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 Title: On the Modularity of Higher-Dimensional Varieties Author(s): Yi, You-Chiang Doctoral Committee Chair(s): Boston, Nigel Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics 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 ). Issue Date: 2005 Type: Text Language: English Description: 91 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2005. URI: http://hdl.handle.net/2142/86856 Other Identifier(s): (MiAaPQ)AAI3199188 Date Available in IDEALS: 2015-09-28 Date Deposited: 2005
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