Workshop on P-adic aspects of differential equations : Crystals, Mirror symmetry, Modular forms

November 5-8, 2007 in room AA.C.006

**Organizers:** V. Golyshev, L. Katzarkov, **D. Van Straten**

This event is part of the semester-long program on Algebraic, Symplectic Geometry and Physics held at Centre Bernoulli, EPFL, Lausanne, Switzerland. See the main program page for information about registration, accomodation, etc. Reminder: all participants (invited or not) should register. Please fill the registration form.

All the talks will take place in **room AAC 006** in **building SG**, near the Bernoulli Center (on the same level).

**ANDRE Yves (ENS, Paris)**

A survey of global crystalsA survey of global crystals

**BATYREV Victor (University of Tübingen, Germany)**

Towards computing quantum cohomology of Fano $3$-folds via conifold degenrationsLet $X$ be a Fano $3$-fold. We intend to compute the small quantum cohomology of $X$ using its conifold degeneration $X_0$ which is toric or a complete intersection in a toric variety. For this purpose, we give a classification of such degenerations. If Fano $3$-fold $X$ has Picard number $1$, then our approach produces all $17$ modular forms appearing in the Golyshev correpondence.

**BEUKERS Frits (Utrecht University, The Netherlands)**

p-adic properties of the Lame differential equation.Beside four singularities and the corresponding local exponents the Lame-differential equation has an extra parameter, known as the accessory parameter. It turns out that the value of this parameter has a very strong influence on the p-adic radii of convergence of the power series solutions. In this lecture we shall discuss this dependence.

**CANDELAS Philip (Oxford University, United Kingdom)**

Complex and p-Adic Special GeometryA complex Calabi-Yau manifold has a holomorphic three form that is unique up to multiplication by a holomorphic function of the parameters of the manifold. The fact that the parameter space of the manifold is the parameter space of the three form modulo this identification imposes a natural structure on the parameter space. This structure is well known in the complex context and is relevant in string theory, where it is known as Special Geometry. Somewhat surprising is the fact that it is possible to develop a largely parallel structure in the p-adic context.

**CORTI Alessio (Imperial College of London, United Kingdom)**

The quantum cohomology of weak Fano toric stacksI describe recent results with Coates, Iritani and Tseng on the calculation of the l-function

**CREW Richard (University of Florida, USA)**

Finiteness questions in the theory of arithmetic D-modulesBerthelot's theory of arithmetic D-modules is an analogue, for algebraic varieties over a field of positive characteristic, of the theory of perverse sheaves on a topological space. The major unsolved problems of this theory concern the preservation of holonomy by cohomological

operations; specifically the direct image by a proper map, and theextraordinary inverse image for a closed immersion. In this talk I shall survey recent results for extraordinary inverse images. In particular, this operation is known to preserve holonomy for the inclusion of a point in a smooth curve, and I will discuss the problem of extending this to higher dimension and codimension.**DE LA OSSA Xenia (Oxford University, United Kingdom)**

Mirror Symmetry and the Zeta-Function for the Quintic ThreefoldThe form of the zeta function, as a function of the parameter, for the Dwork pencil of quintic threefolds will be discussed. Particular reference will be made to the fact that, as classically defined, the zeta-function is almost but not quite mirror symmetric. It is intriguing that, properly understood, the "large parameter limit" of the numerator of the zeta-function is given by the denominator of the zeta-function for the mirror quintic. These same large parameter limits emerge also as the leading term in a 5-adic expansion of the zeta-function.

**DORAN Charles (University of Washington, USA)**

Normal Forms, K3 Moduli, and Modular ParametrizationsWe consider normal forms for certain lattice polarized K3 surfaces, realized as singular quartic hypersurfaces in P3 and generalizing the Weierstrass normal form for elliptic curves. Explicit algebraic correspondences are constructed between these surfaces and Hodge theoretically equivalent abelian surfaces, just as predicted by the Kuga-Satake Hodge conjecture. Furthermore, the Griffiths-Dwork method, when applied to these hypersurfaces, yields Picard-Fuchs equations describing their periods. Parametrizations of curves in moduli supporting further Picard lattice enhancement are then shown to solve a class of auxiliary nonlinear differential equations derived from these Picard-Fuchs equations. The result generalizes also to include modular parametrizations of "moonshine" type not arising from K3 surface moduli in this way.

