Memoirs of the American Mathematical Society, 0065-9266 ; number 1396 [More in this series]
Summary note
"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."--Page v.
Notes
"February 2023, volume 282, number 1396 (fifth of 6 numbers)"
Bibliographic references
Includes bibliographical references (pages 179-182) and index.
Contents
Chapter 1. Introduction
Chapter 2. Grothendieck rings of varieties and motivic vanishing cycles
Chapter 3. Motivic Euler products
Chapter 4. Mixed hodge modules and convergence of Euler products
Chapter 5. The motivic Poisson formula
Chapter 6. Motivic height zeta functions
Bibliography
Index.
ISBN
9781470460211 ((paperback))
1470460211 ((paperback))
OCLC
1372139525
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