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https://hdl.handle.net/2142/86856
Description
Title
On the Modularity of Higher-Dimensional Varieties
Author(s)
Yi, You-Chiang
Issue Date
2005
Doctoral Committee Chair(s)
Boston, Nigel
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
In this thesis, we first introduce Wiles' method to establish that a Calabi-Yau threefold defined over the field Q with 2-dimensional ℓ-adic cohomology is modular, answering a question of Saito & Yui. Second, we show that a quintic threefold with 4-dimensional middle cohomology is Hilbert modular. This answers a question of Consani & Scholten. Let rho : Gal( Q/Q&parl0; 5&parr0; ) → GL4( Q2&parl0; 5&parr0; ) be the representation on H3( X, Q2&parl0;5 &parr0; ). We show that rho corresponds to (f, f sigma), where f is a newform over Q5 of weight (2, 4) and level 30, and sigma is the nontrivial element in the Galois group Gal( Q&parl0;5&parr0;/ Q ).
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