The Bielefeld Algebraic and Arithmetic Geometry Seminar 2024/2025

Welcome to the website of the Bielefeld Algebraic and Arithmetic Geometry Seminar! For more information, please contact one of the organizers: Claudia Alfes, Ana Botero, Eike Lau, Michael Spiess, Charles Vial, and Thomas Zink.

Talks

Wednesday 23 October, 2024
14:15	V2 210/216	Danylo Radchenko (U Lille) Fourier uniqueness pairs, interpolation formulas, and modular forms [Abstract] A Fourier uniqueness pair is a pair of subsets A, B of Euclidean space such that any function can be reconstructed from its values on A and the values of its Fourier transform on B. I will give an overview of recent constructions of Fourier uniqueness pairs, with an emphasis on explicit constructions using modular forms, and discuss their application in the sphere packing and energy minimization problems.
15:45	V2 210/216	Franco Rota (Glasgow) Towards Homological Mirror Symmetry for log Del Pezzo surfaces [Abstract] The homological mirror symmetry conjecture predicts a duality, expressed in terms of categorical equivalences, between the complex (or symplectic) geometry of a variety X and the symplectic (or complex) geometry of its mirror object Y. Motivated by this, we study a series of singular surfaces called log del Pezzo. I will describe the derived category of a series of log del Pezzo’s, using the McKay correspondence and explicit birational geometry. Then, I’ll mention early mirror results, focusing mostly on the special case of smooth low degree del Pezzo surfaces. For these, we will compare several mirror constructions using the language of pseudolattices. This is joint work with Giulia Gugiatti.
Wednesday 6 November, 2024
14:15	V2 210/216	Markus Kirschmer (Bielefeld) Chow groups of one-dimensional noetherian domains [Abstract] We discuss various connections between ideal classes, divisors, Picard and Chow groups of one-dimensional noetherian domains. As a result of these, we give a method to compute Chow groups of orders in global fields and show that there are infinitely many number fields which contain orders with trivial Chow groups. This is joint work with J. Klüners.
15:45	V2 210/216	Florian Breuer (Newcastle) Heights of modular polynomials [Abstract] For every positive integer $N$, the modular polynomial $\Phi_N(X,Y)$ has integer coefficients and vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of order $N$. These polynomials have applications in cryptography and define integral (but singular) models for the modular curves $X_0(N)$. Their coefficients grow rapidly with $N$. In this talk, I will explain recent joint work with Fabien Pazuki and Desir\'ee Gij\'on G\'omez obtaining explicit upper and lower bounds on the size of these coefficients. Our methods also lead to explicit bounds on the heights of Hecke images. If time allows, I can also outline analogous results for Drinfeld modular polynomials.
Wednesday 13 November, 2024
(CANCELLED!)	V2 210/216	Robin de Jong (Leiden) Canonical local heights on abelian varieties: explicit formulas [Abstract] We discuss Néron's canonical local heights on abelian varieties over non-archimedean valued fields from the point of view of Berkovich analytic spaces. We present a refinement of Néron's classical result relating canonical local heights with intersection multiplicities on the Néron model. We show how to reobtain, in our framework, Tate's explicit formulas for Néron's canonical local heights on elliptic curves. Our results can be viewed as extensions of Tate's formulas to higher dimensions. This is joint work with Farbod Shokrieh.
14:15	V2 210/216	Francesco Denisi (Paris Cite) MMP for Enriques pairs and singular Enriques varieties [Abstract] An Enriques manifold is a connected complex manifold that is not simply connected and whose universal covering is an irreducible holomorphic symplectic (IHS) manifold. Given a projective IHS manifold X and an effective R-divisor D such that the pair (X,D) is log canonical, Matsushita and Zhang showed that any sequence of flops starting from (X,D) terminates. Lehn and Pacienza proved, more generally, that any MMP starting from (X,D) terminates. The goal of this talk is twofold. First, we discuss an analogous result for Enriques pairs, which we define as log canonical pairs (Y, D), where Y is an Enriques manifold and D is an effective R-divisor. Second, we characterize the underlying variety of the resulting minimal model (Y’, D’) of (Y,D). This leads naturally to the definition of primitive Enriques varieties, for which we provide examples and explore some of their properties. The talk is based on joint work with Á. D. Ríos Ortiz, N. Tsakanikas and Z. Xie.
Wednesday 20 November, 2024
14:00	V2 210/216	Johannes Krah (Bielefeld) Phantoms and exceptional collections on rational surfaces [Abstract] TBD
16:30	V2 210/216	Alexander Kuznetsov Simultaneous categorical resolutions of singularities [Abstract] It is a classical fact, that for a family of surfaces over a smooth curve with smooth general fiber, if the special fiber has only rational double points as singularities, then after a possible finite base change, one can resolve the singularities of the total space and the special fiber by a common blowup. I will talk about a categorical version of this construction that surprisingly also works in higher dimensions and, if time permits, about its applications.
Wednesday 27 November, 2024
14:15	V2 210/216	Jan-Willem van Ittersum (Cologne University) Arithmetic and algebraic properties of multiple q-zeta values [Abstract] Okounkov conjectured that certain integrals over the Hilbert scheme of points of /any/ projective surface are multiple q-zeta values. We explain this conjecture and the relation of multiple q-zeta values to (quasi)modular forms, which gives two independent reasons to study the space of these series. We then move on by describing several number theoretic properties of the coefficients of these series. Finally, we describe the algebraic structure of this space. In particular, we introduce an sl2-action by derivations extending the action on quasimodular forms.
