Unramified Hilbert modular forms, with examples relating to elliptic curves (with Jude Socrates)
Abstract: We give a method to explicitly determine the space of unramified Hilbert cusp forms of weight two, together with the action of
Hecke, over a totally real number field of even degree and narrow class number one. In particular, one can determine the
eigenforms in this space and compute their Hecke eigenvalues to any degree. As an application we compute this space of cusp forms
for Q(\sqrt{509}), and determine each eigenform in this space which has rational Hecke eigenvalues. We find that not all of
these forms arise via base change from classical forms. To each such eigenform f we attach an elliptic curve with good
reduction everywhere whose L-function agrees with that of f at every place.
Computer programs: The PARI programs used in this paper are below.
Brandt.gp: Used to compute the Brandt matrices
Frob.gp: Used to compute the necessary Frobenii needed to implement Faltings, Serre
Files: [.dvi] [.pdf] [.ps] published version