Commutative algebraic groups and $p$-adic linear forms

Clemens Fuchs¹ ; Duc Hiep Pham²

¹ Department of Mathematics University of Salzburg Hellbrunnerstr. 34 5020 Salzburg, Austria
² Department of Mathematics Hanoi National University of Education 136 Xuan Thuy, Cau Giay Hanoi, Vietnam

Acta Arithmetica, Tome 169 (2015) no. 2, pp. 115-147.

Résumé

Let $G$ be a commutative algebraic group defined over a number field $K$ that is disjoint over $K$ from $\mathbb G_{\rm a}$ and satisfies the condition of semistability. Consider a linear form $l$ on the Lie algebra of $G$ with algebraic coefficients and an algebraic point $u$ in a $p$-adic neighbourhood of the origin with the condition that $l$ does not vanish at $u$. We give a lower bound for the $p$-adic absolute value of $l(u)$ which depends up to an effectively computable constant only on the height of the linear form, the height of the point $u$ and $p$.

Détail

DOI : 10.4064/aa169-2-2

Keywords: commutative algebraic group defined number field disjoint mathbb satisfies condition semistability consider linear form lie algebra algebraic coefficients algebraic point p adic neighbourhood origin condition does vanish lower bound p adic absolute value which depends effectively computable constant only height linear form height point nbsp