# Workshop on Arithmetic and Geometry

Der Workshop findet vom 26. August bis zum 4. September in Cetraro, Italien statt. Organisiert wird er von Fabrizio Catanese (Universität Bayreuth) und Umberto Zannier (Scuola Normale Superiore).

Research funded by

ERC Advanced Grant TADMICAMT

## Programm

### Samstag, 27. August

- 10:45–11:30Michael StollBayreuthRational points on curves contained in the square of an elliptic curve of rank 1
Demyanenko has shown that the rational points on a curve C with n+1 independent morphisms to an elliptic curve E of Mordell-Weil rank n have explicitly bounded height. Recently, Checcoli, Veneziano and Viada have given a different proof of this result for n = 1. I will present an improved version of their result with an elementary proof and compare the two methods (and possibly discuss some generalizations).

- 18:00–18:45Bruno KlinglerJussieu, ParisHodge theory and atypical intersections
Given a variation of Hodge structures $V$ over a smooth quasi-projective base $S$, I will explain the notion of an atypical subvariety for $(S,V)$ and state a simple general conjecture about these atypical subvarieties. When $S$ is a Shimura variety and $V$ a standard variation of Hodge structure on $S$, one recovers the Zilber-Pink conjecture.

### Sonntag, 28. August

- 09:30–10:15Ingrid BauerBayreuthOn rigid compact complex surfaces and manifolds
A compact complex manifold $X$ is rigid if it has no nontrivial deformations. The only rigid complex curve is the projective line; for dimension 2 we prove: Theorem. Let $S$ be a compact complex surface, which is rigid, then:

1) $S$ is minimal of general type, or

2) $S$ is a Del Pezzo surface of degree $\ge 5$, or

3) $S$ is an Inoue surface.

We explain different concepts of rigidity, their relations and give new examples of rigid manifolds, and pose open questions concerning the relation with the property of being a K(\pi,1) space.

This is joint work with F. Catanese.

- 10:45–11:30Yongnam LeeKAIST DaejeonOn rational maps from a hypersurface section of a Fano 3-fold and its double cover
In this talk, we treat dominant rational maps from a very general element in hypersurface sections of a smooth Fano 3-fold and its double cover to smooth projective surfaces. The method of our study combines the classification of algebraic surfaces and smooth Fano 3-folds, Hodge theory, and the deformation theory. This is joint work with Gian Pietro Pirola.

- 18:00–18:45Michael LönneBayreuthOn the stable topology of Hurwitz spaces
We want to discuss the topology of Hurwitz spaces, in particular those which parametrize Galois covers of curves with fixed finite group $G$ and fixed branch datum. First we formulate the problem of homological stability in general and then give a review of results concerning particular aspects. These include the stability range for the first Betti number in case of particular groups and other results obtained in collaboration with F.Catanese and F.Perroni. Also the results of F. Tietz and work in progress will be presented. Time permitting some open problems will be discussed pertaining also to the monodromy group of the Hurwitz spaces over the coloured configuration spaces.

### Montag, 29. August

- 09:30–10:15Yujiro KawamataTokyoDerived McKay correspondence for GL(3, C).
I explain a theorem that the equivariant derived category for a finite subgroup of GL(3, C) has a semi-orthogonal decomposition into the derived category of some partial resolution called a maximal Q-factorial terminalization of the corresponding quotient singularity and a relative exceptional collection. This is a generalization of a result of Bridgeland, King and Reid.

- 10:45–11:30Sönke RollenskeMarburgGorenstein stable surfaces with ${K}_{X}^{2}=1$
The moduli space of stable surfaces is a modular compactification of the Giesecker moduli space of surfaces. I will report on a joint project with Marco Franciosi and Rita Pardini where we explored the world of Gorenstein stable surfaces with ${K}_{X}^{2}=1$ and describe some connections to classical questions.

