Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces
Friday, 12.6.20, 10:30-11:30, virtueller Raum 404
K3 surfaces have been extensively studied over the past decades for\nseveral reasons. For once, they have a rich and yet tractable geometry\nand they are the playground for several open arithmetic questions.\nMoreover, they form the only class which might admit more than one\nelliptic fibration with section. A natural question is to ask if one\ncan classify such fibrations, and indeed that has been done by several\nauthors, among them Nishiyama, Garbagnati and Salgado.\nIn this joint work with A. Garbagnati, C. Salgado, A. Trbović and R.\nWinter we study K3 surfaces defined over a number field k which are\ndouble covers of extremal rational elliptic surfaces. We provide a list\nof all elliptic fibrations on certain K3 surfaces together with the\ndegree of the field extension over which each genus one fibration is\ndefined and admits a section. We show that the latter depends, in\ngeneral, on the action of the cover involution on the fibers of the\ngenus one fibration.
"(Ir)rationality of L-values"
Friday, 26.6.20, 10:30-11:30, virtueller Raum 404
Euler’s beautiful formula\nζ(2n) = −\n(2πi) 2n\nB 2n .\n2(2n)!\ncan be seen as the starting point of the investigation of special values of L-\nfunctions. In particular, Euler’s result shows that all critical zeta values are ra-\ntional up to multiplication with a particular period, here the period is a power of\n(2πi). Conjecturally this is expected to hold for all critical L-values of motives.\nIn this talk, we will focus on L-functions of number fields. In the first part of the\ntalk, we will discuss the ’critical’ and ’non-critical’ L-values exemplary for the\nRiemann zeta function. Afterwards, we will head on to more general number\nfields and explain our recent joint result with Guido Kings on the algebraicity\nof critical Hecke L-values for totally imaginary fields up to explicit periods.
"(Ir)rationality of L-values"
Friday, 26.6.20, 10:30-11:30, v404
Euler’s beautiful formula\nζ(2n) = −\n(2πi) 2n\nB 2n .\n2(2n)!\ncan be seen as the starting point of the investigation of special values of L-\nfunctions. In particular, Euler’s result shows that all critical zeta values are ra-\ntional up to multiplication with a particular period, here the period is a power of\n(2πi). Conjecturally this is expected to hold for all critical L-values of motives.\nIn this talk, we will focus on L-functions of number fields. In the first part of the\ntalk, we will discuss the ’critical’ and ’non-critical’ L-values exemplary for the\nRiemann zeta function. Afterwards, we will head on to more general number\nfields and explain our recent joint result with Guido Kings on the algebraicity\nof critical Hecke L-values for totally imaginary fields up to explicit periods.
Computing discrete invariants of varieties in positive characteristic
Friday, 10.7.20, 10:30-11:30, virtueller Raum 404
For varieties (smooth projective, say) over fields of\npositive characteristic, we can define discrete invariants that have no\nnatural analogue in characteristic 0. A well-known example is that an\nelliptic curve in characteristic p is either ordinary or supersingular.\nI will first review in general terms how this can be generalized to\narbitrary varieties - there is in fact more than one natural\ngeneralization!\n\nAfter this, I will focus on one particular type of discrete invariant;\nfor abelian varieties this is known under the name 'Ekedahl-Oort type'.\nI will address the question how such discrete invariants can be\nconcretely computed. In particular, I will explain a new method that\nallows to explicitly compute the Ekedahl-Oort type of (the Jacobian of)\na complete intersection curve. For plane curves, a magma implementation\nof this method is now available, so if you have a favourite curve of\nwhich you want to know the E-O type, you can ask me and we can let\nmagma calculate the answer.\n\nAt the end of the talk I will try to say a few words about\ngeneralizations for higher-dimensional projective hypersurfaces. There\nis a simple pattern that emerges, but so far I can only prove that it's\ncorrect for varieties of low dimension.\n
Motives of moduli spaces of bundles over a curve
Friday, 17.7.20, 10:30-11:30, virtueller Raum 404
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.
K-Motives and Koszul duality
Friday, 24.7.20, 10:30-11:30, virtueller Raum 404
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.