A Lift of Cohomology Eigenclasses of Hecke Operators

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Brigham Young University - Provo

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https://scholarsarchive.byu.edu/etd/2169; https://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=3168&context=etd
Hansen, Brian Francis
Brigham Young University - Provo
Serre's Conjecture; Hecke operator; cohomology group; lift; eigenvector; Mathematics
thesis / dissertation description
A considerable amount of evidence has shown that for every prime p ≠q; 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.