- 18:00–18:45Matthew StoverTemple Univ. PhiladelphiaBrauer groups of character varieties
Let $G$ be a finitely generated group and $X=X(\mathrm{\Gamma},S{L}_{2}(C))$ be the "$S{L}_{2}(C)$-character variety" of $G$. Given an irreducible curve of characters $C$ on $X$ with smooth projective model $\hat{C}$, I will discuss when, in terms of the Alexander module of $G$, one obtains a canonical element of the Brauer group $Br(\hat{C})$. I will then discuss some applications, e.g., to the arithmetic invariants of hyperbolic 3-manifolds obtained from Dehn surgery on a fixed knot. This is joint work with Ted Chinburg and Alan Reid.

### Dienstag, 30. August

- 10:00–10:45Michael DettweilerBayreuthThe inverse Galois problem, convolution and Fourier transform
The inverse Galois problem is the conjecture that any finite group occurs as Galois group over the rational numbers. We show that arithmetic versions of convolution and Fourier transform can be used to realise many special linear groups and symplectic groups as Galois groups over the rationals.

- 18:00–18:45Gareth JonesManchesterPfaffian functions and elliptic functions
I will discuss work in progress with Harry Schmidt in which we give a pfaffian definition of Weierstrass elliptic functions, refining a result due to Macintyre. The complexity of our definition is bounded by an effective absolute constant. As an application we give an effective version of a result of Corvaja, Masser and Zannier on a sharpening of Manin-Mumford for non-split extensions of elliptic curves by the additive group. We also give a higher dimensional version of their result.

### Mittwoch, 31. August

- 10:00–10:45Alberto PerelliGenovaConverse theorems for L-functions
Converse theorems aim at detecting the nature of L-functions from their analytic properties. After a brief survey of the classical converse theorems of Hamburger, Hecke and Weil, I'll describe more recent work on converse theorems in the general framework of the Selberg class of L-functions.

- 18:00–18:45Fabio PerroniTriesteDihedral Galois covers of algebraic varieties and the simple cases
I will report on a joint work with Fabrizio Catanese in which we study the structure of dihedral Galois covers of algebraic varieties. Explicit constructions of such covers X --> Y in concrete cases rely on the knowledge of the group of divisorial sheaves on double covers of Y. Thus we provide a description of this group in terms of rank 2 vector bundles on Y. We also define two special explicit classes of dihedral covers, the simple and the almost simple dihedral covers. We determine their basic invariants and their natural deformations. We also give applications to fundamental groups of complements of divisors.

### Donnerstag, 01. September

- 10:00–10:45Benjamin CollasDFG - BayreuthArithmetic and motivic considerations for the moduli stack of curves
Considering group theoretic properties of the mapping class group, then geometric properties of the special loci of curves with given automorphisms, I will present how to describe the Galois action on the cyclic stack inertia of the moduli spaces of curves. By a third and last shift of point of view, I will explain how Morel-Voevodski's unstable motivic homotopy theory offers a suitable framework to extend such considerations at a motivic level.

- 18:00–18:45Pietro CorvajaUdineOn the Hilbert Property for algebraic varieties over number fields
In 1892, David Hilbert proved his famous irreducibility theorem. In modern language, it can be rephrased in the following way: given an algebraic curve $C$ defined over a number field $k$ and a morphism $f:C\to {\mathbb{A}}^{1}$ from $C$ to the line ${\mathbb{A}}^{1}$ admitting no rational section, the image $f(C(k))$ cannot cover the set $k={\mathbb{A}}^{1}(k)$. In this talk we shall investigate higher dimensional extensions. While it is well-known that the affine line can be replaced by any rational variety, we shall provide examples of non-rational algebraic surfaces $X$ such that for every morphism $f:YoX$, $Y$ being another (possibly reducible) surface, admitting no rational section, the image $f(Y(k))$ cannot cover $X(k)$. This is a joint work with Umberto Zannier.

### Freitag, 02. September

- Informal talks and mathematical discussions

### Samstag, 03. September

- Informal talks and mathematical discussions