Abstract
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]).
This author was supported in part by a grant from the City University of New York PSCCUNY Research Award Program.
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Brumer, A., Kramer, K. (2004). Semistable Abelian Varieties with Small Division Fields. In: Hashimoto, Ki., Miyake, K., Nakamura, H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0249-0_2
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