Marius Leonhardt

Address:
Ruprecht-Karls-Universität Heidelberg
Institut für Mathematik
Im Neuenheimer Feld 205
69120 Heidelberg

Email: mleonhardt (at) mathi (dot) uni-heidelberg (dot) de
Telephone: +49 6221 54 14841
Room number: 04.232

About me

This term (Wintersemester 2024/25) I am a substitute professor for Katharina Hübner at Goethe-Universität Frankfurt.

While in Frankfurt, I am on leave from my position as a postdoc (wissenschaftlicher Mitarbeiter) at the University of Heidelberg in the group of Alexander Schmidt. But I am still part of the SFB-TRR GAUS and in particular I am the person of contact for young researchers within GAUS.

My research focusses on studying rational points on curves with the Chabauty–Kim method. Other research interests include the arithmetic of abelian varieties, especially the theory of complex multiplication, Shimura varieties, and related topics.

I completed my PhD in 2020 under the supervision of Tony Scholl at the University of Cambridge. In my thesis I studied extra “plectic” Galois actions on Hilbert modular varieties.

Before that I completed my studies at the University of Heidelberg (MSc), the University of Cambridge (MAST) and the Karlsruhe Institute of Technology (BSc).

Here is my CV

Publications

• Bounds on the Chabauty–Kim locus of hyperbolic curves (with L.A. Betts and D. Corwin), Int. Math. Res. Not. IMRN, 2024. https://doi.org/10.1093/imrn/rnae067 arXiv.
• Linear and quadratic Chabauty for affine hyperbolic curves (with M. Lüdtke and J.S. Müller), Int. Math. Res. Not. IMRN, 2023. https://doi.org/10.1093/imrn/rnad185, arXiv.
• Plectic Galois action on CM points and connected components of Hilbert modular varieties, Bull. Lond. Math. Soc., 54(6):2254–2277, 2022. http://doi.org/10.1112/blms.12692, arXiv.
• Plectic arithmetic of Hilbert modular varieties, PhD Thesis, University of Cambridge, version 04/2020.

Teaching

This term (WS 2024/25), I am teaching a course on the arithmetic of elliptic curves. The lectures happen on Wednesdays, 14:00 - 16:00 in room 309, Robert-Mayer-Straße 6-8. The course website is here

On 9 October 2024, I gave a talk on modular arithmetic at the Tag der Mathematik in Heidelberg. Here are the slides.

In the summer, Wasilij Barsukow and I were offering a course about the magic of solving or insolubility of (algebraic, Diophantine, differential) equations at the Deutsche Schülerakademie Torgelow for 11th and 12th grade students.

For details and past teaching, click here.

Reading Groups/Oberseminare/GAUS AGs

During my PhD I gave talks in several reading groups about number-theoretic topics. As a postdoc in Heidelberg, these are now called Oberseminare or GAUS AGs. Details can be found here.

Talks

Here you can find a list of talks I have given at research seminars and conferences:

• Affine Abelian Non-abelian Chabauty, Frankfurt GAUS seminar, Oct 2024.
• Affine Chabauty, The Mordell Conjecture 100 years later, Cambridge MA, July 2024. slides of my lightning talk.
• Affine (Quadratic) Chabauty, Journées Arithmétiques Nancy, July 2023. slides (that I did not use)
• Effective Chabauty–Kim for CM elliptic curves, Groningen Algebra Seminar, May 2022.
• Plectic Galois action on Hilbert modular varieties, British Mathematical Colloquium, Glasgow (online), Apr 2021.
• Plectic phenomena on Hilbert modular varieties, seminar talk, Heidelberg, Jan 2020.
• Plectic phenomena on Hilbert modular varieties, KIT Christmas workshop, Dec 2020.
• Plectic phenomena on Hilbert modular varieties, Y-RANT Warwick,Nov 2019.
• Plectic phenomena on Hilbert modular varieties, Journées Arithmétiques Istanbul, July 2019, slides
• L-functions of CM elliptic curves, Y-RANT Sheffield, Nov 2018.
• Background on modular forms at Kleine AG on Deligne’s Travaux de Shimura, Bonn, October 2018
• The main theorem(s) of complex multiplication and beyond, Bristol Linfoot Number Theory Seminar, May 2018.
• Snapshots of Complex Multiplication, Lancaster Junior Seminar, Apr 2018.
• CFT of IQNF via EC with CM, TMS symposium, Feb 2018. slides
• CFT of IQNF via EC with CM, PhD student colloquium, Cambridge, Apr 2017. notes
• Recovering a local field from its Galois group, Cambridge number theory seminar, Feb 2017; similar talks given at the Copenhagen number theory seminar, Dec 2016, and Cambridge Kinderseminar, Nov 2016. notes

Conferences

Here is a list of conferences and workshops I attended/plan to attend.

