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Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(bar{K})$. More precisely, we prove that there are constants $C_1,C_2>0$ such that the normalized Bernoulli-part of the canonical height is bounded below by $$ hat{h}_A^{mathbb{B}}(P) ge C_1bigl[K(P):Kbigr]^{-2} $$ for all points $Pin{A(bar{K})}$ whose height satisfies $0<hat{h}_A(P)le{C_2}$.
In the paper [On superspecial abelian surfaces over finite fields II. J. Math. Soc. Japan, 72(1):303--331, 2020], Tse-Chung Yang and the first two current authors computed explicitly the number $lvert mathrm{SSp}_2(mathbb{F}_q)rvert$ of isomorphism c
We provide in this paper an upper bound for the number of rational points on a curve defined over a one variable function field over a finite field. The bound only depends on the curve and the field, but not on the Jacobian variety of the curve.
We prove two theorems concerning isogenies of elliptic curves over function fields. The first one describes the variation of the height of the $j$-invariant in an isogeny class. The second one is an isogeny estimate, providing an explicit bound on th
Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $text{Br}, Y/ text{Br}_1, Y$ i
Let $k$ be a field of characteristic $q$, $cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$ or $>2d+1$. L