### Basel Number Theory Seminar: Spring 2019

Time: 14.15 on 28 Feb, 2019

**Yuri Bilu** (Université de Bordeaux)

*Towards effective André-Oort (after Kühne et al.) *

The celebrated André-Oort conjecture about special point on Shimura varieties is now proved conditionally to the GRH in full generality and unconditionally in many important special cases. In particular, Pila (2011) proved it for products of modular curves, adapting a method previously developed by Pila and Zannier in the context of the Manin-Mumford conjecture. Unfortunately, Pila's argument is non-effective, using the Siegel-Brauer inequality.

Since 2012 various special cases of the André-Oort conjecture has been proved effectively, most notably in the work of Lars Kühne. In my talk I will restrict to the case of the "Shimura variety" C^n and will try to explain on some simple examples how the effective approach of Kühne works.

No previous knowledge about André-Oort conjecture is required, I will give all the necessary background.

Time: 14.15 on 11 Apr, 2019

**Sary Drappeau** (Aix-Marseille Université)

*Value distribution of quantum modular forms*

Quantum modular forms are functions on the rationals satisfying a near-modularity relation; they appear naturally as invariants related to modular forms (central values of additive twists), or constructed from q-factorials. This talk will be about recent joint work with Sandro Bettin, where we study their value distribution using dynamical method.

Time: 14.15 on 25 Apr, 2019

**Andrei Yafaev** (University College London)

*$\overline{Q}$ structures on hermitian symmetric spaces*

This is a joint work in progress with Emmanuel Ullmo and consists largely of conjectures and speculations. Inspired by the analogy with the exponential function we define $\overline{Q}$-structures on a hermitian symmetric space $X$uniformising a Shimura variety $S$, formulate a "hyperbolic analytic subgroup theorem" and explore its consequences.

Time: 14.15 on 2 May, 2019

**Vesselin Dimitrov** (University of Cambridge)

*Exponential arithmetic growth in meromorphic dynamics*

A conjecture of Silverman (as extended to any global field $K$) states that every Zariski-dense forward orbit $(f^n(P))_{n \in \mathbb{N}}$ under a dominant rational self-map $f : X \dashrightarrow X$ of an $N$-dimensional projective variety $X$ over $K$ has its height $h(f^n(P))$ growing at the maximal possible exponential rate $\lambda_1(f)$, the first dynamic degree of $f$. I will describe some recent progress around this problem, limiting for the main part to the case of polynomial mappings of affine space, and focusing mostly on cases where an exponential growth can be established for the height along all Zariski-dense orbits. The latter includes all additive polynomial mappings of affine space over $\mathbb{F}_q(t)$ (building on Yu's zero estimate from transcendence theory in positive characteristic), as well as all maps of large topological degree (building on work of Habegger from his paper Special points in fibered powers of elliptic surfaces, and on a theorem of Bell, Ghioca and Tucker on the dynamical Mordell-Lang problem). The proof in the former case yields also a close counterpart, for abelian $t$-modules in function field arithmetic, of Masser's "polynomial in 1/D" canonical height lower bound on abelian varieties. Lastly, and time permitting, I will supplement the exponential growth discussion with a lower bound on height growth for an arbitrary polynomial mapping. This bound comes from a partial progress on a conjecture of Ruzsa, and it is weaker than exponential, yet strong enough to yield as application an "unbounded or eventually periodic" dichotomy for the degree sequence of the iterates of a polynomial mapping of $\mathbb{A}_k^N$ over any field. The dichotomy extends a result of Favre and Jonsson from $N = 2$ to arbitrary dimension.

Time: 14.15 on 9 May, 2019

**Florent Jouve** (Université de Bordeaux)

*Chebyshev's bias in Galois groups of number fields*

In a 1853 letter, Chebyshev observed that in most intervals [2,x] there are more primes of the form 4n+3 than of the form 4n+1. Many generalizations of this bias phenomenon have since been studied. In this talk we will discuss joint work with D. Fiorilli on Chebyshev’s bias in the context of the Chebotarev density theorem. Our focus will be on particular families that either exhibit a surprising behavior as far as Chebyshev's bias is concerned or that are simple enough to enable a very precise computation of the group theoretic and ramification theoretic invariants that come into play in our analysis. Precisely the emphasis will be on some families of abelian, dihedral, or radical extensions of Q as well as families of Hilbert class fields H_d of quadratic fields K_d (of discriminant d) either seen as extensions of Q or of K_d.

Time: 14.15 on 16 May, 2019

**Annette Werner** (Universität Frankfurt am Main)

*Endomorphism of Arrangement Complements*

Let $A$ be an arrangement of finitely many hyperplanes in projective space. We investigate endomorphisms of the complement of $A$. Under mild assumptions we extend any such endomorphism to an endomorphism of the wonderful compactification. We will explain some basic facts on matroids and tropical geometry which are used in the proof.

Time: 14.15 on 23 May, 2019

**Giuseppe Melfi** (University of Neuchâtel)

*Results and conjectures about primitive weird numbers*

A number $ n $ is weird if the sum of its proper divisors is larger than $ n $, and if $ n $ can not be expressed as a sum of some of its proper divisors.In 1972 Benkoski and Erdős published a paper in which they proved some of the properties that will be discussed here, and proposed several conjectures and open questions.One of the most interesting problems is determining whether odd weird numbers exist or not.At present all weird numbers that we know are even, but there are no obvious reasons that would prevent the existence of odd weird numbers.We will see that these problems are related to the study of \textsl{primitive weird numbers}, that is to say those weird numbers that are not multiples of other weird numbers.The most promising approach seems to be that of attacking the study of their prime factors.We will also address the problem of the existence of an infinite number of primitive weird numbers, and the problem of determining how many prime factors a primitive weird number may contain.

Time: 14.15 on 6 Jun, 2019

**Daniel Fiorilli** (Université Paris-Sud)

*Chebyshev’s bias in Galois groups*

In a 1853 letter, Chebyshev noted that there seems to be more primes of the form 4n+3 than of the form 4n+1. Many generalizations of this phenomenon have been studied. In this talk we will discuss Chebyshev’s bias in the context of the Chebotarev density theorem. We will focus on the generic case of S_n extensions, in which the question is strongly linked with the representation theory of this group and the ramification data of the extensions. We will see in detail how to take advantage of the rich representation theory of the symmetric group as well as bounds on characters due to Roichman. This is joint work with Florent Jouve.

Date | Speaker | Title |
---|---|---|

28 Feb, 2019 14.15 | Yuri Bilu(Université de Bordeaux) | Towards effective André-Oort (after Kühne et al.) |

11 Apr, 2019 14.15 | Sary Drappeau(Aix-Marseille Université) | Value distribution of quantum modular forms |

25 Apr, 2019 14.15 | Andrei Yafaev(University College London) | $\overline{Q}$ structures on hermitian symmetric spaces |

2 May, 2019 14.15 | Vesselin Dimitrov(University of Cambridge) | Exponential arithmetic growth in meromorphic dynamics |

9 May, 2019 14.15 | Florent Jouve(Université de Bordeaux) | Chebyshev's bias in Galois groups of number fields |

16 May, 2019 14.15 | Annette Werner(Universität Frankfurt am Main) | Endomorphism of Arrangement Complements |

23 May, 2019 14.15 | Giuseppe Melfi(University of Neuchâtel) | Results and conjectures about primitive weird numbers |

6 Jun, 2019 14.15 | Daniel Fiorilli(Université Paris-Sud) | Chebyshev’s bias in Galois groups |