A finiteness conjecture on abelian varieties with constrained prime power torsion

• Autores: Christopher Rasmussen, Akio Tamagawa
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 15, Nº 5-6, 2008, págs. 1223-1231
• Idioma: inglés
• DOI: 10.4310/mrl.2008.v15.n6.a12
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• Resumen

  • The pro-$\ell$ Galois representation attached to the arithmetic fundamental group of a curve influences heavily the arithmetic of its branched `$\ell$-covers.' In many cases, the $\ell$-power torsion on the Jacobian of such a cover is fixed by the kernel of this representation, giving explicit information about this kernel. Motivated by the relative scarcity of interesting examples for $\ell$-covers of the projective line minus three points, the authors formulate a conjecture to quantify this scarcity. A proof for certain genus one cases is given, and an exact set of curves satisfying the required arithmetic conditions in the base case is determined.