**HAESSIG Douglas (University of Rochester, New York. USA)**

Mirror symmetry and quotients of zeta functionsIn 2002 - 03, Candelas, de la Ossa, and Rodriguez-Villegas considered the quotient of the zeta function of a quintic Calabi-Yau manifold and the zeta function of its mirror. This quotient turned out to be a polynomial, pure of weight 1. The purity suggested that the quotient came from a variety, or motive, of dimension 1. These (Euler) curves were identified. In 2006, using \ell-adic echniques, Wan extended the divisibility and purity result to higher dimensions. In this talk, we will recall these facts and discuss some p-adic aspects of this quotient.

**KATZARKOV Ludmil (University of Vienna, Austria)**

Birational Geometry and HMS SymmetryIn this talk we will consider questions of rationality of some Fano threefolds and fourfolds and will relate them to HMS.

**KEDLAYA Kiran (MIT, USA)**

Effective convergence bounds for p-adic differential equations with Frobenius structureConsider a system of first-order linear differential equations whose coefficients are bounded analytic functions on an open unit p-adic disc, such that the solutions converge in the full disc. The solutions are themselves not bounded, but Dwork and Robba gave effective logarithmic bounds for their growth at the boundary. Assume now that the system also admits a Frobenius structure; then a prediction of Dwork would imply much stronger bounds. For instance, for a Picard-Fuchs system arising from a family of smooth varieties with good reduction modulo p, the stronger bounds depend only on the relative dimension of the family, and not on the rank of the system. Using a method of Chiarellotto and Tsuzuki, we confirm Dwork's prediction.

**SPERBER Steven (University of Minnesota, USA)**

On analytic formulas for eigenvalues of Frobenius forsome families of exponential sumsWe present joint work with Adolphson in which we give p-adic analytic formulas for a class of exponential sums. These are nondegenerate sums on the n-fold torus whose Newton polyhedra contain the origin as an interior point. The proof involves giving a cohomological interpretation to a classical formula due to Schlomilch.

**STIENSTRA Jan (University of Utrecht, The Netherlands)**

Two-variable hypergeometric systems and dessins d'enfantsA recently discovered relation between on the one hand two-variable hypergeometric systems and on the other hand dessin d'enfants (graphs embedded in surfaces) will be discussed.

This includes:

(1) the role of the secondary polytope.

(2) the observation that the characteristic cycle (or principal A-determinant) of the Hypergeometric D-module equals the determinant of the bi-adjacency matrix which describes

the dessin d'enfants.

(3) the size of the bi-adjacency matrix equals the dimension of the solution space of the GKZ hypergeometric system of differential equations.

(4) perfect matchings on dimer models are refinements of the triangulations of the primary polytope in GKZ theory.**WAN Daqing (University of California, Irvine, California, USA)**

Moment Zeta Functions and Mirror SymmetryMoment zeta function provides a bridge between the classical zeta function and the p-adic zeta function for a morphism of schemes of finite type. In this talk, we discuss the moment zeta function for the mirror family of Calabi-Yau hypersurfaces over finite fields, and its connection to Dwork's p-adic unit root zeta function.

**YUI Noriko (Queen's University, Canada)**

Mirror Symmetry for Elliptic Curves**ZUDILIN Vadim (Moscow Lomonosov State University & Steklov Mathematical Institute, Russia)**

An Sp_4 modularity of Picard-Fuchs differential equations for Calabi-Yau threefolds (joint work with Yifan Yang)Motivated by the relationship of classical modular functions and Picard-Fuchs linear differential equations of order 2 and 3, we present an analogous concept for equations of order 4 and 5.