## Tenth Bucharest Number Theory Days |

Home | Schedule & Abstracts | Local information | Registration |

Time | August 3rd | August 4th | August 5th |
---|---|---|---|

9:30-10:20 | Harcos | Ionica | Zudilin |

10:20-10:40 | Coffee break | Coffee break | Coffee break |

10:40-11:30 | Matei | Ţurcaş | Beli |

11:40-12:30 | Risager | Bostan | Dospinescu |

12:30-15:10 | Lunch break | Lunch break | Lunch |

15:10-16:00 | Jones | Anton |

**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 *not*not 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.

**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.

**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.