# Arithmetic and Topology of Differential Equations

@inproceedings{Zagier2016ArithmeticAT, title={Arithmetic and Topology of Differential Equations}, author={Don Zagier}, year={2016} }

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…

## 15 Citations

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

- Mathematics, Physics
- 2021

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

- MathematicsTransactions of the American Mathematical Society
- 2021

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

- Mathematics
- 2021

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

- 2019

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

- MathematicsTexts & Monographs in Symbolic Computation
- 2019

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

- MathematicsResearch in the Mathematical Sciences
- 2018

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…

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

- Mathematics, PhysicsSymmetry, 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…

Taylor series for the Ap\'ery numbers

- Mathematics
- 2020

The sequence of Apery numbers can be interpolated to $\mathbb{C}$ by an entire function. We give a formula for the Taylor coefficients of this function, centered at the origin, as a…

New Representations for all Sporadic Apéry-Like Sequences, With Applications to Congruences

- MathematicsExperimental Mathematics
- 2021

Sporadic Apéry-like sequences were discovered by Zagier, by Almkvist and Zudilin and by Cooper in their searches for integral solutions for certain families of secondand third-order differential…

Apéry's irrationality proof, mirror symmetry and Beukers's modular forms

- Mathematics, Physics
- 2019

In this paper, we will study the connections between Apery's proof of the irrationality of $\zeta(3)$ and the mirror symmetry of Calabi-Yau threefolds. From the mysterious sequences in Apery's proof,…

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