Veranstaltungsübersicht -- Sommersemester 2020

Freitag,
2020-06-12
10:30 Uhr
virtueller Raum 404
Victoria Cantoral Farfan: Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces

K3 surfaces have been extensively studied over the past decades for several reasons. For once, they have a rich and yet tractable geometry and they are the playground for several open arithmetic questions. Moreover, they form the only class which might admit more than one elliptic fibration with section. A natural question is to ask if one can classify such fibrations, and indeed that has been done by several authors, among them Nishiyama, Garbagnati and Salgado. In this joint work with A. Garbagnati, C. Salgado, A. Trbović and R. Winter we study K3 surfaces defined over a number field k which are double covers of extremal rational elliptic surfaces. We provide a list of all elliptic fibrations on certain K3 surfaces together with the degree of the field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution on the fibers of the genus one fibration.

Freitag,
2020-06-26
10:30 Uhr
v404
Dr. Johannes Sprang (Regensburg): "(Ir)rationality of L-values"

Euler’s beautiful formula ζ(2n) = − (2πi) 2n B 2n . 2(2n)! can be seen as the starting point of the investigation of special values of L- functions. In particular, Euler’s result shows that all critical zeta values are ra- tional up to multiplication with a particular period, here the period is a power of (2πi). Conjecturally this is expected to hold for all critical L-values of motives. In this talk, we will focus on L-functions of number fields. In the first part of the talk, we will discuss the ’critical’ and ’non-critical’ L-values exemplary for the Riemann zeta function. Afterwards, we will head on to more general number fields and explain our recent joint result with Guido Kings on the algebraicity of critical Hecke L-values for totally imaginary fields up to explicit periods.

Freitag,
2020-07-10
10:30 Uhr
virtueller Raum 404
Ben Moonen: Computing discrete invariants of varieties in positive characteristic

For varieties (smooth projective, say) over fields of positive characteristic, we can define discrete invariants that have no natural analogue in characteristic 0. A well-known example is that an elliptic curve in characteristic p is either ordinary or supersingular. I will first review in general terms how this can be generalized to arbitrary varieties - there is in fact more than one natural generalization!

After this, I will focus on one particular type of discrete invariant; for abelian varieties this is known under the name 'Ekedahl-Oort type'. I will address the question how such discrete invariants can be concretely computed. In particular, I will explain a new method that allows to explicitly compute the Ekedahl-Oort type of (the Jacobian of) a complete intersection curve. For plane curves, a magma implementation of this method is now available, so if you have a favourite curve of which you want to know the E-O type, you can ask me and we can let magma calculate the answer.

At the end of the talk I will try to say a few words about generalizations for higher-dimensional projective hypersurfaces. There is a simple pattern that emerges, but so far I can only prove that it's correct for varieties of low dimension.

Freitag,
2020-07-17
10:30 Uhr
virtueller Raum 404
Victoria Hoskins: Motives of moduli spaces of bundles over a curve

Following Grothendieck’s vision that a motive of an algebraic variety should capture many of its cohomological invariants, Voevodsky introduced a triangulated category of motives which partially realises this idea. After describing some properties of this category, I will explain how to define motives of certain algebraic stacks. I will then state and prove a formula the motive of the moduli stack of vector bundles on a smooth projective curve; this formula is compatible with classical computations of invariants of this stack due to Harder, Atiyah--Bott and Behrend--Dhillon. The proof involves rigidifying this stack using Quot and Flag-Quot schemes parametrising Hecke modifications as well as a motivic version of an argument of Laumon and Heinloth on the cohomology of small maps. If there is time, I will discuss how this result can be used to also study motives of moduli space of Higgs bundles. This is joint work with Simon Pepin Lehalleur.

Freitag,
2020-07-24
10:30 Uhr
virtueller Raum 404
Jens Eberhardt: K-Motives and Koszul duality

Koszul duality, as first conceived by Beilinson-Ginzburg-Soergel, is a remarkable symmetry in the representation theory of Langlands dual reductive groups. This talks argues that Koszul duality - in it's most natural form - stems from a duality between equivariant K-motives and monodromic sheaves. I will give a short guide to K-motives and monodromic sheaves and then discuss examples of Koszul duality in increasing difficulty: (1) Tori (2) Toric varieties (3) Reductive groups.


Abonnieren

Die Einträge dieser Veranstaltung können im iCalender Format abonniert werden. die URL dazu lautet:
http://wochenprogramm.mathematik.uni-freiburg.de/ical/SS2020/OS-AlgZTalgGeom.ics