Research

Diophantine equations and moduli schemes

Since 2011, my research focuses on the development of an effective strategy (surveyed in [2]) for Diophantine equations related to moduli schemes. To get an idea, see the IAS talk (2011) on a first application to the classical Mordell equations \(x^3-y^2=a\).

[1] Rational points and rational moduli spaces (2025): With S. Fan, 61pp, submitted (abstract, arXiv).
[2] Integral points on moduli schemes (2023): Conference proceedings Biennal JNT 2022, 71pp, JNT 2025 (abstract, journal).
[3] Integral points on (coarse) Hilbert moduli schemes: Joint project with A. Kret consisting of two parts.
    (i) Integral points on Hilbert moduli schemes (2019): With A. Kret, 53pp, submitted (abstract, arXiv).
    (ii) Integral points on coarse Hilbert moduli schemes (2023): With A. Kret, 150pp, submitted (abstract, arXiv).
[4] Solving classical Diophantine equations: Joint project with B. Matschke consisting of three parts.
    (i) Solving \(S\)-unit, Mordell, Thue, Thue–Mahler and generalized Ramanujan–Nagell equations via Shimura–Taniyama conjecture (2016): With B. Matschke, 163pp, Memoirs of AMS 2023 (abstract, arXiv, journal).
    (ii) Construction of rigorous algorithms (source code) which allow to efficiently solve various classical Diophantine problems.
    (iii) Application of the algorithms in (ii) to obtain unconditional databases containing all solutions of large classes of classical Diophantine problems: The data motivates classical conjectures and it led to the discovery of new conjectures in (i).
[5] Modularity and integral points on moduli schemes (2013, abstract, arXiv). This project consists of two parts: Sections 1-7 are published in (i), and the remaining part is published in (ii) except Section 9.4 which was included in [3].
    (i) Integral points on moduli schemes of elliptic curves (2013): 31pp, Trans. London Math. Soc 2014 (abstract, journal).
    (ii) The effective Shafarevich conjecture for abelian varieties of \(\text{GL}_2\)-type (2013): 39pp, FoM Sigma 2021 (abstract, journal).

Effective Shafarevich conjecture for curves and conjectures of Szpiro

Another part of my research is about the effective Shafarevich conjecture for curves over number fields (in particular curves of genus one and two) and about conjectures of Szpiro (small points conjecture and discriminant conjecture).

[6] Szpiro’s small points conjecture for cyclic covers (2012): With A. Javanpeykar, 20pp, Doc. Math. 2014 (abstract, arXiv, journal).
[7] On Szpiro’s discriminant conjecture (2012): 35pp, IMRN 2014 (abstract, arXiv, journal).
[8] An effective proof of the hyperelliptic Shafarevich conjecture (2011): 24pp, JTNB 2014 (abstract, arXiv, journal).
[9] An effective Shafarevich theorem for elliptic curves (2010): With C. Fuchs, G. Wüstholz, 16pp, Acta Arith. 2011 (abstract, journal).
[10] An effective proof of the hyperelliptic Shafarevich conjecture and applications (2010): PhD thesis, published online in Research Collection of ETHZ (abstract, Phd Thesis).