Name: Regulators V
Start: 2024-06-03T10:00:00+02:00
End: 2024-06-13T17:00:00+02:00
Location: Department of Mathematics, University of Pisa

Regulators V

from Monday, 3 June 2024 (10:00) to Thursday, 13 June 2024 (17:00)

Monday, 3 June 2024
10:00 Higher regulators of the ring of integers - Francis Brown (Oxford University)
Higher regulators of the ring of integers
- Francis Brown (Oxford University)
10:00 - 11:00
In keeping with the title of the conference, the first half of my talk will review the celebrated results of Borel, Minkowski, Quillen, and others on the rational K-theory of the integers, the stable cohomology of the general linear group, and the computation of the regulator in terms of odd zeta values. The second half of the talk will cover very recent results due to many authors who have completely transformed our understanding of this field. If time permits, I plan to discuss some of the following topics: how the Borel regulator is related to certain motives of graphs constructed by Bloch, Esnault, and Kreimer; how to construct an algebraic incarnation of the Borel-Serre compactification of $GL_n(\mathbb{Z})$; why the cohomology of $GL_n(\mathbb{Z})$ has additional structures; and finally, why we expect iterated extensions of motives to appear in its unstable cohomology.
11:00 Coffee break
Coffee break
11:00 - 11:30
11:30 Picard–Fuchs Equations of Dimensionally Regulated Feynman Integrals - Pierre Vanhove (Institut de Physique théorique)
Picard–Fuchs Equations of Dimensionally Regulated Feynman Integrals
- Pierre Vanhove (Institut de Physique théorique)
11:30 - 12:30
Feynman integrals are relative period integrals. In this lecture, I will discuss various algorithms for determining the D-module of differential equations satisfied by the Feynman integrals. In integer dimension, we have a (relative) period of a rational differential form. One approach is the use of the Griffiths-Dwork algorithms, which have been adapted to the case of non-isolated singularities, generically present for Feynman integrals. The analysis will be illustrated with a class of two-loop Feynman integrals. The study of the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters will be presented. We will demonstrate that the Hodge structure associated with planar two-loop Feynman graphs decomposes into mixed Tate pieces and the Hodge structures of families of hyperelliptic, elliptic, or rational curves, depending on the space-time dimension. In general dimensions, we have a twisted differential form. We will then explain how to extend the Griffiths-Dwork reduction, making particular use of the properties of the graph polynomials entering the definition of the integrand. We will examine the manner in which the twist is incorporated into the Picard-Fuchs operators.
12:30 Lunch
Lunch
12:30 - 14:30
14:30 Euler's constant and exponential motives - Javier Fresán (Ecole Polytechnique)
Euler's constant and exponential motives
- Javier Fresán (Ecole Polytechnique)
14:30 - 15:30
In the category of exponential motives over $\mathbb{Q}$ there is a new extension of $\mathbb{Q}(-1)$ by $\mathbb{Q}(0)$ which does not come from classical motives. Its period matrix features Euler's constant, which one is tempted to think of as the regularised value of Riemann's zeta function at 1. I will discuss several results and open questions revolving around this extension, for example the role it plays as a "monodromy factor" for differential equations of E-functions. The talk is based on joint work with Peter Jossen.
15:30 Coffee break
Coffee break
15:30 - 16:00
16:00 $K_2$ of elliptic curves over non-Abelian cubic and quartic fields - Rob de Jeu (Vrije Universiteit)
$K_2$ of elliptic curves over non-Abelian cubic and quartic fields
- Rob de Jeu (Vrije Universiteit)
16:00 - 17:00
After a review of some earlier results on (mostly) $K_2$ of curves, we give constructions of families of elliptic curves over certain cubic or quartic fields with three, respectively four, ‘integral’ elements in the kernel of the tame symbol on the curves. The fields are in general non-Abelian, and the elements linearly independent. For their integrality, we discuss a new criterion that does not ignore any torsion. We also verify Beilinson’s conjecture numerically for some of the curves. This is joint work with François Brunault, Liu Hang, and Fernando Rodriguez Villegas.
