Modular Existential Closedness
Abstract
We prove the Modular Existential Closedness conjecture for the modular j-function extended from the upper half-plane to ℂ ∖ ℝ by Schwarz reflection. Specifically, we show that any system of equations involving the field operations and j has solutions in the complex numbers unless it is overdetermined. Such a system is overdetermined if, roughly, there are more independent equations than variables, possibly after applying the transformation rules of the j-function under the linear fractional action of PGL2(ℚ) on ℂ ∖ ℝ. In general, we do not expect overdetermined systems of equations to have solutions due to the algebraic and transcendental properties of the j-function captured by the modular analogue of Schanuel's conjecture. In geometric terms, Modular Existential Closedness states that given a broad and free irreducible algebraic variety V ⊆ ℂn × ℂn -- where broadness and freeness are geometric conditions guaranteeing that the system is not overdetermined -- there is a point of the form (τ1, …, τn, j(τ1), …, j(τn)) ∈ V with τi ∈ ℂ ∖ ℝ. The proof uses Ratner's theorems on the closure and equidistribution of the orbits of unipotent flows in powers of PGL2(ℝ) to describe the asymptotic behaviour of the orbits of certain analytic maps. This is then applied to approximate a relatively simple system of equations using the original system and the action of PGL2(ℤ). The construction of the limiting system makes use of the Ax--Schanuel theorem and guarantees an isolated solution. Rouché's theorem then establishes the existence of solutions for the original system of equations.
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BibTeX
@misc{aslanyan2026modular,
title = {{Modular Existential Closedness}},
author = {Vahagn Aslanyan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35409},
eprint = {2610.35409},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35409},
}
RIS
TY - UNPB
TI - Modular Existential Closedness
AU - Vahagn Aslanyan
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35409
UR - https://arxiv.org/abs/2610.35409
AB - We prove the Modular Existential Closedness conjecture for the modular j-function extended from the upper half-plane to ℂ ∖ ℝ by Schwarz reflection. Specifically, we show that any system of equations involving the field operations and j has solutions in the complex numbers unless it is overdetermined. Such a system is overdetermined if, roughly, there are more independent equations than variables, possibly after applying the transformation rules of the j-function under the linear fractional action of PGL_2(ℚ) on ℂ ∖ ℝ. In general, we do not expect overdetermined systems of equations to have solutions due to the algebraic and transcendental properties of the j-function captured by the modular analogue of Schanuel's conjecture. In geometric terms, Modular Existential Closedness states that given a broad and free irreducible algebraic variety V ⊆ ℂ^n × ℂ^n -- where broadness and freeness are geometric conditions guaranteeing that the system is not overdetermined -- there is a point of the form (τ_1, …, τ_n, j(τ_1), …, j(τ_n)) ∈ V with τ_i ∈ ℂ ∖ ℝ. The proof uses Ratner's theorems on the closure and equidistribution of the orbits of unipotent flows in powers of PGL_2(ℝ) to describe the asymptotic behaviour of the orbits of certain analytic maps. This is then applied to approximate a relatively simple system of equations using the original system and the action of PGL_2(ℤ). The construction of the limiting system makes use of the Ax--Schanuel theorem and guarantees an isolated solution. Rouché's theorem then establishes the existence of solutions for the original system of equations.
ER -