University of Luxembourg
specializes in arithmetic geometry, arithmetic statistics, and in algebraic, analytic, and probabilistic number theory. His main focus is on positive characteristic. He enjoys studying objects connected to the Birch and Swinnerton-Dyer conjecture, such as Abelian varieties, their Mordell-Weil groups and their L-functions. He also takes an interest in the interaction between algebra and combinatorics.
works on the arithmetic of abelian varieties and isogenies. During the first months of his PhD, he got interested in the distribution of Elkies primes for reductions of abelian varieties, which is important to assess the time complexity of the SEA point-counting method and relevant to cryptography. He also worked on two variants of a conjecture of Lang and Trotter on primitive points for the reductions of elliptic curves.
works on problems related to the reductions of elliptic curves and, more generally, abelian varieties. He investigates, in particular, local-global principles and questions related to the arithmetic of points, also with a computational approach.
works on Artin's Conjecture on primitive roots for quadratic fields. Conditionally under GRH, he obtained optimal bounds for the ratio between the density in this conjecture and a suitable Artin constant. He also investigated other questions related to Kummer theory of number fields.
is developing a p-adic approach to both CM theory and RM theory, with the aim to determine the values of modular and other arithmetic functions at special points. This p-adic perspective extracts these values from the deformation theory of the Galois representations attached to various Eisenstein series and theta series, and has applications to explicit class field theory.
works on the interplay between Galois representations and modular forms. More precisely, her current research focuses on a Katz newform theory for the Eisenstein space in positive characteristic.
works broadly in arithmetic geometry. In particular, he studies explicit Galois representations, focusing on the representations of unit groups of number rings. He also works on curves over finite fields, aiming to construct curves with many rational points for applications in code-based cryptography. Additionally, he is involved in Hopf Galois theory, where he studies the construction of Hopf Galois objects via abelian varieties and Drinfeld modules.
works on the growth of Mordell–Weil ranks of non-split Cartan modular curves under base change from ℚ to number fields. She has also studied genus 5 "maximal" curves over finite fields with discriminant −19, proving non-existence in some cases. Additionally, she has worked on the Galois action on torsion and division points of connected commutative algebraic groups over number fields.
is developing Kummer theory, a classical and foundational theory which concerns field extensions generated by radicals. She has also extensively worked on the reductions of algebraic groups, and in particular on arithmetic questions related to the divisibility of points. Recently, she has investigated Artin's Conjecture on primitive roots.
works in cryptography, with a focus on Fully Homomorphic Encryption (FHE), which enables computations on encrypted data without decryption. His main interest lies in studying and developing algorithms for efficient bootstrapping in the CKKS scheme.
mostly works in Algebraic Number Theory and Arithmetic Geometry, particularly, on the mathematics issued from the proof of Fermat's Last Theorem, i.e. the interplay between Galois representations and modular forms. He combines theoretical methods with computational and experimental approaches, and also takes interest in cryptography.