Formal Fourier Jacobi expansions and special cycles of codimension two

2013/02/28 by Martin Raum, Martin Westerholt-Raum
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Codimension #Degree (music) #Fourier series #Modular form #Modularity (biology) #Siegel modular form #Theta function #math.AG #math.NT #msc:11F30 #msc:11F46 #msc:11G18 #msc:11Y40

paper · pdf · doi:10.1112/s0010437x15007514

published as Compositio Mathematica 151 (2015) 2187-2211 · 22 pages

arxiv created 2013/09/22 · openalex publication_date 2015/08/06 · arxiv updated 2015/12/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We prove that formal Fourier Jacobi expansions of degree two are Siegel modular forms. As a corollary, we deduce modularity of the generating function of special cycles of codimension two, which were defined by Kudla. A second application is the proof of termination of an algorithm to compute Fourier expansions of arbitrary Siegel modular forms of degree two. Combining both results enables us to determine relations of special cycles in the second Chow group.

Citations

• Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I
• Intersection numbers of cycles on locally symmetric spaces and Fourier coefficients of holomorphic modular forms in several complex variables
• �ber eine Spezialschar von Modulformen zweiten Grades