# 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…

## 22 Citations

### Picard–Fuchs equations for Shimura curves over Q

- MathematicsBulletin of the London Mathematical Society
- 2020

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

- Mathematics
- 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

- Mathematics
- 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…

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

- Mathematics
- 2022

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

- MathematicsSymmetry, 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

- Computer Science, MathematicsResearch 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.

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