Last week (!) April 16 – 20

* Thursday, April 19th, 14:00

Differential equations in characteristic 0 and p
Marius van der Put (Groningen University)

Linear differential equations over the field of complex numbers are largely governed by differential Galois theory, monodromy and Stokes matrices. The good notion of linear differential equation over a field of positive characteristic is called “iterative differential module” or ”stratified bundle”. Differential Galois theory exists in this case. The Tannaka group of the stratified bundles on a variety is the algebraic fundamental group in characteristic p. The main problem is producing stratified bundles and the inverse problem for differential Galois group. We will compare characteristic 0 and p. Further methods for the construction of stratified bundles on curves are presented.

* Thursday, April 19th, 11:00

Cohomology of arithmetic groups, periods and special values II
Günter Harder (MPIM)

abstract

* Wednesday, April 18th, 14:30 (Venue: MPIM)

First order differential equations
Marius van der Put (Groningen University)

We consider a first order differential equation of the form f(y′;y)=0 with f∈K[S;T] and K a differential field either complex or of positive characteristic. We investigate several properties of f, namely the ‘Painlevé property’ (PP), solvability and stratification. A modern proof of the classication of first order equations with PP is presented for all characteristics. A version of the Grothendieck-Katz conjecture for first order equations is proposed and proven for special cases. Finally the relation with Malgrange’s Galois groupoids and model theory is discussed.

* Tuesday, April 17th, 14:30

Modular forms in Pari/GP
Henri Cohen (Université de Bordeaux)

The aim of this talk is to describe the new modular forms package available in Pari/GP which has a number of features not available in other packages, in particular expansion at all cusps and computation of arbitrary Petersson products.

* Tuesday, April 17th, 11:00
Cohomology of arithmetic groups, periods and special values I
Günter Harder (MPIM)

* Monday, April 16th, 16:30
Serre-Tate theory for Calaby-Yau varieties
Piotr Achinger (Instytut Matematyczny PAN)

Classical Serre–Tate theory describes deformations of ordinary abelian varieties. It implies that every such variety has a canonical lift to characteristic zero. Variants of this theory have been obtained for ordinary K3 surfaces by Nyygard and Ogus, and for ordinary curves by Mochizuki. I will report on a joint project in progress with Maciej Zdanowicz (EPFL), in which we construct canonical liftings of ordinary Calabi–Yau varieties modulo p^2. Consequently, we obtain a Frobenius
lifting on the moduli space of such varieties, and a weak form of canonical coordinates. The construction uses Frobenius splittings and a ‘relative’ version of Witt vectors of length two, and seems closely related to results of Deligne-Illusie on the decomposition of the de Rham complex.

Advertisements

This week (April 9 – 13) update and recap

* Tuesday April 10, 16:30

Elliptic dilogarithm (Discussion)

* Wednesday April 11, 14:30
@MPI

MPI/HIM Number Theory Lunch Seminar

p-adic multiple zeta values and p-adic pro-unipotent harmonic actions I

David Jarossay (University of Geneve)

Multiple zeta values are periods of the pro-unipotent fundamental groupoid of the projective line minus three points. We will explain a way to compute their p-adic analogues, which keeps track of the motivic Galois action, and which has an application to the finite multiple zeta values recently studied by Kaneko and Zagier. The computation will be expressed by means of new objects which we will call p-adic pro-unipotent harmonic actions.

* Thursday April 12, 14:30
@HIM

p-adic multiple zeta values and p-adic pro-unipotent harmonic actions II

* Friday April 13,

Two talks by V. Vologodsky

11:00 Canonical coordinates

14:00 Prismatic site (after Bhatt-Schotze)

This week (April 9 – 13)

* Thursday, April 12th, 14:30

P-adic multiple zeta values, pro-unipotent harmonic actions and multiple harmonic values

David Jarossay (University of Geneva)

Abstract
We will compute the p-adic analogues of multiple zeta values in a way which keeps track of the motivic Galois action on the pro-unipotent fundamental groupoid of the projective line minus three points. This will be expressed by means of new objects which we will call pro-unipotent harmonic actions.

Talks this week (April 3 – 6)

* April 5 (Thu):
11:00 Henri Cohen
Computing Peterson products in half-integral weight (after Nelson and Collins)

Abstract: It is now well-known that one can compute the petersson product of modular forms of integral weight k >= 2 by using formulas similar to Haberland’s formula,
essentially by using modular symbols. These are not available in weight 1 and 1/2-integral weight. Following Nelson and Collins we will show how to compute Petersson products also in these cases using a different method. In the integral weight k >= 2 this method is also applicable but considerably slower.