Tuesday, 4 June 2024
10:00 Modular regulators and multiple modular values - François Brunault (École Normale Supérieure de Lyon)
Modular regulators and multiple modular values
- François Brunault (École Normale Supérieure de Lyon)
10:00 - 11:00
I will present newly constructed elements in the $K_4$ group of modular curves, and explain how to compute their regulators in terms of L-functions of modular forms. One crucial tool is the theory of multiple modular values by Manin and Brown. This is joint work with Wadim Zudilin.
11:00 Coffee break
Coffee break
11:00 - 11:30
11:30 Linear relations between 1-periods - Emre Sertöz (Leiden University)
Linear relations between 1-periods
- Emre Sertöz (Leiden University)
11:30 - 12:30
I will sketch a modestly practical algorithm to compute all linear relations with algebraic coefficients between any given finite set of 1-periods. As a special case, we can algorithmically decide transcendence of 1-periods. This is based on the “qualitative description” of these relations by Huber and Wüstholz. We combine their result with the recent work on computing the endomorphism ring of abelian varieties. This is a work in progress with Jöel Ouaknine and James Worrell.
12:30 Lunch
Lunch
12:30 - 14:30
14:30 Heights and periods of limit mixed Hodge structures - Robin de Jong (University of Leiden)
Heights and periods of limit mixed Hodge structures
- Robin de Jong (University of Leiden)
14:30 - 15:30
This is a report on joint work with Spencer Bloch and Emre Sertöz. We aim to establish connections between the arithmetic height of certain cycles found in the resolution of odd-dimensional nodal projective hypersurfaces defined over the rationals, and periods of limit mixed Hodge structures found in the smoothening of such hypersurfaces.
15:30 Coffee break
Coffee break
15:30 - 16:00
16:00 Generalized Cross-Ratios - Spencer Bloch (University of Chicago)
Generalized Cross-Ratios
- Spencer Bloch (University of Chicago)
16:00 - 17:00
For $\mathsf{P}$ smooth projective variety of dim. $n$ over $\mathbb{C}$ complex numbers; $Y=Y_1-Y_2 $codim r algebraic cycle, $Z=Z_1-Z_2$ dim $r-1$ algebraic cycle, $Y$, $Z$ disjoint support and homologous to 0. Biextension $\mathsf{B} := H^{2r-1}(\mathsf{P}\smallsetminus Y,Z;\mathbb{Q}(r))$ mixed $\mathbb{Q}_{HS}$ with weights 0,-1,-2 and weight graded $W_{-2}\mathsf{B}=\mathbb{Q}(1)$, $gr^W_{-1}\mathsf{B}=H^{2r-1}(\mathsf{P},\mathbb{Q}(r))$, and $gr^W_0 = \mathbb{Q}(0)$. Degenerate case $gr^W_{-1}\mathsf{B}=(0)$ yields a Kummer extension $0 \to \mathbb{Z}(1) \to \mathsf{B} \to \mathbb{Z}(0)\to 0$. Such a Kummer extension carries a generalized cross-ratio $\lambda(\mathsf{B}) \in \mathbb{C}^*$. Examples and conjectures about generalized cross-ratios will be discussed.
17:30 Reception
Reception
17:30 - 19:00
Wednesday, 5 June 2024
10:00 Height pairings of cycles on modular varieties and L-functions - Wei Zhang (MIT)
Height pairings of cycles on modular varieties and L-functions
- Wei Zhang (MIT)
10:00 - 11:00
The Gross-Zagier formula relates the Neron-Tate height of Heegner divisors to the first central derivative of the L-function of elliptic curves over the rationals. This talk will focus on presenting several generalizations of this formula to higher dimensional modular varieties (often parameterizing abelian varieties with decorations), including conjectures made by Gan-Gross-Prasad, Kudla-Rapoport, and Yifeng Liu, and some recent developments towards them.