• Instructional workshop on rational points, Groningen, Nov 2024
• Anabelian and Motivic Homotopy Theory, Heidelberg, Sept 2024
• The Mordell conjecture 100 years later, Cambridge (MA), July 2024
• Winter Workshop Chabauty–Kim, Feb 2024, organised by Tim Holzschuh and myself. Here is the website.
• Journées Arithmétiques, Nancy, July 2023
• An Expedition into Arithmetic Geometry, Leiden, May-June 2023
• Arithmetic Algebraic Geometry, Darmstadt, Oct 2022
• Women in Arithmetic Geometry, Heidelberg, Sept 2022
• Mordell at 100, Cambridge, Aug 2022
• Cohomology of Varieties, Warsaw, Apr 2022
• British Mathematical Colloquium, Glasgow (online), Apr 2021
• Masterclass on Condensed Mathematics, Copenhagen (online), November 2020
• Lean for the curious mathematician, Online, July 2020
• British Mathematical Colloquium, Glasgow, UK, April 2020 (deferred due to the pandemic)
• Arithmetic Geometry, Darmstadt, Germany, March 2020 (deferred due to the pandemic)
• Kleine AG on Lawrence-Venkatesh’s proof of Siegel’s theorem, Bonn, November 2019
• Journées Arithmétiques, Istanbul, Turkey, July 2019
• CMI-HIMR Summer School in Computational Number Theory, Bristol, UK, June 2019
• Kleine AG on Serre’s Modularity Conjecture, Bonn, May 2019 (organised by Christoph Spenke and myself)
• Y-RANT, Sheffield, UK, November 2018
• CMI at 20, Oxford, UK, September 2018
• Kleine AG on Falting’s Endlichkeitssätze für Abelsche Varietäten über Zahlkörpern, Bonn, October 2017
• Journées Arithmétiques, Caen, France, July 2017
• Arizona Winter School, Tucson, USA, March 2017
• Kleine AG on Tate’s p-divisible groups, Heidelberg, February 2017
• $p$-adic methods for Galois representations and modular forms, Barcelona, Spain, February 2017 (including Payman Kassaei’s course on $p$-adic Hilbert modular forms)
• Christmas workshop for Geometry and Number Theory, Karlsruhe, Germany, December 2016
• Galois Representations and Automorphic Forms, Bedlewo, Poland, August 2016
• Crashcourse on Shimura Varieties, Leiden, Netherlands, June 2016
• Christmas workshop for Geometry and Number Theory, Karlsruhe, Germany, December 2015
• Kleine AG on The Neukirch-Uchida theorem, Heidelberg, June 2015

Other stuff I have written

(contact me for the pdfs if they are not linked)

• Hodge-Tate sections, summary of my talk at the GAUS AG on Mochizuki’s proof of the Hom-conjecture, 19/12/2024, Frankfurt.
• Kleine Sätze und große Geheimnisse, slides of my talk at the Tag der Mathematik, 09/10/2024 in Heidelberg.
• Complex Multiplication, notes of an overview talk I gave as part of the GAUS AG on Rigid Meromorphic Cocycles, 27/10/2023.
• The main theorems of complex multiplication, Smith-Knight and Rayleigh Knight Prize Essay, University of Cambridge, 01/2017.
• Galois characterization of local fields, master thesis, University of Heidelberg, supervised by Alexander Schmidt, 09/2015; in German; English summary here.
• The Tate-module and the Weil pairing of an elliptic curve, seminar write-up, University of Heidelberg, supervised by Oliver Thomas and Kay Wingberg, 07/2015; in German; notes here. The rank calculation of the $\mathbb{Z}$-module of isogenies between two elliptic curves (as presented in Silverman) contains a little gap – one needs to use a bit more about the finitely generated submodules. This is necessary since otherwise something like $\mathbb{Z}[\frac{1}{p}]$ would have rank $1$. Thanks to Lennart Gehrmann for pointing it out. The issue is solved here.
• p-adic L-functions, part III essay, University of Cambridge, supervised by Tony Scholl, 05/2014.
• Minkowski’s existence and uniqueness theorem for surface area measures, bachelor thesis, University of Karlsruhe, supervised by Daniel Hug, 07/2012.

Thanks to Sam Power for his help and advice on creating this webpage.