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# Gästeseminar "Arithmetische Geometrie"

Zeit: Donnerstags, 14-16 Uhr

Ort: SR 032 Arnimallee 6

## Vorträge:

Datum Vortragende(r)
20. October

10:15 (SR 032/A6)

Disputation von Elena

14:15

Titel: Where do little elliptic curves go?

Abstrakt: Let C be a curve over Q provided with an integral model, an ample line  bundle on the model and a semipositive metric. To these data we can associate the height of the curve and the height of every algebraic point of the curve. The essential minimum of the curve is the minimal accumulation point of the height of the algebraic points. The essential minimum is a mysterious and elusive invariant. A result of Zhang shows that the essential minimum has a lower bound in terms of the height of the curve, and an example of Zagier shows that there can be several isolated values of the height below the essential minimum. When C is the modular curve, the line bundle agrees with the bundle of modular forms and the metric is the Weil-Petersson metric, then the height of an algebraic point agrees with the stable Faltings height  of the corresponding elliptic curve. In this talk we will discuss methods of proving lower and upper bounds for the essential minimum and apply them to the modular curve, giving a partial description of the spectrum of the stable Faltings height of elliptic curves. This is joint with with Ricardo Menares and Juan Rivera-Letelier.

15:30

Tomohide Terasoma (Tokyo)

Titel: Exotic S6 action on the cohomogy of abelian coverings of M05

Abstrakt: The period integrals of abelian coverings of M0n are expressed by Selberg integrals. If n=5, they are also expressed as a special values of hypergeometric function. In this talk, we explain that symmetry generated by these two expressions gives rise to an exotic symmetry of S6, described via Grothendieck's bi-icosahedra. As a consequence, we give an examples of algebraic Weil Hodge cycles. Combinatorial part is joint work with Y.Hashimoto.

Montag

14. November

16:00 (SR 031/A6)

Disputation von Inder

17. November

14:15

Olivier Wittenberg (Paris)

Titel: On the integral Hodge conjecture for real threefolds

Abstrakt: (Joint work with Olivier Benoist.) We formulate an analogue of the integral Hodge conjecture for real algebraic varieties. This analogue turns out to be closely related to more classical properties: existence of a curve of even genus, algebraicity of the homology of the real locus. As in the complex case, the real integral Hodge conjecture can fail; but it is plausible for 1-cycles on varieties whose geometry is simple enough. We establish it for several families of uniruled threefolds.

24. November

14:15

Peter Jossen (Zürich)

Titel: Exponential motives and exponential periods (joint work with Javier Fresán)

Abstrakt: What motives are to varieties, exponential motives are to varieties endowed with a potential, that is, to pairs (X,f) consisting of an algebraic variety X and a regular function f on X. Our primary motivation for studying exponential motives is that they provide a framework for a Galois theory for special values of the gamma function, of Bessel functions and for other interesting numbers which are not expected to be periods in the usual sense of algebraic geometry. In my talk, I aim to explain how to construct, following ideas of Kontsevich and Nori, a Q-linear neutral tannakian category of exponential motives over a subfield of the complex numbers, and how to calculate periods and Galois groups of a few particular exponential motives.

15:30
Atsushi Shiho (Tokyo)

Titel: Comparison of relatively unipotent log de Rham fundamental groups

1. Dezember

14:15

Lars Hesselholt (Copenhagen)

Titel: Topological Hochschild homology

15:30

Michael Groechenig (Berlin)

Titel: Adèles and the geometry of schemes

Abstrakt: By a result of André Weil vector bundles on a curve can be described as a double quotient of the set of invertible matrices over the ring of adèles. In this talk we will discuss an extension of this description to arbitrary Noetherian schemes, and perfect complexes. As a corollary we obtain that every Noetherian scheme can be reconstructed from Beilinson’s cosimplicial ring of adèles.

5. Januar

14:15

Jérôme Poineau (Caen)

Titel: Berkovich spaces over Z

Abstrakt: Although Berkovich spaces usually appear in a non-archimedean setting, their general definition actually allows arbitrary Banach rings as base rings, e.g. Z endowed with the usual absolute value. Over the latter, Berkovich spaces look like fibrations that contain complex analytic spaces as well as p-adic analytic spaces for every prime number p. We will try to show what those spaces look like and explain their main properties: topological (local path-connectedness, etc.) or algebraic (noetherianity of the local rings, etc.).

We will also spend some time talking about global functions on those spaces: they are typically convergent arithmetic power series, i.e. power series with coefficients in Z with a positive radius of convergence. We will explain how the nice properties of Berkovich spaces over Z translate into nice properties of rings of convergent arithmetic power series.

12. Januar

14:15

Shane Kelly (Berlin)

Titel: tba

15:30

Raju Krishnamoorthy (Berlin)

Titel: Étale correspondences without a core

Mittwoch!

1. Februar

13:00 (SR 031/A6)

Christian Liedtke (München)

Titel: Crystalline Galois Representations arising from K3 Surfaces
Abstract:
Let K be a p-adic field, let X be a K3 surface over K, and assume that X has potential semi-stable reduction. Then, we show that the following are equivalent:

1) the l-adic Galois representation on H^2(\bar{X},Q_l) is unramified for one l different from p
2) the l-adic Galois representation on H^2(\bar{X},Q_l) is unramified for all l different from p
3) the p-adic Galois representation on H^2(\bar{X},Q_p) is crystalline
4) the surface has good reduction after an unramified extension of K

This is an analog of the classical Serre-Tate theorem for Abelian varieties. We also show by counter-examples that neither 1), nor 2), nor 3) implies that X has good reduction of X over K. However, in this case X admits a proper model over O_K, whose special fiber X_0 has at worst canonical singularities. Now, if the Galois-representation on H^2(\bar{X},Q_p) is crystalline, then functors of Fontaine and Kisin provide us with an F-crystal over W(k) that looks like the crystalline cohomology of some smooth K3 surface - in fact, we will show that this is the crystalline cohomology of the minimal resolution of singularities of X_0. In my talk, I will introduce all the above notions and functors (which will not give me much time to give proofs). Part of this is joint with Matsumoto, part of this is joint with Chiarellotto and Lazda.