15:45	V2 210/216	David Ter-Borch Gram Lilienfeldt (Leiden University) Derivatives of Rankin-Selberg L-functions and heights of generalized Heegner cycles [Abstract] In the 1980s, Gross and Zagier obtained a formula expressing the heights of CM points on modular curves in terms of derivatives of certain L-functions, leading to applications towards the Birch and Swinnerton-Dyer conjecture for elliptic curves. In this talk, I will present a formula for the heights of certain algebraic cycles first introduced by Bertolini, Darmon, and Prasanna. This formula generalizes the Gross-Zagier formula to higher dimensions and has applications to the Beilinson-Bloch-Kato conjectures, notably in the case of Jacobians with CM. This is joint work with Ari Shnidman.
Wednesday December 11, 2024
14:15	V2 210/216	Martin Raum (Chalmers) Mod p^2 theta cycles of modular forms [Abstract] Based on joint work with Scott Ahlgren, Mike Hanson, and Olav Richter. We start by explaining modular forms modulo primes, and illustrate their relevance. Building up on this we define and investigate the theta operator on modular forms, which yields the concept of theta cycles. Modular forms modulo prime powers are much less understood than those modulo primes. In particular, their theta cycles are more erratic, as showcased by experimental data, but nonetheless reveals some patterns. We present congruences for some key modular forms, and show how these explain some of those patterns.
15:45	V2 210/216	Rızacan Çiloğlu (TU Darmstadt) Perverse sheaves on twisted affine flag varieties and Langlands duality [Abstract] The Geometric Satake equivalence, which asserts an equivalence of categories between perverse sheaves on the affine Grassmannian of a split reductive group and algebraic representations of its Langlands dual group, was recently extended to the setting of non-split reductive groups by Xinwen Zhu (tamely ramified groups) and Timo Richarz (arbitrary group). It is a natural question to ask whether a similar extension holds in the case of affine flag varieties. I will be talking about a first step in this direction, by extending work of Arkhipov-Bezrukavnikov on describing Iwahori-Whittaker equivariant perverse sheaves on affine flag variety of a split group to the setting of non-split reductive groups.
Wednesday 15 January, 2025
14:15	V2 210/216	Sebastián Herrero (University of Santiago de Chile/ETH Zurich) Special values of Green's functions on hyperbolic 3-space [Abstract] In 1986, Gross and Zagier proposed a conjecture on the algebraicity of special values of higher Green's functions on modular curves, which mirrors the properties of the modular j-function in the classical theory of complex multiplication. This conjecture was recently proven by Bruinier, Li, and Yang. In this talk, I will present new results on an analogous problem for special values of Green's functions on hyperbolic 3-space. We show that certain finite sums of these special values can be expressed in terms of logarithms of primes and logarithms of units in real quadratic fields. This is joint work with Özlem Imamoglu, Anna von Pippich, and Markus Schwagenscheidt.
15:45	V2 210/216	Robert Laterveer (Strasbourg) On the Chow ring of double EPW sextics [Abstract] Chow rings of Hyper-Kaehler varieties are expected to have particular properties. Notably, a famous conjecture made by Beauville and Voisin predicts that the subring (of the Chow ring of a Hyper-Kaehler variety) generated by divisors and Chern classes should inject into cohomology. Building on the work of many other people, I proved that this conjecture is true for double EPW sextics. I will also report on a recent joint work with M. Bolognesi, concerning double EPW sextics and their covering involution. In this work, we relate the action of the covering involution on zero-cycles to Voisin’s rational orbit filtration, in accordance with general conjectures.
Wednesday 29 January, 2025
14:15	V2 210/216	Giada Grossi (Universite Sorbonne Paris Nord) Kolyvagin systems and Selmer group bounds [Abstract] Since their origins in the works of Thaine and Kolyvagin, Euler systems have been used to produce bounds on certain Selmer groups (e.g. class groups of totally real fields, Tate-Shafarevich groups of rational elliptic curves, etc). In this talk, I will explain how improving such bounds for twists of the Galois representation, combined with some known results in the Iwasawa theory of elliptic curves, leads to the proof of Kolyvagin's conjecture on the non-vanishing of the Heegner point Kolyvagin system (as well as the analogous results for Kato's Euler system). This is joint work with A. Burungale, F. Castella, and C. Skinner.
15:45	V2 210/216	Josefien Kuijper (Utrecht) Uniqueness for six-functor formalisms [Abstract] In recent years, efforts have been made to formalise, in the most efficient way, Grothendieck's six operations of sheaves (tensor product and internal hom, inverse and direct image, and exceptional inverse and direct image) and their properties. In addition to these operations, one might want to encode natural isomorphisms between the inverse image and exceptional inverse image for a certain class of "étale" morphisms, and between the direct image and the exceptional direct image for a class of "proper morphisms". These can be encoded as extra data, using (infinity,2)-categories, but this is not necessary. More recently, it has become clear that these natural isomorphisms arise canonically from a property of the six operations themselves. We formulate this condition in an efficient way, leading to the notion of a "Nagata six-functor formalism". We show that Nagata six-functor formalisms are uniquely determined by the data of the tensor product and the inverse image functor alone, giving a positive answer to a conjecture by Scholze. This talk is based on joint work with Adam Dauser.

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The Bielefeld Algebraic and Arithmetic Geometry Seminar 2024/2025

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27 November, 2024

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December 11, 2024

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29 January, 2025