11:00 Coffee break
Coffee break
11:00 - 11:30
11:30 Hypergeometric families and Beilinson’s conjectures - Matt Kerr (Washington University, St. Louis)
Hypergeometric families and Beilinson’s conjectures
- Matt Kerr (Washington University, St. Louis)
11:30 - 12:30
I will describe the construction of motivic cohomology classes on hypergeometric families of Calabi-Yau 3-folds using Hadamard convolutions. These are analogous to elements of the Mordell-Weil group for families of elliptic curves, and produce solutions to certain inhomogeneous Picard-Fuchs equations. This is part of a joint project with Vasily Golyshev in which we numerically verify Beilinson’s conjectures in some new cases.
12:30 Lunch
Lunch
12:30 - 14:30
14:30 Stable I-surfaces of index 2 and generalized spin curves of genus 2 - Rita Pardini (Università di Pisa)
Stable I-surfaces of index 2 and generalized spin curves of genus 2
- Rita Pardini (Università di Pisa)
14:30 - 15:30
An I-surface (also called a (1,2)-surface) is a complex projective surface with $K^2=1$, $h^2(O)=2$ and ample canonical class. Gorenstein stable I-surfaces are hypersurfaces of degree 10 in $\mathbb{P}(1,1,2,5)$. In order to study stable I-surfaces of index 2 we introduce generalized Gorenstein spin curves, namely pairs $(C,L)$ where $C$ is a Gorenstein curve with ample canonical class and $L$ is a torsion-free rank 1 sheaf on $C$ with $\chi(L)=0$ admitting a generically injective map $L\otimes L\to\omega_C$. We obtain a complete classification of such pairs with C reduced of genus 2 and derive from it the classification of stable I-surfaces of index 2 with a reduced canonical curve.
15:30 Coffee break
Coffee break
15:30 - 16:00
16:00 Atypical Hodge Loci - Phillip Griffiths (Institute for Advanced Study)
Atypical Hodge Loci
- Phillip Griffiths (Institute for Advanced Study)
16:00 - 17:00
In the recent works of a number of people there has emerged a new perspective on Hodge loci. A central result in that development appears in a paper by Baldi, Klinger, and Ullmo*. In this talk, we will explain their result and give a proof. The essential step it to use the integrability conditions associated to a Pfaffian PDE system. *Baldi, Gregorio; Klingler, Bruno; Ullmo, Emmanuel. On the distribution of the Hodge locus. Invent. Math. 235 (2024), no. 2, 441–487
Thursday, 6 June 2024
09:30 Algebraizability of vector bundles and motivic homotopy theory - Tom Bachmann (University of Mainz)
Algebraizability of vector bundles and motivic homotopy theory
- Tom Bachmann (University of Mainz)
09:30 - 10:30
I will outline a program envisioned by Mike Hopkins to prove that all topological vector bundles on certain varieties are algebraizable, and I will report on recent progress on implementing this program. In particular I will explain how to combine convergence theorems of Levine and Bousfield--Kan to construct an unstable Novikov spectral sequence in motivic homotopy theory.
10:30 Coffee break
Coffee break
10:30 - 11:00
11:00 Evaluations of areal Mahler measure - Matilde Lalín (University of Montreal)
Evaluations of areal Mahler measure
- Matilde Lalín (University of Montreal)
11:00 - 12:00
The (logarithmic) Mahler measure of a non-zero rational function $P$ in $n$ variables is defined as the mean of $\log |P|$ (with respect to the normalized arclength measure) restricted to the standard $n$-dimensional unit torus. It has been related to special values of L-functions via regulators. Pritsker (2008) defined the areal Mahler measure, which is obtained by replacing the normalized arclength measure on the standard $n$-torus by the normalized area measure on the product of $n$ open unit disks. In this talk, we will investigate some similarities and differences between the two versions of Mahler measure. We will also discuss some evaluations of the areal Mahler measure of multivariable polynomials, which also yields special values of L-functions.
12:00 Break
Break
12:00 - 12:15
12:15 Motives, Periods and Species - Annette Huber-Klawitter (University of Freiburg)
Motives, Periods and Species
- Annette Huber-Klawitter (University of Freiburg)
12:15 - 13:15
The Period Conjecture makes a qualitative prediction about all linear relations between the periods of motives. It is a theorem in the case of 1-motives, e.g. for numbers like logarithms or algebraic numbers or periods of elliptic curves over number fields. In joint work with Martin Kalck, we explain how to deduce dimension formulas via the structure theory of finite dimensional algebras over perfect fields.
