Tenth Bucharest Number Theory Days August 3-5, 2022

Schedule

Abstracts

Wednesday, August 3rd

Gergely Harcos, The sequence of prime gaps is graphic
Let us call a simple graph on \(n>1\) vertices a prime gap graph if its vertex degrees are 1 and the first \(n-1\) prime gaps (we need the 1 so that the sum of these numbers is even). We can show that such a graph exists for every large \(n\), and under the Riemann hypothesis for every \(n>1\). Moreover, a sequence of such graphs can be generated by a so-called degree preserving growth process: in any prime gap graph on \(n\) vertices, we can find \((p_{n+1}-p_n)/2\) independent edges, delete them, and connect the ends to a new, \((n+1)\)-th vertex. This creates a prime gap graph on \(n+1\) vertices, and the process never ends. Joint work with P. L. Erdős, S. R. Kharel, P. Maga, T. R. Mezei, and Z. Toroczkai.

Vlad Matei, Counting polynomials with a prescribed Galois group
An old problem, dating back to Van der Waerden, asks about counting irreducible degree \(n\) polynomials with coefficients in the box \([-H,H]\) and prescribed Galois group. Van der Waerden was the first to show that \(H^n+O(H^{n-\delta})\) have Galois group \(S_n\) and he conjectured that the error term can be improved to \(o(H^{n-1})\). Recently, Bhargava almost proved van der Waerden conjecture showing that there are \(O(H^{n-1+\varepsilon})\) non \(S_n\) extensions, while Chow and Dietmann showed that there are \(O(H^{n-1.017})\) non \(S_n\), non \(A_n\) extensions for \(n\ge 3\) and \(n\neq 7,8,10\). In joint work with Lior Bary-Soroker, and Or Ben-Porath we use a result of Hilbert to prove a lower bound for the case of \(G=A_n\), and upper and lower bounds for \(C_2\) wreath \(S_{n/2}\). The proof for \(A_n\) can be viewed, on the geometric side, as constructing a morphism \(\varphi\) from \(A^{n/2}\) into the variety \(z^2=\Delta(f)\) where each \(\varphi_i\) is a quadratic form. For the upper bound for \(C_2\) wreath \(S_{n/2}\) we improve on the monic version of Widmer's result for counting polynomials with imprimitive Galois group. We also pose some open problems/conjectures.

Morten Risager, Distributions of Manin's iterated integrals
We recall the definition of Manin's iterated integrals of a given length. We then explain how these generalise modular symbols and certain aspects of the theory of multiple zeta-values. In length one and two these iterated integrals have a limiting distribution, which, maybe surprisingly, we cannot conclude in higher length. This is joint work with Petridis and with Matthes.

Nathan Jones, Elliptic curves with acyclic reductions modulo primes in arithmetic progressions
Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) and, for a prime \(p\) of good reduction for \(E\) let \(\tilde{E}_p\) denote the reduction of \(E\) modulo \(p\). Inspired by an elliptic curve analogue of Artin's primitive root conjecture posed by S. Lang and H. Trotter in 1977, J-P. Serre adapted methods of C. Hooley to prove a GRH-conditional asymptotic formula for the number of primes \(p \leq x\) for which the group \(\tilde{E}_p(\mathbb{F}_p)\) is cyclic. More recently, Akbal and Gülo\(\breve{\text{g}}\)lu considered the question of cyclicity of \(\tilde{E}_p(\mathbb{F}_p)\) under the additional restriction that \(p\) lie in a fixed arithmetic progression. In this note, we study the issue of which elliptic curves \(E\) and which arithmetic progressions \(a \bmod n\) have the property that, for all but finitely many primes \(p \equiv a \bmod n\), the group \(\tilde{E}_p(\mathbb{F}_p)\) is notnot cyclic, answering a question of Akbal and Gülo\(\breve{\text{g}}\)lu on this issue. This is based on joint work with Sung Min Lee.

