Arithmetic and Topology of Differential Equations

  title={Arithmetic and Topology of Differential Equations},
  author={Don Zagier},
The talk will describe arithmetic properties of solutions of linear differential equations, especially Picard-Fuchs equations, and their connections with modular forms, mirror symmetry, and invariants of algebraic varieties. (No prior knowledge of any of these topics will be assumed.) There will be cross-connections to many of the great achievements of Friedrich Hirzebruch, including his proportionality principle, his Riemann-Roch theorem, his resolution of the cusps of Hilbert modular surfaces… 

Picard–Fuchs equations for Shimura curves over Q

We show that Picard–Fuchs equations of periods of certain families of abelian surfaces with quaternionic multiplication have fractional powers of algebraic modular forms as solutions. We give several

Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality

For each of the simple Lie algebras g = Al, Dl or E6, we show that the all-genera one-point FJRW invariants of g-type, after multiplication by suitable products of Pochhammer symbols, are the

Geometric Langlands for hypergeometric sheaves

Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler--Gauss

Interpolated Apéry numbers, quasiperiods of modular forms, and motivic gamma functions

We extend the definitions of the sequences used by Apéry in his proof of the irrationality of ζ(3) to non-integral values of the index and relate the value with index −1/2 to the central value of the

On Gromov – Witten invariants of P 1

Here, (Σg, p1, . . . , pn) denotes an algebraic curve of genus g with at most double-point singularities as well as with the distinct marked points p1, . . . , pn, and the equivalence relation ∼ is

Interpolated Sequences and Critical L-Values of Modular Forms

Recently, Zagier expressed an interpolated version of the Apery numbers for \(\zeta (3)\) in terms of a critical L-value of a modular form of weight 4. We extend this evaluation in two directions. We

Goursat rigid local systems of rank four

We study the general properties of certain rank 4 rigid local systems considered by Goursat. We analyze when they are irreducible, give an explicit integral description as well as the invariant

D-brane masses at special fibres of hypergeometric families of Calabi-Yau threefolds, modular forms, and periods

We consider the fourteen families W of Calabi-Yau threefolds with one complex structure parameter and Picard-Fuchs equation of hypergeometric type, like the mirror of the quintic in P. Mirror

A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds

  • W. Zudilin
  • Mathematics
    Symmetry, Integrability and Geometry: Methods and Applications
  • 2018
We examine instances of modularity of (rigid) Calabi-Yau manifolds whose periods are expressed in terms of hypergeometric functions. The $p$-th coefficients $a(p)$ of the corresponding modular form

On congruence schemes for constant terms and their applications

  • A. Straub
  • Computer Science, Mathematics
    Research in Number Theory
  • 2022
This approach to algorithmically determine the modulo p r reductions of values of combinatorial sequences representable as constant terms is revisited, and a third natural type of scheme that combines benefits of automatic and linear ones is suggested.



Differential equations associated with nonarithmetic Fuchsian groups

We describe globally nilpotent differential operators of rank 2 defined over a number field whose monodromy group is a nonarithmetic Fuchsian group. We show that these differential operators have an

Linear Differential Equations and Group Theory from Riemann to Poincare

The origins of the theory of modular and automorphic functions are found in the work of Legendre, Gauss, Jacobi, and Kummer on elliptic functions and the hypergeometric equation. Riemann's work on

Exponential sums and di?erential equations

This book is concerned with two areas of mathematics, at first sight disjoint, and with some of the analogies and interactions between them. These areas are the theory of linear differential

Principles of Algebraic Geometry

A comprehensive, self-contained treatment presenting general results of the theory. Establishes a geometric intuition and a working facility with specific geometric practices. Emphasizes applications


This is the final project for Prof. Bhargav Bhatt’s Math 613 in the Winter of 2015 at the University of Michigan. The purpose of this paper is to prove a result of Grothendieck [Gro66] showing, for a

Simple Lie algebras and topological ODEs

For a simple Lie algebra $\mathfrak g$ we define a system of linear ODEs with polynomial coefficients, which we call the topological equation of $\mathfrak g$-type. The dimension of the space of


(writing fij; for Qj,/c) which satisfies the usual product rule and which extends to define a structure of complex on toy ®GV M, the " absolute de Rham complex " of (M, V). Now let S be a proper and

Modular embeddings of Teichmüller curves

Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle

Euler and algebraic geometry

Euler’s work on elliptic integrals is a milestone in the history of algebraic geometry. The founders of calculus understood that some algebraic functions could be integrated using elementary

Rational curves on hypersurfaces [after A. Givental]

This article accompanies my June 1998 seminaire Bourbaki talk on Givental's work. After a quick review of descendent integrals in Gromov-Witten theory, I discuss Givental's formalism relating