14:00 Wadim Zudilin
Many (more) odd zeta values are irrational

Abstract: We considerably improve the asymptotic lower bound on the number of irrational odd zeta values as originally given in the Ball–Rivoal theorem. The proof is based on the construction of several linear forms in odd zeta values with related coefficients.

* April 6 (Fri):

14:30 Dali Shen

Title: Interpreting Lauricella hypergeometric system as a Dunkl system

Abstract: In the 80’s of last century, Deligne and Mostow studied the monodromy problem of Lauricella hypergeometric functions and gave a rigorous treatment on the subject, which provides ball quotient structures on $\mathbb{P}^n$ minus a hyperplane configuration of type $A_{n+1}$. Then some 20 years later Couwenberg, Heckman and Looijenga developed it to a more general setting by means of the Dunkl connection, which deals with the geometric structures on projective arrangement complements. In this talk, I will briefly review the Lauricella system first and then explain how to fit it into the picture of Dunkl system.

Updated abstract for J. Voight’s talk

The abstract of John Voight’s talk tomorrow got somehow messed up in the announcements. Here is the actual abstract for his talk.

Title: On the hypergeometric decomposition of symmetric K3 quartic pencils

Abstract: We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchs differential equations; we count points using Gauss sums and rewrite this in terms of finite field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write
this in terms of global $L$-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives.

This is joint work with Charles F. Doran, Tyler L. Kelly, Adriana Salerno, Steven Sperber, and Ursula Whitcher.

Workshop III: “Picard-Fuchs Equations and Hypergeometric Motives” Abstracts

Abstracts

Frits Beukers: Some supercongruences of arbitrary length

In joint work with Eric Delaygue it is shown that truncated hypergeometric sums with parameters ½,…,½ and 1,…,1 and evaluated at the point 1 are equal modulo p to Dwork’s unit-root eigenvalue modulo p2. Congruences modulo p follow directly from Dwork’s work, the fact that the congruence holds modulo p2 accounts for the name ‘supercongruence’.

R. Paul Horja: Spherical Functors and GKZ D-modules

Some classical mirror symmetry results can be recast using the more recent language of spherical functors. In this context, I will explain a Riemann-Hilbert type conjectural connection with the GKZ D-modules naturally appearing in toric mirror symmetry.

Kiran S. Kedlaya: Frobenius structures on hypergeometric equations: computational methods

Current implementations of the computation of L-functions associated to hypergeometric motives in Magma and Sage rely on a p-adic trace formula. We describe and demonstrate (in Sage) an alternate approach based on computing the right Frobenius structure on the hypergeometric equation. This gives rise to a conjectural formula for the residue at 0 of this Frobenius structure in terms of p-adic Gamma functions, related to Dwork’s work on generalized hypergeometric functions.

Robert Kucharczyk: The geometry and arithmetic of triangular modular curves

In this talk I will take a closer look at triangle groups acting on the upper half plane. Except for finitely many special cases, which are highly interesting in themselves, these are non-arithmetic groups. However, a notion of congruence subgroup is well-defined for these, and there are natural moduli problems that are classified by quotients of the upper half plane by such subgroups, giving rise to models over number fields. These curves have much to do with very classical mathematics, and they build a bridge between the hypergeometric world and the world of Shimura varieties. This is ongoing joint work with John Voight, who is also present at this conference.

Bartosz Naskręcki: Elliptic and hyperelliptic realisations of low degree hypergeometric motives

In this talk we will discuss what are the so-called hypergeometric motives and how one can approach the problem of their explicit construction as Chow motives in explicitely given algebraic varieties. The class of hypergeometric motives corresponds to Picard-Fuchs equations of hypergeometric type and forms a rich family of pure motives with nice L-functions. Following recent work of Beukers-Cohen-Mellit we will show how to realise certain hypergeometric motives of weights 0 and 2 as submotives in elliptic and hyperelliptic surfaces. An application of this work is the computation of minimal polynomials of hypergeometric series with finite monodromy groups and proof of identities between certain hypergeometric finite sums, which mimics well-known identities for classical hypergeometric series. This is a part of the larger program conducted by Villegas et al. to study the hypergeometric differential equations (special cases of differential equations ‘”coming from algebraic geometry'”) from the algebraic perspective.

