Galois Theory and Modular Forms

Volume 11 of the series Developments in Mathematics pp 13-37

Semistable Abelian Varieties with Small Division Fields

  • Armand BrumerAffiliated withDepartment of Mathematics, Fordham University
  • , Kenneth KramerAffiliated withDepartment of Mathematics, Queens College (CUNY)

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The conjecture of Shimura-Taniyama-Weil, now proved through the work of Wiles and disciples, is only part of the Langlands program. Based on a comparison of the local factors ([And], [Seri]), it also predicts that the L-series of an abelian surface defined over ℚ should be the L-series of a Hecke eigen cusp form of weight 2 on a suitable group commensurable with Sp4 (ℤ). The only decisive examples are related to lifts of automorphic representations of proper subgroups of Sp4, for example the beautiful work of Yoshida ([Yos], [BSP]).