Decidability of some complicated structures definable in ℂ(t)

2025/08/24 by Scanlon, Thomas
#Algebraic Geometry (math.AG) #FOS: Mathematics #Logic (math.LO) #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2508.17485

Abstract

Several properly countable unions of algebraic sets in ℂⁿ are definable in ℂ(t) including the set CM of j-invariants of complex elliptic curves with complex multiplication. It has been suggested that one could prove the undecidability of Th(ℂ(t)) by showing that the theory of the structure CM := (ℂ,+,⋅,0,1,CM) of the field of complex numbers considered with a unary predicate picking out CM is undecidable. We show using an effective version of the André-Oort conjecture that to the contrary Th(CM) is stable and decidable. We discuss some related structures on the complex numbers definable in ℂ(t) and how their theories may be connected to the Zilber-Pink conjectures.

Citations

• Generalised André-Pink-Zannier Conjecture for Shimura varieties of abelian type
• Effective transcendental Zilber-Pink for variations of Hodge structures
• Some effective estimates for André-Oort in Y(1)ⁿ
• Strong minimality and the j-function

Related

• A Note on Hilbert's "Geometric" Tenth Problem
• Birch and Swinnerton-Dyer conjecture in the complex multiplication case and the congruent number problem
• Undecidability of infinite algebraic extensions of \mathbbF_p(t)
• The conjecture of Birch and Swinnerton-Dyer for certain elliptic curves with complex multiplication
• An arithmetic intersection for squares of elliptic curves with complex multiplication