13th Bucharest Number Theory Days July 14-15, 2025

Schedule

Abstracts

Monday, July 14th

Stevan Gajović, Ratio sets of images of integral forms and p-adic polynomials, and the question of their denseness in \(\mathbb{Q}_p\)
For a subset of integers, we define its ratio set and ask if the ratio set is dense in \(\mathbb{Q}_p\) for \(p\) a prime that can be fixed, or vary in a family. For integral forms in many variables, we consider the ratio set of the set of their values and investigate whether it is dense in \(\mathbb{Q}_p\). Later, we restrict our attention to polynomials in \(\mathbb{Z}_p[x]\), and we explore if we can solve the equation \(f(x)/f(y)=s\) for all \(s\) in \(\mathbb{Q}_p\), where \(x\) and \(y\) are in \(\mathbb{Z}_p\). Miska, Murru, and Sanna proved that the answer is yes if \(f\) has a simple root or, more generally, if \(f\) has two roots with coprime multiplicities. We consider classes of polynomials which do not belong to the situations where we already know the asnwer, and give criteria for such families. We show that these criteria are the best possible, and relate this statement to a conjecture in additive combinatorics. We give a criterion for denseness for polynomials of small degree. This is joint work with Deepa Antony, Rupam Barman, and Daniel Širola.

Petru Constantinescu, Effective local limit laws for homology classes
The study of closed geodesics and the homology of Riemannian manifolds is a rich and beautiful subject at the intersection of geometry, topology, and arithmetic. In this work, we develop a method for hyperbolic quotient surfaces to study the distribution of homology classes of closed and infinite geodesics, based on the perturbation theory of the Laplacian. Our approach recovers several important results from the literature and yields new applications to the distribution of closed geodesic periods and modular symbols-more precisely, to simultaneous non-vanishing, large values, and mixed moments. Joint work with Asbjørn Nordentoft.

Claudia Schoemann, The kernel of the Gysin homomorphism for smooth projective curves
Let \(S\) be a smooth projective connected surface over an algebraically closed field \(k\) inside a projective space \(\mathbb{P}^d\) and let \(C\) be a smooth projective curve on \(S\). Let \(\mathrm{CH}_0(S)_{\deg=0}\) and \(\mathrm{CH}_0(C)_{\deg=0}\) be the Chow groups of zero-cycles of degree \(0\) on \(S\) and \(C\), respectively. The kernel of the Gysin homomorphism from \(\mathrm{CH}_0(C)_{\deg=0}\) to \(\mathrm{CH}_0(S)_{\deg=0}\) is a countable union of translates of an abelian subvariety \(A\) inside the Jacobian \(J\) of the curve \(C\). Further there is a countable open subset \(U_0\) contained in the set \(U \subset (\mathbb{P}^d)^*\) parametrizing the smooth curves such that \(A=B\) or \(A=0\) for all curves parametrized by \(U_0\), where \(B\) is the abelian subvariety of \(J\) corresponding to the vanishing cohomology \(H^1(C, k')_{\text{van}}\) of \(C\). Hence the kernel of the Gysin homomorphism either is a countable union of shifts of \(B\) or, if \(A=0\), it is countable.
Passing to finite type smooth schemes as a generalisation of algebraic varieties we extend this result to the set \(U \subset (\mathbb{P}^d)^*\), i.e. to all smooth curves \(C\). Using the language of algebraic stacks this is done by constructing an increasing filtration of Zariski countable open substacks \(U_i, i \in I,\) of \(U,\) where \(I\) is a countable set, that respects the required isomorphism between geometric generic and special point and by applying a convergence argument.

Mabud Ali Sarkar, Two-dimensional Lubin-Tate formal group over \(\mathbb{Z}_p\) and consequences
The 1-dimensional Lubin-Tate formal groups play a central role in the formulation of local class field theory by providing explicit descriptions of the maximal abelian extensions of local fields. However, a systematic generalization of this theory to higher-dimensional formal groups remains undeveloped.
In this proposed talk, I will report a recent construction of a class of 2-dimensional formal groups over the ring of p-adic integers that serve as a natural higher-dimensional analogue of the classical Lubin-Tate theory. These formal groups exhibit rich arithmetic structure and offer new perspectives in non-abelian local Galois representations. We will explore some immediate consequences of this construction and outline potential future applications.

Tuesday, July 15th

Zhenghui Li, Duality for arithmetic p-adic pro-etale cohomology of rigid analytic varieties
Let \(K\) be a finite extension of \(\mathbb{Q}_p\). We prove that the arithmetic p-adic pro-etale cohomology of smooth partially proper spaces over \(K\) satisfies a duality, as conjectured by Colmez-Gilles-Niziol. I will start from some recent history to motivate this question and explain the difference with algebraic case via some examples. If time permits, I will explain how it is related to quasi-coherent sheaves (defined via condensed maths) on the Fargues-Fontaine curve and how to deduce the result from the Poincare duality of syntomic sheaves.

Alexandru Pascadi, On the exponents of distribution of primes and smooth numbers
Results on the equidistribution of primes and related sequences in arithmetic progressions, going beyond GRH on average, have been a crucial input to sieve theory methods over the past decades. We'll discuss recent improvements to such results, for both primes (using triply-well-factorable weights) and smooth numbers (using arbitrary weights). These rely on new inputs from the spectral theory of automorphic forms, and completely eliminate the dependency on Selberg's eigenvalue conjecture from previous results. As applications, we'll mention refined upper bounds for the counts of twin primes and consecutive smooth numbers up to x.

Răzvan Bărbulescu, Regev's attack on hyperelliptic cryptosystems
Hyperelliptic curve cryptography (HECC) is a candidate to standardization which is a competitive alternative to elliptic curve cryptography (ECC). We extend Regev's algorithm to this setting. For genus-two curves relevant to cryptography, this yields a quantum attack up to nine times faster than the state-of-the-art. This implies that HECC is slightly weaker than ECC. In a more theoretical direction, we show that Regev's algorithm obtains its full speedup with respect to Shor's when the genus is high, a setting which is already known to be inadequate for cryptography. This is a joint work with Gaëtan Bisson.