14th Bucharest Number Theory Days June 16-17, 2026

Schedule

Abstracts

Tuesday, June 16th

Gergely Harcos, Tatuzawa's theorem for Rankin-Selberg L-functions
In 2023, Jesse Thorner and I established a new zero-free region for all GL(1)-twists of GL(m)xGL(n) Rankin-Selberg L-functions, generalizing Siegel's celebrated work on Dirichlet L-functions. In the talk, I will discuss our recent strengthening of this result, which generalizes Tatuzawa's refinement of Siegel's theorem.

Elisa Lorenzo Garcia, An arithmetic intersection for squares of elliptic curves with complex multiplication
Let C be a genus 2 curve with Jacobian isomorphic to the square of an elliptic curve with complex multiplication by a maximal order in an imaginary quadratic field of discriminant \(-d\leq 0\). We show that if the stable model of C has bad reduction over a prime p then \(p \leq d/4\). We give an algorithm to compute the set of such \(p\) using the so-called refined Humbert invariant introduced by Kani. Using results from Kudla-Rapoport and the formula of Gross-Keating, we compute for each of these primes \(p\) its exponent in the discriminant of the stable model of C. We conclude with some explicit computations for \(d\leq 100\) and compare our results with an unpublished arithmetic intersection formula by Rodriguez Villegas.

Christophe Ritzenthaler, A story which converges
Consider a finite family of curves, typically all curves of a given genus over a finite field. What can be said about the proportion of the curves in this family with a given number of rational points? This will be our starting point to explore some results of arithmetic statistics, from the very general framework of Katz-Sarnak established in the 90's to a series of recent results pointing towards a more precise knowledge of the asymptotic distribution.

Diana Mocanu, Local points on twists of X(p)
Let \(E\) be a rational elliptic curve and \(p\) an odd prime. The modular curve \(X_E^{-}(p)\) parametrizes elliptic curves with \(p\)-torsion modules anti-symplectically isomorphic to \(E[p]\). In this talk, I present my recent work with Nuno Freitas on a complete classification for when these curves admit points everywhere locally. In the second half of the talk, we will see a Diophantine application of this result. More precisely, we use the modular method, together with our classification theorem to prove that certain generalized Fermat equations have no non-trivial, coprime solutions.

Alex Pascadi, On bilinear forms with Kloosterman sums
We introduce a new approach to bounding bilinear (Type II) sums of Kloosterman sums beyond the Pòlya-Vinogradov range. Such sums play a crucial role in analytic number theory, with connections to combinatorics, algebraic geometry, and automorphic forms. Our approach is based on a fourth and less explored connection, to words in \(\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})\) - which in turn leads to representation theory and quadratic character sums. We briefly mention applications to moments of twisted cuspidal L-functions and to large sieve inequalities for exceptional cusp forms. This is based on arxiv.org/abs/2511.08445 and on work in progress with Valentin Blomer.

Cristian Popescu, Determinantal Ideals in Equivariant Iwasawa Theory and Applications
I will give a brief overview of the prominent role played in recent years by determinantal (Fitting) ideals and Grothendieck determinants of complexes of Iwasawa modules in the study of special values of equivariant motivic \(L\)-functions. Applications to recent progress on several central conjectures in the area will be discussed.

Wednesday, June 17th

Oana Padurariu, Shimura curve Atkin-Lehner quotients of genus at most two
Shimura curves are the compact cousins of modular curves, yet finding defining equations for them is challenging. In this talk, I will explain how one may be able to find explicit models for some quotients of Shimura curves when the genus is either one or two. This is joint work with Freddy Saia.

David Kohel, Supersingular isogeny graphs with level structure
The isogeny graphs of supersingular elliptic curves play an increasing role in post-quantum cryptography. For given prime \(\ell\), the adjacency matrix of \(\ell\)-isogeny graphs can be interpreted as a Hecke operator associated to the modular curve \(X_0(p)\), acting on points on the \(j\)-line \(X(1)\). We introduce analogous isogeny graphs with level structure associated to arbitrary covers \(X(\Gamma) \to X(1)\) of modular curves. The original application of such supersingular modules (as introduced by Mestre in his "Méthode des graphes") was to the study of modular forms. We describe how this construction permits one to prove existence of modular elliptic curves with prescribe reduction type beyond existing databases. In the application to cryptography, certain higher level structure permits one to efficiently compute isogeny relations in these graphs using smaller modular polynomials. This is joint work with Leonardo Colò.

Sorina Ionica, Computing isogenies between ordinary elliptic curves for cryptography
Computing isogenies between elliptic curves defined over finite fields is an essential problem in cryptography. Most recent algorithms for isogeny computation rely on Kani's work on isogenies between products of elliptic curves and on the efficient computation of small-degree isogenies between higher-dimensional abelian varieties. Once the Kani isogeny has been computed, these methods simply extract the elliptic curve isogeny from its higher-dimensional counterpart. These algorithms are effective provided that the rational torsion of the elliptic curve is smooth. Consequently, they apply only to supersingular curves with smooth rational torsion, which are easy to construct. We show how to adapt these methods to ordinary elliptic curves by removing the smoothness constraint on the torsion. This is joint work with Maxime Louvet.

Alin Bostan, Algorithmic determination of algebraic values of E-functions
E-functions form a large class of transcendental entire functions, introduced by Siegel in 1929, which generalize the exponential function. They are extensively studied in number theory because their values enjoy important transcendence results, thanks to a theory developed over decades by mathematicians such as Siegel, Shidlovskii, Nesterenko, André and Beukers. One of the culminations of this theory is a very general version of the Hermite-Lindemann-Weierstrass theorem for E-functions. In this talk, the primary focus will be on effective versions of these results. I will present a recent algorithm that, for any given E-function, computes the set of all algebraic numbers to which it takes algebraic values. Joint work with Bruno Salvy and Tanguy Rivoal.

Anamaria Costache, Fully Homomorphic Encryption: a gentle introduction
Fully Homomorphic Encryption (FHE) allows to compute on encrypted data. It remained an open problem for nearly 30 years, until Gentry'09 provided a first construction. After that, many constructions followed, but they remained too impractical to be considered for real-life deployment. Recent advances have changed this, and we have begun to see applications in industry which deploy FHE. In this talk, we will introduce FHE, and discuss the current most relevant questions in the field.