Gauss--Heilbronn Sums and Coverings of Deligne--Lusztig Type Curves

2025/12/12 by Ito, Tetsushi, Takeuchi, Daichi, Tsushima, Takahiro
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.11288

openalex publication_date 2025/12/12 · openalex created_date 2025/12/16 · openalex updated_date 2026/07/28

Abstract

We study exponential sums on Witt vectors, known as Gauss--Heilbronn sums, and the curves whose Frobenius traces realize these sums via a Deligne--Lusztig type construction. For 3-typical Witt vectors of length two, we analyze Gauss--Heilbronn sums, from which we fully determine the Frobenius slopes of the associated curves.

Citations

• Finite monodromy of some two-parameter families of exponential sums
• Newton Polygons of Sums on Curves I: Local-to-Global Theorems
• Newton Polygons of Sums on Curves II: Variation in p-adic Families
• p-adic estimates of abelian Artin L-functions on curves
• Finite monodromy of some families of exponential sums
• Newton slopes for Artin-Schreier-Witt towers
• The L-functions of twisted Witt extensions
• The L-functions of Witt coverings

Related

• The L-functions of twisted Witt extensions
• The Deligne-Lusztig curve associated to the Suzuki group
• A New Bound for Kloosterman Sums
• Quotient curves of the Deligne-Lusztig curve of Suzuki type
• Geometric Gauss Sums and Gross-Koblitz Formula over Function Fields