A note on polarizations of finitely generated fields

A generalization of conjectures of Bogomolov and Lang over finitely generated fields

Let K be a finitely generated field over Q, and A an abelian variety over K. Let <, > : A(K^a) x A(K^a) --> R be an arithmetic height pairing on A, where K^a is the algebric closure of K. For x_1,..., x_l \in A(K^a), we denote det(<x_i, x_j>) by d(x_1,..., x_l). Let G be a subgroup of finite rank in A(K^a), and X a subvariety of A_{K^a}. Fix a basis {g_1,..., g_n} of G_Q. In this note, we prove a generalization of Poonen's theorem: If the set {x \in X(K^a) | d(g_1,..., g_n, x) <= e} is Zariski dense in X for every positive number e, then X is a translation of an abelian subvariety by an elemen…

A simple proof that any additive basis has only finitely many essential subsets

Let $A$ be an additive basis. We call ``essential subset'' of $A$ any finite subset $P$ of $A$ such that $A \setminus P$ is not an additive basis and that $P$ is minimal (for the inclusion order) to have this property. A recent theorem due to B. Deschamps and the author states that any additive basis has only finitely many essential subsets (see ``Essentialité dans les bases additives, J. Number Theory, 123 (2007), p. 170-192''). The aim of this note is to give a simple proof of this theorem.

有限生成域上小点的等分布

Equidistribution of small points over finitely generated fields

本文研究有限生成域上小点的等分布，给出反例表明数值小性不保证所有赋值处的等分布，并证明在完全超越赋值处等分布成立，关联Yuan--Zhang猜想。

Supersingularity of Motives with Complex Multiplication and a Twisted Polarization

In this note, we prove a criteria for supersingularity when the variety has a large automorphism group and a perfect bilinear pairing. This criteria unifies and extends many known results on the supersingularity of curves and varieties and in particular, applies to a large family of Artin-Schreier curves.

迈向adelic曲线上Lehmer问题的推广

Toward a generalization of Lehmer's problem to adelic curves

本文探讨了Lehmer问题在adelic曲线上的推广，研究其在有限生成字段中的应用。

A note on Larsen's conjecture and ranks of elliptic curves

Let E be an elliptic curve defined over a number field K. Michael Larsen conjectured that for any finitely generated subgroup G of Gal(\bar K/K), the Mordell-Weil rank of E is unbounded in number fields fixed by G. We prove that the conjecture holds over K=Q for both the analytic rank and the p-infinity Selmer rank of E for every odd prime p. For arbitrary E/K, we show that Larsen's conjecture follows from the standard conjectures for ranks of elliptic curves, provided K has a real place or E has non-integral j-invariant.

Generic elements of a Zariski-dense subgroup form an open subset

Let G be a semi-simple algebraic group over a finitely generated field K of characteristic zero, and let Γ< G(K) be a finitely generated Zariski-dense subgroup. In this note we prove that the set of K-generic elements of Γ(whose existence was established earlier in [9]) is open in the profinite topology of Γ. We then extend this result to the fields of positive characteristic, and also prove the existence of generic elements in this case.