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<p>Let <i>K</i> be a number field, and <i>G</i> a commutative algebraic group defined over <i>K</i>. Let <i>l</i> be a linear form on the Lie algebra of <i>G</i> with coefficients in <i>K</i>, and <i>u</i> an algebraic point in a <i>p</i>-adic neighbourhood of the origin with the condition that <i>l</i> does not vanish at <i>u</i>. Denote by <i>W</i> the linear space associated with <i>l</i> in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>K</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. In this paper, we give a lower bound for the <i>p</i>-adic absolute value of <i>l</i>(<i>u</i>) in the case where the pair (<i>G</i>,&#xa0;<i>W</i>) is semistable. This lower bound depends up to an effectively computable constant only on the height of the linear form <i>l</i>, the height of the point <i>u</i> and the prime <i>p</i>.</p>

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Linear forms on commutative algebraic groups over p-adic fields

  • 期刊论文

  • 发表日期：

    2025年12月14日

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  • Duc Hiep Pham

摘要

Let K be a number field, and G a commutative algebraic group defined over K. Let l be a linear form on the Lie algebra of G with coefficients in K, and u an algebraic point in a p-adic neighbourhood of the origin with the condition that l does not vanish at u. Denote by W the linear space associated with l in \(K^n\) K n . In this paper, we give a lower bound for the p-adic absolute value of l(u) in the case where the pair (G, W) is semistable. This lower bound depends up to an effectively computable constant only on the height of the linear form l, the height of the point u and the prime p.