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On relative integral monodromy of abelian logarithms and normal functions

Yves André

TL;DR

The paper investigates relative integral monodromy for abelian schemes with a section, comparing it to the relative algebraic monodromy and showing that, under the lattice condition that the (global) monodromy $\Gamma$ is a lattice in $H_{\mathbb{R}}$, the relative integral monodromy $\Delta$ is a lattice in the relative vector group $U$ (i.e., of finite index in $U(\mathbb{Z})$). The author develops aHodge-theoretic framework to interpret holomorphic and algebraic sections, reduces to the modular (elliptic) case via Graber-Starr and Mok-To theory, and then leverages vanishing results for $H^1$ of arithmetic groups (e.g., Margulis-Starkov, Bass–Milnor–Serre) as well as pro-$\ell$ techniques to prove finiteness statements for Mordell-Weil groups of Kuga fiber spaces. The core contribution is a general method showing that, in many cases, the relative integral monodromy is a lattice, clarifying the relationship between integral and algebraic monodromy and extending to normal functions. The results have implications for Betti maps, the theory of 1-motives over $S$, and the study of sections in families of abelian varieties, offering a Hodge-theoretic path to lattice-ness and potential generalizations to Lefschetz pencils and normal functions.

Abstract

The relative algebraic monodromy of abelian logarithms (defined as the kernel of a map between algebraic monodromy groups attached to an abelian scheme with and without a section) was computed in \cite{A1}: under natural assumptions, this vector group turns out to be maximal. The relative integral monodromy of abelian logarithms is defined similarly as a kernel of integral monodromy groups, without taking Zariski closures. We show that if the integral monodromy of the abelian scheme is a lattice in the algebraic monodromy (which is not always the case), then the relative integral monodromy of the abelian logarithm is also a lattice in the relative algebraic monodromy. The proof uses a Hodge-theoretic interpretation of sections of abelian schemes. We also consider relative integral monodromy groups in the more general context of normal functions.

On relative integral monodromy of abelian logarithms and normal functions

TL;DR

The paper investigates relative integral monodromy for abelian schemes with a section, comparing it to the relative algebraic monodromy and showing that, under the lattice condition that the (global) monodromy is a lattice in , the relative integral monodromy is a lattice in the relative vector group (i.e., of finite index in ). The author develops aHodge-theoretic framework to interpret holomorphic and algebraic sections, reduces to the modular (elliptic) case via Graber-Starr and Mok-To theory, and then leverages vanishing results for of arithmetic groups (e.g., Margulis-Starkov, Bass–Milnor–Serre) as well as pro- techniques to prove finiteness statements for Mordell-Weil groups of Kuga fiber spaces. The core contribution is a general method showing that, in many cases, the relative integral monodromy is a lattice, clarifying the relationship between integral and algebraic monodromy and extending to normal functions. The results have implications for Betti maps, the theory of 1-motives over , and the study of sections in families of abelian varieties, offering a Hodge-theoretic path to lattice-ness and potential generalizations to Lefschetz pencils and normal functions.

Abstract

The relative algebraic monodromy of abelian logarithms (defined as the kernel of a map between algebraic monodromy groups attached to an abelian scheme with and without a section) was computed in \cite{A1}: under natural assumptions, this vector group turns out to be maximal. The relative integral monodromy of abelian logarithms is defined similarly as a kernel of integral monodromy groups, without taking Zariski closures. We show that if the integral monodromy of the abelian scheme is a lattice in the algebraic monodromy (which is not always the case), then the relative integral monodromy of the abelian logarithm is also a lattice in the relative algebraic monodromy. The proof uses a Hodge-theoretic interpretation of sections of abelian schemes. We also consider relative integral monodromy groups in the more general context of normal functions.
Paper Structure (15 sections, 3 theorems, 29 equations)

This paper contains 15 sections, 3 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. The case of families of elliptic curves: Hodge-theoretic interpretation of CZ
  3. Hodge-theoretic interpretation of sections of abelian schemes
  4. Holomorphic sections
  5. Algebraic sections
  6. A "Hodge-theoretic proof" of Theorem 1.2.1
  7. General setting
  8. Reduction to the case $\Delta =0$
  9. Reduction to finiteness of the Mordell-Weil groups of Kuga fiber spaces
  10. Variants and complements
  11. Vanishing theorems of $H^1$ for arithmetic groups
  12. The Margulis-Starkov vanishing theorem
  13. Reduction to the case where $H$ is $\mathbb Q$-simple
  14. The pro-$\ell$ picture; congruence subgroup property and vanishing theorems for $H^1$
  15. Vista: Integral relative monodromy of normal functions

Key Result

Theorem 1.1.1

Assume that $i)$$A/S$ has no fixed part, even after pull-back to a finite etale cover of $S$, $ii)$$\mathbb Z.\sigma(S)$ is Zariski-dense in $A$ ( i.e. the image of the section is not contained in any proper $S$-subgroup of $A$). Then there is a natural isomorphism $U(\mathbb Z)\cong H_1(A_s, \mathb

Theorems & Definitions (6)

  • Theorem 1.1.1
  • Theorem 1.2.1
  • Remark 2.1.1
  • Theorem 3.2.1
  • proof
  • Remark 5.2.1