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Three vignettes on hypergeometric normal functions

Matt Kerr

TL;DR

The paper develops a Hodge-theoretic framework to study normal functions in families of algebraic varieties, linking them to Hodge classes via the Lefschetz correspondence and Beilinson–Hodge theory. It focuses on hypergeometric variations of Hodge structure and provides three vignettes that connect Lefschetz theory to Ramanujan-type identities, algebraicity results for exponentials of integrals, and the theory of quasi-periods. In particular, it proves that balanced, interlaced hypergeometric data yield algebraic exponentials and elucidates how integral structures and Ur-convolution encode the Lefschetz relationship between normal functions and Hodge classes. The results bridge arithmetic, algebraic geometry, and special function theory, offering explicit constructions and computational paths for quasi-periods and regulators in hypergeometric settings.

Abstract

We use Hodge-theoretic methods to (i) explain number-theoretic identities of a type recently considered by Guillera and Zudilin, (ii) describe the Frobenius dual of Abel-Jacobi period functions, and (iii) offer a new proof of Golyshev's conjecture on algebraic hypergeometrics aided by an argument in the spirit of Lefschetz's (1,1) theorem.

Three vignettes on hypergeometric normal functions

TL;DR

The paper develops a Hodge-theoretic framework to study normal functions in families of algebraic varieties, linking them to Hodge classes via the Lefschetz correspondence and Beilinson–Hodge theory. It focuses on hypergeometric variations of Hodge structure and provides three vignettes that connect Lefschetz theory to Ramanujan-type identities, algebraicity results for exponentials of integrals, and the theory of quasi-periods. In particular, it proves that balanced, interlaced hypergeometric data yield algebraic exponentials and elucidates how integral structures and Ur-convolution encode the Lefschetz relationship between normal functions and Hodge classes. The results bridge arithmetic, algebraic geometry, and special function theory, offering explicit constructions and computational paths for quasi-periods and regulators in hypergeometric settings.

Abstract

We use Hodge-theoretic methods to (i) explain number-theoretic identities of a type recently considered by Guillera and Zudilin, (ii) describe the Frobenius dual of Abel-Jacobi period functions, and (iii) offer a new proof of Golyshev's conjecture on algebraic hypergeometrics aided by an argument in the spirit of Lefschetz's (1,1) theorem.
Paper Structure (7 sections, 8 theorems, 79 equations)

This paper contains 7 sections, 8 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Normal functions
  3. Hypergeometrics
  4. Vignette #1: Identities
  5. Lefschetz correspondence revisited
  6. Vignette #2: Algebraicity
  7. Vignette #3: Quasi-periods

Key Result

Proposition 2.10

The inhomogeneity$L\mathfrak{V}=:g$ is a polynomial in $z$ with $\deg(g)\leq d-h$. Further, if $\nu$ is nonsingular at $0$ then we have $z\mid g$; and if $\nu=\nu_{\mathcal{Z}}$ is motivic (with $\mathcal{Z}$ defined/$K\subset\overline{\mathbb{Q}}$) then $g\in K[z]$.

Theorems & Definitions (20)

  • Proposition 2.10
  • Proposition 3.2
  • Example 3.4
  • Proposition 3.8
  • proof
  • Remark 3.9
  • Remark 4.4
  • Theorem 4.11
  • Example 4.14
  • Remark 4.15
  • ...and 10 more