13:15 Free afternoon
Free afternoon
13:15 - 17:00
Friday, 7 June 2024
10:00 A topological formula for the central value of symplectic L-functions and Reidemeister torsion - Amina Abdurrahman (IHES)
A topological formula for the central value of symplectic L-functions and Reidemeister torsion
- Amina Abdurrahman (IHES)
10:00 - 11:00
We give a global cohomological formula for the central value of the L-function of a symplectic representation on a curve up to squares. It involves a map C. Soule defined in his work on the Lichtenbaum conjectures. The proof relies crucially on a similar formula for the Reidemeister torsion of 3-manifolds. We sketch both analogous arithmetic and topological pictures. This is based on joint work with A. Venkatesh.
11:00 Coffee break
Coffee break
11:00 - 11:30
11:30 Continuity principle in ramification theory - Tomoyuki Abe (Kavli IPMU)
Continuity principle in ramification theory
- Tomoyuki Abe (Kavli IPMU)
11:30 - 12:30
One of the goals of ramification theory is to compute the Euler-Poincare characteristic of a given sheaf by using an invariant measuring the ramification. Bloch's revolutionary approach to this problem is to measure using CH_0. After a long sought, this element of CH_0 had finally been constructed by T. Saito, by constructing the characteristic cycle. In the first half of the talk, I'll explain an alternative construction using "continuity principle". In the second half, using the continuity principle in another way, I'll give a proof to a conjecture of Serre on the construction of Artin representation in the equal characteristic case.
12:30 Lunch
Lunch
12:30 - 14:30
14:30 Unlikely Intersections and applications to Diophantine Geometry - Laura Capuano (Università di Roma III)
Unlikely Intersections and applications to Diophantine Geometry
- Laura Capuano (Università di Roma III)
14:30 - 15:30
The Zilber-Pink conjectures on unlikely intersections deal with intersections of subvarieties of a (semi)abelian variety or, more in general, of a Shimura variety, with “special” subvarieties of the ambient space. These conjectures generalize many classical results such as Faltings’ Theorem (Mordell Conjecture), Raynaud’s Theorem (Manin-Mumford Conjecture) and André-Oort Conjecture and have been studied by several authors in the last two decades. Most proofs of results in this area follow the well-established Pila-Zannier strategy, first introduced by the two authors in 2008 to give an alternative proof of Raynaud’s theorem as a combination of results coming from o-minimality (Pila-Wilkie’s theorem) with other Diophantine ingredients. The talk will focus on a general introduction to these problems, on some results for semi-abelian varieties and families of abelian varieties, and on applications to other problems of Diophantine nature.
15:30 Coffee break
Coffee break
15:30 - 16:00
16:00 Sato-Tate groups of Fermat Jacobians - Davide Lombardo (Università di Pisa)
Sato-Tate groups of Fermat Jacobians
- Davide Lombardo (Università di Pisa)
16:00 - 17:00
The arithmetic of abelian varieties is often studied through the lens of their Galois representations. Given an abelian variety $A$ over a number field $K$, an important invariant is the so-called Sato-Tate group $\operatorname{ST}(A)$, a compact Lie group which conjecturally describes the asymptotic distribution of the characteristic polynomials of Frobenius acting on the Tate modules of $A/K$. The group of connected components of $\operatorname{ST}(A)$ has particular arithmetic significance, and there exists a unique minimal extension $L/K$ such that $\operatorname{ST}(A_L)$ is connected. There is currently no general technique to determine $\operatorname{ST}(A)$, nor the extension $L/K$. In this talk I will describe how to compute these two invariants for the Jacobian of the curve $y^2=x^m+1$ by relating them to the cohomology of (several) Fermat hypersurfaces $X_m^n : Y_0^m + \cdots + Y_{n+1}^m=0$. The structure of this cohomology has been studied extensively by Deligne; our application, however, will require a more detailed analysis of the action of the absolute Galois group of $\mathbb{Q}$ on the étale cohomology of $X_m^n$.
Saturday, 8 June 2024
Sunday, 9 June 2024