January, 2024 Analytic Ax–Schanuel for semi-abelian varieties and Nevanlinna theory
Author Affiliations +
J. Math. Soc. Japan 76(1): 1-22 (January, 2024). DOI: 10.2969/jmsj/89588958


Let $A$ be a semi-abelian variety with an exponential map $\exp : \mathop{\mathrm{Lie}}(A) \to A$. The purpose of this paper is to explore Nevanlinna theory of the entire curve $\hskip1pt{\widehat{\mathrm{exp}}}\hskip2pt f := (\exp f, f) : \mathbf{C} \to A \times \mathop{\mathrm{Lie}}(A)$ associated with an entire curve $f : \mathbf{C} \to \mathop{\mathrm{Lie}}(A)$. Firstly we give a Nevanlinna theoretic proof to the analytic Ax–Schanuel Theorem for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series (Ax–Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that the elements of the vector-valued function $f(z) - f(0) \in \mathop{\mathrm{Lie}}(A) \cong \mathbf{C}^n$ are $\mathbf{Q}$-linearly independent in the case of $A = (\mathbf{C}^*)^n$. Our proof is based on the Log Bloch–Ochiai Theorem and a key estimate which we show.

Our next aim is to establish a Second Main Theorem for $\hskip1pt{\widehat{\mathrm{exp}}}\hskip2pt f$ and its $k$-jet lifts with truncated counting functions at level one. We give some applications to a problem of a type raised by S. Lang and the unicity. The results clarify a relationship between the problems of Ax–Schanuel type and Nevanlinna theory.

Funding Statement

Research supported in part by JSPS KAKENHI Grant Number JP19K03511.


Download Citation

Junjiro NOGUCHI. "Analytic Ax–Schanuel for semi-abelian varieties and Nevanlinna theory." J. Math. Soc. Japan 76 (1) 1 - 22, January, 2024. https://doi.org/10.2969/jmsj/89588958


Received: 1 June 2022; Published: January, 2024
First available in Project Euclid: 4 April 2023

Digital Object Identifier: 10.2969/jmsj/89588958

Primary: 32H30
Secondary: 11D61 , 11J89

Keywords: analytic Ax–Schanuel , Log Bloch–Ochiai , Nevanlinna theory , semi-abelian Schanuel , value distribution theory

Rights: Copyright ©2024 Mathematical Society of Japan


This article is only available to subscribers.
It is not available for individual sale.

Vol.76 • No. 1 • January, 2024
Back to Top