F-structures and integral points on semiabelian varieties over finite fields

使用全局赋值域的新间隙原理研究半阿贝尔簇

New gap principle for semiabelian varieties using globally valued fields

本文基于 Hrushovski 的观察，通过将 Bogomolov 猜想归约，证明了半阿贝尔簇在正特征下的新间隙原理，其中阿贝尔商为椭圆曲线。

The theory of the exponential differential equations of semiabelian varieties

The complete first order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arises from an amalgamation-with-predimension construction in the style of Hrushovski. The theory includes necessary and sufficient conditions for a system of equations to have a solution. The necessary condition generalizes Ax's differential fields version of Schanuel's conjecture to semiabelian varieties. There is a purely algebraic corollary, the "Weak CIT" for semiabelian varieties, which concerns the intersections of algebraic subgroups with …

On function field Mordell-Lang: the semiabelian case and the socle theorem

We here aim to complete our model-theoretic account of the function field Mordell-Lang conjecture, avoiding appeal to dichotomy theorems for Zariski geometries, where we now consider the general case of semiabelian varieties. The main result is a reduction, using model-theoretic tools, of the semiabelian case to the abelian case.

Semiabelian varieties over separably closed fields, maximal divisible subgroups, and exact sequences

Given a separably closed field K of positive characteristic and finite degree of imperfection we study the # functor which takes a semiabelian variety G over K to the maximal divisible subgroup #G of G(K). We show that the # functor need not preserve exact sequences. The main result is an example where #G does not have "relative Morley rank", yielding a counterexample to a claim of Hrushovski. The methods involve studying preservation of exact sequences by the # functor as well as issues of descent. We also develop the notion of an iterative D-structure on a group scheme over an iterative Hass…

The Dynamical Mordell-Lang Conjecture for endomorphisms of semiabelian varieties defined over fields of positive characteristic

Let K be an algebraically closed field of prime characteristic p, let X be a semiabelian variety defined over a finite subfield of K, let f be a regular self-map on X defined over K, let V be a subvariety of X defined over K, and let x be a K-point of X. The Dynamical Mordell-Lang Conjecture in characteristic p predicts that the set S consisting of all positive integers n for which f^n(x) lies on V is a union of finitely many arithmetic progressions, along with finitely many p-sets, which are sets consisting of all integers of the form c_1 p^{k_1 n_1} + ... + c_m p^{k_m n_m} (as we vary n_1,..…

Projectivity in (bounded) integral residuated lattices

In this paper we study projective algebras in varieties of (bounded) commutative integral residuated lattices from an algebraic (as opposed to categorical) point of view. In particular we use a well-established construction in residuated lattices: the ordinal sum. Its interaction with divisibility makes our results have a better scope in varieties of divisibile commutative integral residuated lattices, and it allows us to show that many such varieties have the property that every finitely presented algebra is projective. In particular, we obtain results on (Stonean) Heyting algebras, certain v…