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Volume 47, Issue 1, 2011, pp. 29–98
DOI: 10.2977/PRIMS/31

Published online: 2011-03-24

A Family of Calabi–Yau Varieties and Potential Automorphy II

Tom Barnet-Lamb[1], David Geraghty[2], Michael Harris and Richard Taylor[3]

(1) Brandeis University, Waltham, USA
(2) Institute for Advanced Study, Princeton, USA
(3) Harvard University, Cambridge, USA

We prove new potential modularity theorems for n-dimensional essentially self-dual l-adic representations of the absolute Galois group of a totally real eld. Most notably, in the ordinary case we prove quite a general result. Our results suffice to show that all the symmetric powers of any non-CM, holomorphic, cuspidal, elliptic modular newform of weight greater than one are potentially cuspidal automorphic. This in turns proves the Sato–Tate conjecture for such forms. (In passing we also note that the Sato–Tate conjecture can now be proved for any elliptic curve over a totally real eld.)

Keywords: automorphic representation, Galois representation, Dwork family, Sato–Tate conjecture

Barnet-Lamb Tom, Geraghty David, Harris Michael, Taylor Richard: A Family of Calabi–Yau Varieties and Potential Automorphy II. Publ. Res. Inst. Math. Sci. 47 (2011), 29-98. doi: 10.2977/PRIMS/31