On the Modularity of Higher-Dimensional Varieties

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 ).

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Yi, You-Chiang

Contributors dc:contributor

• Boston, Nigel

• Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3199188

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86856