#### Abstract

A considerable amount of evidence has shown that for every prime p &neq; N observed, a simultaneous eigenvector v_0 of Hecke operators T(l,i), i=1,2, in H^3(Γ_0(N),F(0,0,0)) has a “lift” v in H^3(Γ_0(N),F(p−1,0,0)) — i.e., a simultaneous eigenvector v of Hecke operators having the same system of eigenvalues that v_0 has. For each prime p>3 and N=11 and 17, we construct a vector v that is in the cohomology group H^3(Γ_0(N),F(p−1,0,0)). This is the first construction of an element of infinitely many different cohomology groups, other than modulo p reductions of characteristic zero objects. We proceed to show that v is an eigenvector of the Hecke operators T(2,1) and T(2,2) for p>3. Furthermore, we demonstrate that in many cases, v is a simultaneous eigenvector of all the Hecke operators.

#### Degree

PhD

#### College and Department

Physical and Mathematical Sciences; Mathematics

#### Rights

http://lib.byu.edu/about/copyright/

#### BYU ScholarsArchive Citation

Hansen, Brian Francis, "A Lift of Cohomology Eigenclasses of Hecke Operators" (2010). *All Theses and Dissertations*. 2169.

http://scholarsarchive.byu.edu/etd/2169

#### Date Submitted

2010-05-24

#### Document Type

Dissertation

#### Handle

http://hdl.lib.byu.edu/1877/etd3597

#### Keywords

Serre's Conjecture, Hecke operator, cohomology group, lift, eigenvector