The Bielefeld Arithmetic Geometry Seminar 2016-2017

Welcome to the website of the Bielefeld Arithmetic Geometry Seminar! The seminar is supported by the SFB CRC 701: Spectral Structures and Topological Methods in Mathematics, and related to the projects B5 and C7. For more information, please contact the organizers: Jeanine Van Order and Charles Vial.

Talks

Wednesday 2 November, 2016
15:15	U2-135	Ulrike Riess (Bonn) On Beauville’s conjectural weak splitting property [Abstract] We present a result on the Chow ring of irreducible symplectic varieties. The main object of interest is Beauville’s conjectural weak splitting property, which predicts the injectivity of the cycle class map restricted to a certain subalgebra of the rational Chow ring (the subalgebra generated by divisors). For special irreducible symplectic varieties we relate it to a conjecture on the existence of rational Lagrangian fibrations. After deducing that this implies the weak splitting property in many new cases, we present parts of the proof.
16:30	U2-135	Robert Pollack (MPIM-Bonn) On mu-invariants and congruences with Eisenstein series [Abstract] For any irregular prime p, one has a Hida family of cuspidal eigenforms of level 1 whose residual Galois representations are all reducible. This family has already played a starring role in Wiles’ proof of Iwasawa’s main conjecture for totally real fields. In this talk, we instead focus on the Iwasawa theory of these modular forms in their own right. We will discuss new phenomena that occur in this residually reducible case including the fact that analytic mu-invariants are unbounded in this family and directly related to the p-adic zeta-function. This is a joint work with Joel Bellaiche.
Wednesday 30 November, 2016
15:15	C01-226	Yichao Tian (Bonn) On the super-singular locus of Hilbert modular varieties and arithmetic applications [Abstract] We will study the super-singular locus of odd dimensional Hilbert modular varieties at an inert prime. The main result says that the super-singular locus is an iterated P^1 bundle over a zero-dimensional Shimura variety associated to a definite quaternion algebra. This has some applications to congruences of Hilbert modular forms, which can be viewed as an analogue of Ribet classical theorem on level raising of modular forms. This is a joint work in progress with Y. Liu and L. Xiao.
16:30	C01-226	Anna Medvedovsky (MPIM-Bonn) The Nilpotence Growth Theorem, and applications to dimensions of mod-p Hecke algebras [Abstract] We will discuss a theorem about the annihilating behavior of certain operators defined by linear recursions on the polynomial algebra Fp[x]. Such operators capture algebraic properties of Hecke operators, and this theorem has applications to dimensions of big mod-p Hecke algebras, generalizing recent work of Nicolas and Serre on the case p = 2 in level one.
Wednesday 25 January, 2017
15:15	T2-214	Tian An Wong (MPIM-Bonn) Beyond endoscopy and refinements of the stable trace formula [Abstract] In 2001 Langlands proposed a strategy to approach his Functoriality Conjectures, now referred to as Beyond Endoscopy. The endoscopic cases of Functoriality are addressed by the stable Arthur-Selberg trace formula, following the solution of the Fundamental Lemma in 2009. A key to the Beyond Endoscopy strategy is several proposed refinements of the stable trace formula, involving the analytic theory of L-functions. In this talk, I will give a general introduction to these ideas, and survey recent work of Frenkel-Langlands-Ngo, Altug, and Arthur in this direction. I will also report on generalizing to GL(n) Altug's method of smoothing of singularities real elliptic orbital integrals on GL(2), and discuss directions and obstructions to further work. This is joint work with Gonzalez, Kwan, Miller, and van Peski.
Wednesday 26 April, 2017
15:15	V3-204	Ana Maria Brecan (Bayreuth) Cycle transversal flag domains of Hodge type [Abstract] Flag domains are open orbits of real semisimple Lie groups in flag manifolds of their complexifications. A special class of flag domains constitute the classifying spaces for variations of Hodge structure, namely period domains or more generally Mumford-Tate domains. In this talk I will consider the problem of classifying all equivariant embeddings of an arbitrary flag domain in a period domain satisfying a certain transversality condition. Satake studied this problem in the weight 1 case in connection to the study of (algebraic) families of abelian varieties where the transversality condition is trivial. In this talk I will describe certain combinatorial structures at the Lie algebra level, called Hodge triples, which are a generalization of sl(2)-triples and show how this structures provide a solution to the classification problem.
Wednesday 10 May, 2017
