Commutative algebraic groups and $p$-adic linear forms

Volume 169 / 2015

Clemens Fuchs, Duc Hiep Pham Acta Arithmetica 169 (2015), 115-147 MSC: Primary 11G99; Secondary 14L10, 11J86. DOI: 10.4064/aa169-2-2


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 $p$.


  • Clemens FuchsDepartment of Mathematics
    University of Salzburg
    Hellbrunnerstr. 34
    5020 Salzburg, Austria
  • Duc Hiep PhamDepartment of Mathematics
    Hanoi National University of Education
    136 Xuan Thuy, Cau Giay
    Hanoi, Vietnam

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