Thursday, August 4th

Sorina Ionica, Isogenous hyperelliptic and non-hyperelliptic Jacobians with maximal complex multiplication
We analyze complex multiplication for Jacobians of curves of genus 3, as well as the resulting Shimura class groups and their subgroups corresponding to Galois conjugation over the reflex field. We combine our results with numerical methods to find CM fields for which there exist both hyperelliptic and non-hyperelliptic curves whose Jacobian have maximal complex multiplication. More precisely, we find all sextic CM fields in the LMFDB data base for which (heuristically) Jacobians of both types exist. There turn out to be 14 such fields among the 547,156 sextic CM fields that the LMFDB contains. This supports the conjecture that the list of CM fields of this kind is finite. We show some cryptographic applications on the hardness of the discrete logarithm problem for genus 3 hyperelliptic curves. This is joint work with Bogdan Dina and Jeroen Sijsling.

George Ţurcaş, Absolute irreducibility of mod p representations of Frey curves defined over number fields
Mazur's rational isogeny theorem plays a role in the "modular approach" to Diophantine equations. To be precise, non-existence of a rational p-isogeny is one of the hypotheses for applying Ribet's level-lowering theorem. In recent years, the modular method has been applied in a context in which the Frey elliptic curves are defined over number fields K for which no analogue of Mazur's theorem is known. In this talk, we will explore various results for the (absolute) irreducibility of the mod p representations of Frey curves and their applications to Diophantine equations. A substantial part of the results are joint work with Filip Najman.

Alin Bostan, On deciding transcendence of D-finite power series
A formal power series in \(\mathbb{Q}[[t]]\) is said to be D-finite ("differentially finite"), or holonomic, if it satisfies a linear differential equation with polynomial coefficients. D-finite power series are ubiquitous in number theory and combinatorics. In a seminal article (1980), Richard Stanley asked whether it is possible to decide if a given D-finite power series is algebraic or transcendental. Several very useful sufficient criteria for transcendence exist, e.g., using asymptotics, but none of them is also a sufficient condition. Characterizing the transcendence of a D-finite power series is highly nontrivial even if its coefficient sequence satisfies a recurrence of first order: this question was completely solved only in 1989 by Frits Beukers and Gert Heckman. In this talk, I will present answers to Stanley's question and illustrate them through several examples coming from number theory and combinatorics.

Marian Anton, Computational Aspects of Arithmetic Group Cohomology
The integral cohomology of \(GL_n\) and its stable version \(GL\) over a ring of integers \(O_F\) in a number field \(F\) has been a source of rich mathematical ideas. One approach is to localize the problem at each prime \(p\). In this talk, we look at the mod \(p\) cohomology of \(GL_n O_F[1/p]\) where \(F\) is the cyclotomic field of \(p\)-roots of unity. In particular, we describe some explicit cycles in the mod \(p\) homology of \(SL_2 O_F[1/p]\) which are relevant for the general case.

Friday, August 5th

Wadim Zudilin, uNTitled
The talk discusses an identity for the product of two special hyperelliptic integrals and related differential equations. It is based on joint work with Mark van Hoeij and Duco van Straten.

Nicolae Beli, Rational biquadratic reciprocity laws via Galois cohomology
The term "rational biquadratic (or quartic) reciprocity laws" applies to various results involving Legendre symbols of the type \((a+b\sqrt m\mid p)\), defined whenever \((m\mid p)=(a^2-b^2m\mid p)=1\). A particular case of such symbols are the biquadratic (quartic) symbols \((m\mid p)_4:=(\sqrt m\mid p)\), defined whenever \((m\mid p)=(-1\mid p)=1\). Such results have been obtained by Gauss, Dirichlet, Burde, Frölich, Lemmermeyer etc. In this talk, given a number field \(F\), we introduce a morphism from a subgroup of a quotient of \(T^3(F^*/F^{*2})\) to \(\{\pm 1\}\). In the case \(F=\mathbb Q\), this morphism is given in terms of the Legendre symbols \((a+b\sqrt m\mid p)\). From its properties we were able to recover all rational biquadratic reciprocity laws we encountered and to produce new ones. These results were initially obtained while searching for invariants for the spinor genera of quadratic forms. But they can be deduced in an alternative way, using Galois cohomology and the properties of the Brauer group.

Gabriel Dospinescu, p-adic and mod p cohomology of the Drinfeld tower for \(GL_2\)
We will survey recent and not-so-recent results in our joint project with Pierre Colmez and Wieslawa Niziol, whose aim is understanding the mod \(p\) and p-adic cohomology of the coverings of the Drinfeld upper half-plane and their link with p-adic representations of \(GL_2(\mathbb{Q}_p)\), in particular with the p-adic local Langlands correspondence for this group.