Boundedness of trace fields of rank two local systems

2022/10/24 by Yeuk Hay Joshua Lam, Lam, Yeuk Hay Joshua
Computer Science · Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2210.13563

openalex publication_date 2022/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a fixed prime number, and q a power of p. For any curve over \mathbbF_q and any local system on it, we have a number field generated by the traces of Frobenii at closed points, known as the trace field. We show that as we range over all pointed curves of type (g,n) in characteristic p and rank two local systems satisfying a condition at infinity, the set of trace fields which are unramified at p and of bounded degree is finite. This proves observations of Kontsevich obtained via numerical computations, which are in turn closely related to the analogue of Maeda's conjecture over function fields. The key ingredients of the proofs are Chin's theorem on independence of ℓ of monodromy groups, and the boundedness of abelian schemes of GL₂-type over curves in positive characteristics, obtained using partial Hasse invariants; the latter is an analogue of Faltings' Arakelov theorem for abelian varieties in our setting.

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