Motivic Euler Products and Motivic Height Zeta Functions / Margaret Bilu.

Format:

Book

Author/Creator:

Bilu, Margaret, 1990- author.

Series:

Memoirs of the American Mathematical Society ; Volume 282.

Memoirs of the American Mathematical Society Series ; Volume 282

Language:

English

Subjects (All):

Arithmetical algebraic geometry.

Functions, Zeta.

Geometry, Algebraic.

Physical Description:

1 online resource (198 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"A motivic height zeta function associated to a family of varieties parametrised by a curve is the generating series of the classes, in the Grothendieck ring of varieties, of moduli spaces of sections of this family with varying degrees. This text is devoted to the study of the motivic height zeta function associated to a family of varieties with generic fiber having the structure of an equivariant compactification of a vector group. Our main theorem describes the convergence of this motivic height zeta function with respect to a topology on the Grothendieck ring of varieties coming from the theory of weights in cohomology. We deduce from it the asymptotic behaviour, as the degree goes to infinity, of a positive proportion of the coefficients of the Hodge-Deligne polynomial of the above moduli spaces: in particular, we get an estimate for their dimension and the number of components of maximal dimension. The main tools for this are a notion of motivic Euler product for series with coefficients in the Grothendieck ring of varieties, an extension of Hrushovski and Kazhdan's motivic Poisson summation formula, and a motivic measure on the Grothendieck ring of varieties with exponentials constructed using Denef and Loeser's motivic vanishing cycles"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Manin's problem in the arithmetic setting

1.2. Manin's problem via harmonic analysis

1.3. Manin's problem over function fields

1.4. Main result

1.5. Sketch of proof

Acknowledgments

Chapter 2. Grothendieck Rings of Varieties and Motivic Vanishing Cycles

2.1. Grothendieck rings of varieties

2.2. Motivic vanishing cycles

2.3. The motivic vanishing cycles measure

2.4. The Thom-Sebastiani theorem: an explicit example

Chapter 3. Motivic Euler Products

3.1. Symmetric products

3.2. Iteration of the symmetric product construction

3.3. Cutting into pieces

3.4. Symmetric products and affine spaces

3.5. Symmetric products of non-effective classes

3.6. Symmetric products of varieties with exponentials

3.7. Symmetric products in localised Grothendieck rings

3.8. Euler products

3.9. Double products

3.10. Allowing other constant terms

Chapter 4. Mixed Hodge Modules and Convergence of Euler Products

4.1. Mixed Hodge modules

4.2. Vanishing cycles and mixed Hodge modules

4.3. Symmetric products and vanishing cycles

4.4. Compatibility with motivic vanishing cycles

4.5. Weight filtration on Grothendieck rings of mixed Hodge modules

4.6. Weight filtration on Grothendieck rings of varieties

4.7. Convergence of power series

Chapter 5. The Motivic Poisson Formula

5.1. Symmetric products

5.2. Motivic Schwartz-Bruhat functions and Poisson formula

5.3. Families of Schwartz-Bruhat functions

5.4. Fourier transformation in families

5.5. Summation over ( )ⁿ

5.6. Poisson formula in families

Chapter 6. Motivic Height Zeta Functions

6.1. Geometric setting

6.2. Height zeta functions

6.3. Analysis of local factors and convergence

6.4. Proof of the main theorem and its corollary

Bibliography

Index.

Back Cover.

Notes:

Includes bibliographical references and index.

Description based on print version record.

Other Format:

Print version: Bilu, Margaret Motivic Euler Products and Motivic Height Zeta Functions

ISBN:

9781470473525