Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 1

Special Issue in honor of Don Zagier

Guest editors: Benedict H. Gross, Ken Ono, and Fernando Rodriguez Villegas

On $p$-integrality of instanton numbers

Pages: 7 – 44

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n1.a2


Frits Beukers (Department of Mathematics, Utrecht University, Utrecht, Netherlands)

Masha Vlasenko (Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland)


We show integrality of instanton numbers in several key examples of mirror symmetry. Our methods are essentially elementary, they are based on our previous work in the series of papers called Dwork crystals I, II and III.


instanton number, Picard–Fuchs equation, Frobenius structure, $p$-adic cohomology

2010 Mathematics Subject Classification

Primary 12H25, 14N35. Secondary 14F30.

The work of Frits Beukers was supported by the Netherlands Organisation for Scientific Research (NWO), grant TOP1EW.15.313.

The work of Masha Vlasenko was supported by the National Science Centre of Poland (NCN), grant UMO-2020/39/B/ST1/00940.

Received 21 September 2021

Accepted 13 May 2022

Published 3 April 2023