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An upper bound for the conductor of an abelian variety. (English) Zbl 0816.14021

Let \(K/ \mathbb{Q}_ p\) be a \(p\)-adic field, \(A\) an abelian variety over \(K\) of dimension \(d\) and \(f(A/K)\) the exponent conductor. Let \(v_ k : K^* \to \mathbb{Z}\) be the normalized discrete valuation on \(K\). The authors prove the following upper bound: \[ f(A/K) \leq 2d + 2dv_ K(p) \left[ {1 \over p - 1} + \sum_{e \geq 0} \left( {2d \over p^ e (p - 1)} \right) \right]. \] The authors prove that this is the best possible bound for \(p = 2 + 1\) showing that the Jacobian of the curve \(y^ 2 = x^ p + \pi\) \((\pi\) being an uniformizer of \(K)\) has an exponent conductor \(f(J/K)\) of maximal value.
Proposition 0.3 is devoted to improve the main bound in the cases \({2 \over 3} d + 1 < p \leq d + 1\) \((p \neq 2)\); in this case the authors find the bound: \(f(A/K) \leq 2d + 2d(1 + {1 \over p - 1}) v_ k(p)\). Finally, the authors apply these results to prove bounds for the exponent conductor of elliptic curves over 2-adic fields.

MSC:

14K15 Arithmetic ground fields for abelian varieties
14G20 Local ground fields in algebraic geometry
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