关于函数域算术中的Manin-Mumford猜想及广义Pila-Wilkie估计的几点注记

亨泽尔极小曲线的皮拉 - 威尔基定理

A Pila--Wilkie theorem for Hensel minimal curves

研究亨泽尔极小结构中可定义的超越曲线，通过引入与剩余域上维数计数相关的新点计数概念，证明皮拉 - 威尔基型定理，给出多类示例并证明界的最优性。

Point counting and Wilkie's conjecture for non-archimedean Pfaffian and Noetherian functions

We consider the problem of counting polynomial curves on analytic or definable subsets over the field ${\mathbb{C}}(\!(t)\!)$, as a function of the degree $r$. A result of this type could be expected by analogy with the classical Pila-Wilkie counting theorem in the archimean situation. Some non-archimedean analogs of this type have been developed in the work of Cluckers-Comte-Loeser for the field ${\mathbb{Q}}_p$, but the situation in ${\mathbb{C}}(\!(t)\!)$ appears to be significantly different. We prove that the set of polynomial curves of a fixed degree $r$ on the transcendental part of a s…

Wilkie's conjecture for restricted elementary functions

We consider the structure ${\mathbb R}^{\mathrm{RE}}$ obtained from $({\mathbb R},<,+,\cdot)$ by adjoining the restricted exponential and sine functions. We prove Wilkie's conjecture for sets definable in this structure: the number of rational points of height $H$ in the transcendental part of any definable set is bounded by a polynomial in $\log H$. We also prove two refined conjectures due to Pila concerning the density of algebraic points from a fixed number field, or with a fixed algebraic degree, for ${\mathbb R}^{\mathrm{RE}}$-definable sets.

A note on o-minimal flows and the Ax--Lindemann--Weierstrass theorem for abelian varieties over $\mathbb{C}$

In this short note we present an elementary proof of the Ax-Lindemann-Weierstrass theorem for abelian and semi-abelian varieties. The proof uses ideas of Pila, Ullmo, Yafaev, Zannier and is based on basic properties of sets definable in o-minimal structures. It does not use the Pila-Wilkie counting theorem.

A Schanuel Property for $j$

I give a model-theoretic setting for the modular $j$ function and its derivatives. These structures, here called $j$-fields, provide an adequate setting for interpreting the Ax-Schanuel theorem for $j$ (Pila-Tsimerman 2015). Following the ideas of M. Bays, J. Kirby and A.J. Wilkie for exponential fields, I prove a generic transcendence property for the $j$ function.

Applications of the hyperbolic Ax-Schanuel conjecture

In 2014, Pila and Tsimerman gave a proof of the Ax-Schanuel conjecture for the $j$-function and, with Mok, have recently announced a proof of its generalization to any (pure) Shimura variety. We refer to this generalization as the hyperbolic Ax-Schanuel conjecture. In this article, we show that the hyperbolic Ax-Schanuel conjecture can be used to reduce the Zilber-Pink conjecture for Shimura varieties to a problem of point counting. We further show that this point counting problem can be tackled in a number of cases using the Pila-Wilkie counting theorem and several arithmetic conjectures. Our…

Rational points in periodic analytic sets and the Manin-Mumford conjecture

We present a new proof of the Manin-Mumford conjecture about torsion points on algebraic subvarieties of abelian varieties. Our principle, which admits other applications, is to view torsion points as rational points on a complex torus and then compare (i) upper bounds for the number of rational points on a transcendental analytic variety (Bombieri-Pila-Wilkie) and (ii) lower bounds for the degree of a torsion point (Masser), after taking conjugates. In order to be able to deal with (i), we discuss (Thm. 2.1) the semi-algebraic curves contained in an analytic variety supposed invariant for tra…

An application of the value distribution theory for semi-abelian varieties to problems of Ax-Lindemann and Manin-Mumford types

The aim of this paper is to prove a theorem of Ax-Lindemann type for complex semi-abelian varieties as an application of a big Picard theorem proved by the author in 1981, and then apply it to prove a theorem of classical Manin-Mumford Conjecture for semi-abelian varieties, which was proved by M. Raynaud 1983, M. Hindry 1988, ....., and Pila-Zannier 2008 by a different method from others, which is most relevant to our's. The present result might be a first instance of a direct connection between the value distribution theory of holomorphic maps and the arithmetic (Diophantine) theory over alge…

相对马宁-芒福德猜想

The Relative Manin-Mumford Conjecture

本文在特征0下证明阿贝尔簇族的相对马宁-芒福德猜想，采用皮拉-赞尼尔方法，结合新的高度不等式与退化轨迹研究，还强化结果给出复数域上阿贝尔概法子簇挠点稠密判据，并给出一致马宁-芒福德猜想的无等分布新证明。