[Paper Review] Non-Archimedean Big Picard Theorems

William B. Cherry|ArXiv.org|Jul 10, 2002

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper establishes a non-Archimedean analog of the classical Big Picard Theorem using Berkovich’s theory of analytic spaces, proving that holomorphic maps from a punctured disc to a non-Archimedean algebraic curve of genus ≥2 extend across the puncture. The key contribution is a uniformization-based extension result for maps into curves with totally degenerate reduction, generalizing Picard-type theorems to the non-Archimedean setting and providing a foundation for characteristic p analogs of Lang’s conjecture.

ABSTRACT

A non-Archimedean analog of the classical Big Picard Theorem, which says that a holomorphic map from the punctured disc to a Riemann surface of hyperbolic type extends accross the puncture, is proven using Berkovich's theory of non-Archimedean analytic spaces.

Motivation & Objective

• To establish a non-Archimedean analog of the classical Big Picard Theorem for holomorphic maps into algebraic curves.
• To apply Berkovich’s theory of analytic spaces to prove extension theorems for maps into curves of genus ≥2.
• To provide a foundational step toward proving Lang’s conjecture in positive characteristic without model-theoretic methods.
• To generalize Buium’s approach to Raynaud’s theorem by proving a non-Archimedean version of the Big Picard Theorem.

Proposed method

• Utilizes Berkovich’s theory of non-Archimedean analytic spaces, particularly the theory of analytic curves and their uniformization.
• Applies the concept of Mumford curves, which are algebraic curves with totally degenerate reduction, to enable uniformization via the Berkovich projective line minus a perfect set.
• Employs lifting techniques: analytic maps from the punctured disc lift to the universal cover of the curve when the curve is a Mumford curve.
• Uses the extension of bounded analytic maps to the unit disc via the non-Archimedean version of Montel’s theorem, relying on the sup-norm and compactness.
• Applies Theorem 2.1 (extension of maps omitting two points) and Theorem 2.2 (image density in the punctured disc) to control the behavior near the puncture.
• For non-Mumford curves, uses a rational function to reduce the problem to boundedness, then applies extension theorems to lift the map to the normalization.

Experimental results

Research questions

• RQ1 Can a non-Archimedean analog of the Big Picard Theorem be established using Berkovich’s analytic geometry?
• RQ2 Under what conditions does a holomorphic map from a punctured disc to a non-Archimedean curve of genus ≥2 extend across the puncture?
• RQ3 How can the uniformization theory of Mumford curves be used to prove extension theorems in the non-Archimedean setting?
• RQ4 Can this result be used to generalize Buium’s proof of Raynaud’s theorem to positive characteristic?
• RQ5 What role does totally degenerate reduction play in enabling extension theorems for non-Archimedean maps?

Key findings

• A holomorphic map from a punctured disc to a non-Archimedean algebraic curve of genus ≥2 extends to a holomorphic map on the full disc.
• If a map to a genus 1 curve omits at least one point or the normalization has good reduction, it extends across the puncture.
• For genus 0 curves, a map omitting at least two points extends across the puncture, generalizing the non-Archimedean version of the classical Picard theorem.
• The extension result holds for all irreducible algebraic curves of genus ≥2, regardless of reduction type, via uniformization or boundedness arguments.
• The extension of analytic maps to the projective completion implies that such maps are rational, as shown by the GAGA principle in Berkovich geometry.
• The result provides a non-model-theoretic path toward proving Lang’s conjecture in positive characteristic, bypassing Hrushovski’s method.