Madhav Nori: Semi-Abelian Motives

joint work with Deepam Patel

Danylo Radchenko: Goursat rigid local systems of rank 4

I will talk about certain rigid local systems of rank 4 considered by Goursat, with emphasis on explicit constructions and examples. The talk is based on joint work with Fernando Rodriguez Villegas.

Damian Rössler: The arithmetic Riemann-Roch theorem and Bernoulli numbers

(with V. Maillot) We shall show that integrality properties of the zero part of the abelian polylogarithm can be investigated using the arithmetic Adams-Riemann-Roch theorem. This is a refinement of the arithmetic Riemann-Roch theorem of Bismut-Gillet-Soulé-Faltings, which gives more information on denominators of Chern classes than the original theorem. We apply this theorem to the Poincaré bundle on an abelian scheme and and the final calculation involves a variant of von Staudt’s theorem.

On a canonical class of Green currents for the unit sections of abelian schemes. Documenta Math. 20 (2015), 631–668

Jan Stienstra: Zhegalkin zebra motives, digital recordings of Mirror Symmetry

I present a very simple construction of doubly-periodic tilings of the plane by convex black and white polygons. These tilings are the motives in the title. The vertices and edges in the tiling form a quiver (=directed graph) which comes with a so-called potential, provided by the polygons. Dual to this graph one has the bipartite graph formed by the black/white polygons and the edges in the tiling. We deform this structure by putting weights on the edges and connect this with representations of the Jacobi algebra of the quiver with potential and with the Kasteleyn matrix of the bi-partite graph.

Duco van Straten: Frobenius structure for Calabi-Yau operators

This is a report on joint work in progress with P. Candelas and X. de la Ossa on the (largly conjectural) computation of Euler factors from Calabi-Yau operators. The method uses Dworks deformation method starting from a simple Frobenius matrix at the MUM-point that involves a p-adic version of ζ(3). We give some new applications, in particular to the determination of congruence levels.

Alexander Varchenko: Solutions of KZ differential equations modulo p

Polynomial solutions of the KZ differential equations over a finite field Fp will be constructed as analogs of multidimensional hypergeometric solutions.

Roberto Villaflor Loyola: Periods of linear algebraic cycles in Fermat varieties

In this talk we will show how a theorem of Carlson and Griffiths can be used to compute periods of linear algebraic cycles inside Fermat varieties of even dimension. As an application we prove that the locus of hypersurfaces containing two linear cycles whose intersection is of low dimension, is a reduced component of the Hodge locus in the underlying parameter space. Our method can be used to verify similar statements for other kind of algebraic cycles (for example complete intersection algebraic cycles) by means of computer assistance. This is joint work with Hossein Movasati.

Masha Vlasenko: Dwork crystals and related congruences

In the talk I will describe a realization of the p-adic cohomology of an affine toric hypersurface which originates in Dwork’s work and give an explicit description of the unit-root subcrystal based on certain congruences for the coeficients of powers of a Laurent polynomial. This is joint work with Frits Beukers.

John Voight: On the hypergeometric decomposition of symmetric K3 quartic pencils

We study the hypergeometric functions associated to five one-parameter deformations of Delsarte K3 quartic hypersurfaces in projective space. We compute all of their Picard–Fuchs differential
equations; we count points using Gauss sums and rewrite this in terms of finite field hypergeometric sums; then we match up each differential equation to a factor of the zeta function, and we write this in terms of global $L$-functions. This computation gives a complete, explicit description of the motives for these pencils in terms of hypergeometric motives.
This is joint work with Charles F. Doran, Tyler L. Kelly, Adriana Salerno, Steven Sperber, and Ursula Whitcher.

Mark Watkins: Computing with hypergeometric motives in Magma

We survey the computational vistas that are available for computing with hypergeometric motives in the computer algebra system Magma. Various examples that exemplify the theory will be highlighted.

Wadim Zudilin: A q-microscope for hypergeometric congruences

(https://arxiv.org/abs/1803.01830)

By examining asymptotic behavior of certain infinite basic (q
-) hypergeometric sums at roots of unity (that is, at a “q-microscopic” level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a q-analogue of Ramanujan’s formula of two supercongruences valid for all primes p>3, where S(N) denotes the truncation of the infinite sum at the N-th place and (−3⋅) stands for the quadratic character modulo 3.