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.

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