorphism ring of a Jacobian (with Edgar Costa, Nicolas Mascot and John Voight) accepted by Math. Comp, 37 pp. 13 [1701.06489] Plane quartics over $\Q$ with complex multiplication (with Pınar Kılıçer, Hugo Labrande, Reynald Lercier, Christophe Ritzenthaler and Marco Streng), accepted by Acta Arithmetica, 25 pp. 12 [1606.05594] Reconstructing plane quartics from their invariants (with Reynald Lercier and Christophe Ritzenthaler), preprint, 36 pp. 11 [1602.03715] A database of genus 2 curves over the rational numbers (with Andrew Booker, Andrew Sutherland, John Voight and Dan Yasaki), LMS J. Comput. Math. 19 (2016), suppl. A, 235–254. 10 [1601.00126] On some bounds for symmetric tensor rank of multiplication in finite fields (with Stéphane Ballet, Julia Pieltant and Matthieu Rambaud), Arithmetic, geometry, cryptography and coding theory, 93–121, Contemp. Math., 686, Amer. Math. Soc., pp. 93--121, 2017. 9 [1510.05601] Distributions of traces of Frobenius for smooth plane curves over finite fields (with Reynald Lercier, Christophe Ritzenthaler, Florent Rovetta and Ben Smith), Exp. Math. (July 2017), 12 pp. 8 [1504.02814] On explicit descent of marked curves and maps, (with John Voight), Res. Number Theory 2 (2016), Art. 27, 35 pp. 7 [1403.0562] Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields (with Reynald Lercier, Christophe Ritzenthaler and Florent Rovetta), LMS Journal of Computation and Mathematics, Volume 17, Special Issue A (ANTS XI), LMS, London, pp. 128--147, 2014. 6 [1311.2529] On computing Belyi maps (with John Voight) Pub. Math. de Besançon 2014 (1), pp. 73--131, 2014. 5 [1301.0695] Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group (with Reynald Lercier and Christophe Ritzenthaler), Math. Comp. 85 (2016), 2011-2045. 4 [1211.1327] An explicit expression of the Lüroth invariant (with Romain Basson, Reynald Lercier and Christophe Ritzenthaler), Proceedings of ISSAC 2013. 3 [1203.5440] Fast computation of isomorphisms of hyperelliptic curves and explicit descent (with Reynald Lercier and Christophe Ritzenthaler), ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium, pp. 463--486, 2013. 2 [PDF] Canonical models of arithmetic (1;e)-curves, Math. Z., 273, pp. 173--210, 2013. 1 [PDF] Arithmetic (1;e)-curves and Belyi maps, Math. Comp., 81 (279), pp. 1823--1855, 2012.

## Thesis: Equations for arithmetic pointed tori

My Ph.D. thesis is concerned with the following questions:

1. Calculating Belyi maps associated with arithmetic Fuchsian groups commensurable with a triangle group;
2. Finding equations for arithmetic curves of genus 1 with a single elliptic point;
3. Explicitly calculating dual graphs for Shimura curves at primes dividing the discriminant of the associated quaternion algebra.

It is in this context that the Escher-like tiling of the disc on the right made its appearance. You can click here to download the thesis as submitted at the time. It contains a somewhat broader and chattier exposition on Shimura curves. Experts may prefer the articles above.

## History of interests

As a Ph.D student in Utrecht under the supervision of Frits Beukers, my research was centered around explicit aspects of Shimura curves. More on this can be found in my thesis below; on the left, a Dirichlet domain for a genus 1 Shimura curve with a single elliptic point of order 3 is given.

I remain interested in the following questions centered around Shimura curves:

1. Finding algorithmic methods to determine the correctness of a conjectural equation;
2. Finding necessary and sufficient conditions on a Belyi map to arise from an inclusion of arithmetic Fuchsian groups that are congruence;
3. Finding a way to bring the power of p-adic geometry and p-adic differential equations to bear on Lamé equations, both for determining accessory parameters and to compare the monodromy groups in the classical and the p-adic situations.

I have been fortunate enough to work on these questions at the MPI in Bonn and after that at the IMPA in Rio de Janeiro (on invitation of Hossein Movasati).

After these projects, I spent a year at the IRMAR in Rennes in the context of Project CHIC. This research, conducted together with Reynald Lercier and Christophe Ritzenthaler, was concerned with covariants and their applications to hyperelliptic and plane curves, as well as the problem of Galois descent, which we managed to solve completely for hyperelliptic curves and plane quartics; these results will soon appear in preprint form. Furthermore, together with Romain Basson we managed to solve a century-old problem on quartics admitting an inscribed pentalateral, as well as disproving an equally old conjecture due to Morley.

This period in France was followed by a Marie Curie grant with Samir Siksek at the University of Warwick on a project concerning dessins d'enfants, thus revisiting the subject of my master's thesis, written under the supervision of Jaap Top. The project was concerned with dessins as a subject in itself, but in fact dessins are intimately related with many areas of mathematics. For example, the question of Galois descent for hyperelliptic curves mentioned above shows that even the theory of degree 2 covers of the projective line is already very rich from an arithmetic point of view.

Before coming to Ulm, I worked at Dartmouth College with John Voight on many questions related to computational aspects of curves. This included both the aforementioned themes of descent and dessins, but also dealt with other questions, such as computational aspects modularity and endomorphisms of Jacobians of curves of small genus. Some of these results and algorithms have been included in the genus 2 part of the LMFDB.

More recently Lercier, Ritzenthaler and I again took up the subject of invariants of curves and we managed to write algorithms to reconstruct a plane quartic from given Dixmier-Ohno invariants. The corresponding preprint can be found below.

I remain interested in basically any questions concerning automorphisms of curves and Galois descent of varieties and dessins, also in positive characteristic. Results obtained so far on Galois descent seem to suggest that there is a more formal theory building on Galois cohomology that would explain their surprising simplicity, a subject that I hope to continue to consider in the future. I also remain very interested in curves of small genus and their arithmetic aspects.