Serre's conjecture over \Bbb F_9

• Autores: Jordan S. Ellenberg
• Localización: Annals of mathematics, ISSN 0003-486X, Vol. 161, Nº 3, 2005, págs. 1111-1142
• Idioma: inglés
• DOI: 10.4007/annals.2005.161.1111
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• Resumen

  • In this paper we show that an odd Galois representation ¿P¿Ï : Gal( ¿PQ/Q) ¿¿ GL2(F9) having nonsolvable image and satisfying certain local conditions at 3 and 5 is modular. Our main tools are ideas of Taylor [21] and Khare [10], which reduce the problem to that of exhibiting points on a Hilbert modular surface which are defined over a solvable extension of Q, and which satisfy certain reduction properties. As a corollary, we show that Hilbert-Blumenthal abelian surfaces with ordinary reduction at 3 and 5 are modular.