Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A motivic circle method

Margaret Bilu, Tim Browning

TL;DR

The paper develops a motivic analogue of the Hardy–Littlewood circle method over the complex function field, encoding exponential sums in the Grothendieck ring of varieties with exponentials and organizing their contributions via a motivic major/minor arc decomposition. It introduces a weight topology based on mixed Hodge modules to control convergence and proves a main asymptotic for the class of the moduli space $M_e$ as $e o ty$: $[M_e] = oldsymbol{L}^{(e)}ig(oldsymbol{ rak S}(f)oldsymbol{L}^{-N(n-1)}[ ext{...}]+R_eig)$ in the completed ring, where $oldsymbol{ rak S}(f)$ is a motivic Euler product and $R_e$ has strictly controlled weight. The approach yields a motivic analogue of local densities and integrals, relates to jet spaces $oldsymbol{ m abla}_N(f,ullet)$, and provides explicit results for the spaces of morphisms $ ext{Mor}_e(P^1,Z)$ and, in particular, the Fano variety of lines $F_1(Z)$ for smooth hypersurfaces, with information about their Hodge–Deligne polynomials. This framework achieves geometric information about $M_e$ and related moduli without field counting, and connects motivic phenomena to adelic-type structures envisaged in Manin–Peyre conjectures.

Abstract

The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results about moduli spaces of rational curves on hypersurfaces. In this paper a version of the circle method is implemented in the setting of the Grothendieck ring of varieties. This allows us to approximate the classes of these moduli spaces directly, without relying on point counting, and leads to a deeper understanding of their geometry.

A motivic circle method

TL;DR

The paper develops a motivic analogue of the Hardy–Littlewood circle method over the complex function field, encoding exponential sums in the Grothendieck ring of varieties with exponentials and organizing their contributions via a motivic major/minor arc decomposition. It introduces a weight topology based on mixed Hodge modules to control convergence and proves a main asymptotic for the class of the moduli space as : in the completed ring, where is a motivic Euler product and has strictly controlled weight. The approach yields a motivic analogue of local densities and integrals, relates to jet spaces , and provides explicit results for the spaces of morphisms and, in particular, the Fano variety of lines for smooth hypersurfaces, with information about their Hodge–Deligne polynomials. This framework achieves geometric information about and related moduli without field counting, and connects motivic phenomena to adelic-type structures envisaged in Manin–Peyre conjectures.

Abstract

The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results about moduli spaces of rational curves on hypersurfaces. In this paper a version of the circle method is implemented in the setting of the Grothendieck ring of varieties. This allows us to approximate the classes of these moduli spaces directly, without relying on point counting, and leads to a deeper understanding of their geometry.
Paper Structure (53 sections, 44 theorems, 400 equations)

This paper contains 53 sections, 44 theorems, 400 equations.

Table of Contents

  1. Introduction
  2. The Grothendieck ring of varieties with exponentials
  3. Functional interpretation
  4. A cohomological realisation
  5. Motivic functions and integrals
  6. Motivic Euler products
  7. The motivic circle method
  8. Approximation by rational functions
  9. Activation of the circle method
  10. The weight function
  11. Mixed Hodge modules
  12. The category of mixed Hodge modules
  13. The complex $\mathbf{Q}_X^{\mathrm{Hdg}}$
  14. Mixed Hodge modules with monodromy
  15. The weight filtration on Hodge modules
  16. ...and 38 more sections

Key Result

Theorem 1.1

Let $f\in \mathbf{C}[x_1,\dots,x_n]$ be a non-singular homogeneous polynomial of degree $d\geq 3$, defining a hypersurface $X\subset \mathbf{A}^n$. Assume $n > 2^d(d-1)$ and let $e\geq 1$. Then in $\widehat{\mathscr{M}_\mathbf{C}}$, where is a motivic Euler product and $R_e$ is an error term satisfying

Theorems & Definitions (98)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Remark 2.1: Interpretation as exponential sums
  • Lemma 2.2
  • proof
  • Remark 2.3
  • ...and 88 more