数论：2025–2026 年度综述

An annual survey of the math.NT arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 132 papers and organised into 37 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• Diophantine Geometry and Arithmetic of Curves (38.5%) — Focuses on rational and quadratic points on algebraic curves, Diophantine equations, and dynamical criteria for arithmetic properties.
• Galois Representations and Modular Arithmetic (33.6%) — Investigates Galois representations, Selmer groups, modular Jacobians, and arithmetic of Drinfeld modules in local and global fields.
• Modular Forms and L-Function Moments (10.5%) — Analyzes coefficients, moments, and asymptotic behaviors of modular forms, Dirichlet characters, and associated L-functions.
• Special Functions and q-Series Identities (5.6%) — Explores mock theta functions, continued fractions, and generalized zeta functions through algebraic identities and recurrence relations.
• Diophantine Approximation and Additive Combinatorics (5.1%) — Studies Diophantine triples, character sums, and multiplicative approximation problems using set-theoretic and analytic methods.
• Partition Theory and Combinatorial Number Theory (2.4%) — Examines partition functions, arithmetic progressions, and combinatorial proofs related to integer partitions and hypernatural numbers.
• Explicit Formulas and Class Number Asymptotics (1.7%) — Derives explicit formulas and asymptotic estimates for class numbers, L-function moments, and error terms in quadratic fields.
• Erdős Problems and Diophantine Equations (1.4%) — Addresses specific Erdős conjectures, reducibility of polynomials, and existence problems in Diophantine pairs and perfect numbers.
• Analytic Bounds and Height Functions (0.8%) — Establishes uniform bounds for zeta moments, height functions, and lattice point counts using analytic and geometric techniques.
• Residue Patterns and Non-Archimedean Invariants (0.2%) — Classifies quadratic residue patterns, Wilson theorem extensions, and invariants in non-Archimedean and modular contexts.
• Digit Expansions and Fractal Geometry (0.2%) — Analyzes digit frequencies, Kaprekar dynamics, and fractal properties of number representations in various bases.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.