Level structures on abelian varieties, Kodaira dimensions, and Lang's conjecture

2016/01/11 by Dan Abramovich, Anthony Várilly-Alvarado, Anthony Várilly‐Alvarado +2
Mathematics · #14K15 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 14K10 #Secondary 11G18 #math.AG #math.NT #msc:11G18 #msc:14K10 #msc:14K15

paper · pdf · doi:10.48550/arxiv.1601.02483

17 pages. References to new work of Brunebarbe added; discussion of implications arising from Lang's geometric conjecture suppressed in light of Brunebarbe's new results. Section 4 recast in more general terms; see Proposition 4.3 and Theorem 1.13

openalex publication_date 2016/01/11 · arxiv created 2016/11/13 · arxiv updated 2016/11/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Assuming Lang's conjecture, we prove that for a fixed prime p, number field K, and positive integer g, there is an integer r such that no principally polarized abelian variety A/K of dimension g has full level p^r structure. To this end, we use a result of Zuo to prove that for each closed subvariety X in the moduli space A_g of principally polarized abelian varieties of dimension g, there exists a level m_X such that the irreducible components of the preimage of X in A_g^[m] are of general type for m > m_X.

Citations

• A strong hyperbolicity property of locally symmetric varieties

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• Irreducibility of the Siegel moduli spaces with parahoric level structure