15:15	V3-204	Mikhail Borovoi (Tel Aviv) Cayley Groups [Abstract] I will start the talk with the classical "Cayley transform" for the special orthogonal group SO(n) defined by Arthur Cayley in 1846. A connected linear algebraic group G over C is called a Cayley group if it admits a Cayley map, that is, a G-equivariant birational isomorphism between the group variety G and its Lie algebra Lie(G). For example, SO(n) is a Cayley group. A linear algebraic group G is called stably Cayley if G x S is Cayley for some torus S. I will consider semisimple algebraic groups, in particular, simple algebraic groups. I will describe classification of Cayley simple groups and of stably Cayley semisimple groups. (Based on joint works with Boris Kunyavskii and others.) See also this answer: http://mathoverflow.net/a/101343/4149
16:30	V3-204	Hsueh-Yung Lin (Bonn) Constant cycle subvarieties and zero-cycles in some hyper-Kähler varieties [Abstract] We will discuss progress on Beauville’s question on the existence of a multiplicative splitting of the conjectural Bloch-Beilinson filtration for hyper-Kähler varieties, with a special emphasis on Voisin’s approach using constant cycle subvarieties to study the Chow group of 0-cycles. We will provide some evidence on some of her conjectures related to this approach.
Wednesday 24 May, 2017
15:15	V3-204	Abhishek Saha (Bristol) Integral representation and critical L-values for the standard L-function of a Siegel modular form [Abstract] I will talk about some of my recent work with Pitale and Schmidt where we prove an explicit pullback formula that gives an integral representation for the twisted standard L-function for a holomorphic vector-valued Siegel cusp form of degree n and arbitrary level. In contrast to all previously proved pullback formulas in this situation, our formula involves only scalar-valued functions despite being applicable to L-functions of vector-valued Siegel cusp forms. Further, by specializing our integral representation to the case n = 2, we prove an algebraicity result for the critical L-values in that case (generalizing previously proved critical-value results for the full level case).
16:30	V3-204	Yonghwa Cho (Bayreuth) Ulrich bundles on prime Fano threefolds of index 2 and degree 4 [Abstract] In this talk, we investigate the moduli space of Ulrich bundles on a prime Fano threefold X of index two and of degree 4, following the method developed by Kuznetsov and Lahoz-Macrì-Stellari. More precisely, we consider the semiorthogonal decomposition of D^b(X) into two line bundles and D^b(C), the derived category of a certain hyperelliptic curve C, and show that the projection functor to D^b(C) maps an Ulrich bundle of rank r to a vector bundle of rank r and degree 2r over C. We prove a partial converse of this: if F is a vector bundle of rank r and degree 2r over C with some vanishing conditions, then the inclusion functor to D^b(X) maps F to an Ulrich bundle of rank r. In this way, we prove that the moduli space of stable Ulrich bundles of rank r over X is isomorphic to an open subscheme (nonempty if and only if r>1) of the moduli space of stable bundles of rank r and degree 2r over C. This is a joint work with Yeongrak Kim and Kyoung-Seog Lee.
Wednesday 7 June, 2017
15:15	V3-204	Mingmin Shen (Amsterdam) Hyperkähler geometry and the rationality problem of cubic fourfolds [Abstract] In 1970's, Clemens and Griffiths proved that a smooth cubic threefold is not rational via a relation between the rationality of a threefold and its intermediate Jacobian. In some aspects, hyperkähler varieties behave in a similar way as abelian varieties. In the case of a cubic fourfold, such an analogy suggests a connection between the rationality problem and the geometry of the variety of lines which is a hyperkähler variety. In this talk I will explain how such a relation can be obtained using universal generation of 1-cycles on cubic hypersurfaces.
16:30	V3-204	Charles Vial (Bielefeld) Distinguished models of intermediate Jacobians [Abstract] Let X be a smooth projective variety defined over a subfield K of the complex numbers. It is natural to ask whether the complex abelian variety that is the image of the Abel-Jacobi map defined on algebraically trivial cycles admits a model over K. I will show that it admits a unique model making the Abel-Jacobi map equivariant with respect to the action of the automorphism group of the complex numbers fixing K. As a corollary, we answer a question of Mazur by showing that this model over the base field K is dominated by the Albanese variety of a product of components of the Hilbert scheme of X. We also recover a result of Deligne on complete intersections of Hodge level one. This is joint work with Jeff Achter and Yano Casalaina-Martin.
Wednesday 14 June, 2017
15:15	V3-204	Martin Raum (Gothenburg) The skew Maass lift [Abstract] This is joint work in progress with Ozlem Imamoglu and Olav Richter. The Maass lift is a lifting from Jacobi forms to Siegel modular forms of genus 2. Historically the first counterexamples to the naive generalization of the Ramanujan conjecture originated from it. The endoscopic classification shows that the A-packet for the corresponding Saito-Kurokawa lift is, in general, a doublet. We generalize the Maass lift to certain non-holomorphic Jacobi forms (skew-holomorphic Jacobi forms) to obtain the missing, real-analytic Siegel modular form. As a corollary, we discover generalized Maass relations. The key feature of our approach is a passing from real-analytic Siegel modular forms to holomorphic elliptic modular forms
