Graphs, Elliptic and Modular curves is an institutional research project funded by NextGenerationEU through the call for institutional research projects – Projects for Raising Scientific Excellence. The project deals with number theory and discrete mathematics.
The project team consists of five members of the Faculty of Civil Engineering and two external collaborators:
Nikola Adžaga, project leader
Tomislav Došlić
Alan Filipin
Timo Keller, IBM Deutschland Research & Development
Oana Padurariu, Max-Planck-Institut für Mathematik
Luka Podrug
Bruno Zadravec
We study elliptic and modular curves, the dual graphs of their reductions, and other Diophantine equations and systems. We use a combination of classical methods (such as linear forms in logarithms, Diophantine approximation and Baker–Davenport reduction) and algorithmic methods of arithmetic geometry (such as the Chabauty method). We have completed the classification of torsion groups of elliptic curves over sextic fields. We are developing a modular algorithm for determining the dual graph of a quotient of the classical modular curve. We will then apply this algorithm to compute local heights and to rigorously determine the set of rational points on these curves. We study Diophantine m-tuples and their generalizations, as well as a generalization of Euler numbers in the context of walks on graphs. The algorithms, results and papers we develop will be made publicly available.
Papers (preprints):
• N. Adžaga, T. Gužvić: Completing the classification of torsion subgroups for rational elliptic curves over sextic fields, https://arxiv.org/abs/2602.14718
• N. Adžaga: Arithmetic of elliptic curves induced by regular Diophantine triples, https://arxiv.org/abs/2608.01057
• N. Adžaga, M. Derickx, T. Keller: Semi-stable models, local heights and quadratic Chabauty for X0(N)*, https://arxiv.org/abs/2609.40207
The code for these papers is publicly available on GitHub: github.com/NikolaAdzaga/SexticTorsion, github.com/NikolaAdzaga/TorsionRegularTriples, github.com/nt-lib/local-heights-X0Nstar
Talks:
• ChaBONNty, Max-Planck-Institut für Mathematik, Bonn, 29 June – 3 July 2026
• Modular curves and Applications of AI to Number theory, Opatija, 14–18 September 2026
Organized events:
• CroCoDays, a meeting on combinatorics and graph theory, September 2026
Pristupačnost