17:15 -- COLLOQUIUM	H6	Raghuram (IISER-Pune) CANCELLED [Abstract] TBA
Wednesday 21 June, 2017
15:15	V3-204	Farrell Brumley (Paris 13) Counting cusp forms by analytic conductor [Abstract] A famous theorem of Drinfeld counts the number of rank 2 irreducible l-adic local systems of a curve over a finite field, by reducing the statement to one about cusp forms on GL2 over the corresponding function field. The resulting expression is reminiscent of a Lefschetz fixed point theorem on a smooth algebraic variety over a finite field. We study an analogous problem over number fields, posed directly on the automorphic side: count the number of cusp forms on GLn of bounded analytic conductor. In joint work with Djordje Milicevic, we provide an asymptotic for this quantity (subject to an additional archimedean condition when n>2). The answer is similar in shape to a well-known result of Schanuel, which counts rational points of bounded height on projective spaces.
Wednesday 28 June, 2017
15:15	V3-204	Guiseppe Ancona (Strasbourg) Standard conjectures for abelian fourfolds [Abstract] Let X be a smooth projective variety and V the finite-dimensional Q-vector space of algebraic cycles on X modulo numerical equivalence. Grothendieck defined a quadratic form on V (basically using the intersection product) and conjectured that it is positive definite. This conjecture is a formal consequence of Hodge theory in characteristic zero, but almost nothing is known in positive characteristic. Instead of studying the quadratic form at archimedean place (the signature), we will study it at the p-adic places. It turns out that this question is more tractable thanks to p-adic comparison theorems and the Shimura-Taniyama formula. Moreover, using a classical product formula for quadratic forms, the p-adic information will give us non-trivial information on the archimedean place. For instance, we will prove the original conjecture when X is an abelian variety of dimension (up to) 4.
16:30	V3-204	Bjorn Poonen (MIT) Bertini irreducibility theorems over finite fields [Abstract] The classical Bertini irreducibility theorem states that if X is a geometrically irreducible subvariety of P^n over an infinite field k, and dim X >= 2, then there exists a hyperplane H in P^n over k whose intersection with X is geometrically irreducible. This can fail if k is finite, but certain variants are true. For instance, we prove that if X is as above but k is finite, then the fraction of degree d hypersurfaces H whose intersection with X is geometrically irreducible tends to 1 as d tends to infinity. This result, which is more difficult than the Bertini smoothness theorem over finite fields proved in 2004, is joint work with François Charles
Wednesday 5 July, 2017
15:15	V3-204	Robert Laterveer (Strasbourg) Algebraic cycles on hyperkaehler varieties with automorphism. [Abstract] Let $X$ be a hyperkaehler variety, and let $\sigma$ be a non-symplectic automorphism. The Bloch-Beilinson conjectures predict the action of $\sigma$ on certain ``pieces’’ of the Chow groups of $X$. We will present some series of examples where this can be verified. This has some interesting consequences for the Chow ring of the quotient $X/\sigma$.
Wednesday 12 July, 2017
14:00	V3-204	Michael Rapoport (Bonn) On the tame coverings of the Drinfeld upper half plane [Abstract]
15:15	V3-204	Andreas Langer (Exeter) Higher Displays arising from filtered de Rham-Witt complexes [Abstract] For a smooth projective scheme X over a ring R on which p is nilpotent that meets some general assumptions we prove that the crystalline cohomology is equipped with the structure of a higher display which is a relative version of Fontaine's strongly divisible lattices. Frobenius-divisibilty is induced by the Nygaard filtration on the relative de Rham-Witt complex. For a nilpotent PD-thickening S/R we also consider the associated relative display and can describe it explicitly by a relative version of the Nygaard filtration on the de Rham-Witt complex associated to a lifting of X over S.
16:30	V3-204	Eike Lau (Paderborn) Display crystals for p-divisible groups [Abstract] We define a category of display crystals which classify p-divisible groups over F-finite noetherian schemes with arbitrary singularities.

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The Bielefeld Arithmetic Geometry Seminar 2016-2017

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2 November, 2016

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30 November, 2016

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25 January, 2017

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26 April, 2017

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10 May, 2017

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24 May, 2017

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7 June, 2017

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21 June, 2017

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28 June, 2017

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5 July, 2017

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12 July, 2017