The main result of this thesis is a matching between an elliptic curve E overF = Q(√509) which has good reduction everywhere, and a normalized holomorphicHilbert modular eigenform f for F of weight 2 and full level. The curve E is notF-isogenous to its Galois conjugate E^σ and does not possess potential complex multiplication.The eigenform f has rational eigenvalues, does not come from the basechange of an elliptic modular form, and does not satisfy f = f ⊗ ε for any quadraticcharacter ε of F associated to a degree 2 imaginary extension of F. We show that a_ρ(E) = a_ρ(f) for a large set Ʃ of σ invariant primes in F. This provides the first known non-trivial example of the conjectural Langlands correspondence (see Section1.1) in the everywhere unramified case.
The method we use exploits the isomorphism between the spaces of holomorphicHilbert modular cusp forms and quaternionic cusp forms. The construction of finvolves explicity constructing a maximal order O in the quaternion algebra B/Fwhich ramified precisely at the finite primes. We determine the type number T_1of B as well as the class number H_1 for O, which equals T_1 in our case of interest.We found that for Q(√509), T_1 = H_1 = 24. One sees that the space of weight 2 full level cusp forms for F has dimension 23.
The main tools are θ-series attached to ideals and Brandt matrices B(ξ) for anorder in B for quadratic fields Q (√m) with class number 1 and whose fundamentalunit u has nor -1. (Q(√509) is such a field.) The θ-series gives a way to obtainrepresentatives of left O-ideal classes and hence representatives of maximal orders ofdifferent type. The Hecke action on quaternionic cusp forms is given by the modifiedBrandt matrices B'(ξ), hence a set of simultaneous eigenvectors for these matricescorresponds to the normalized eigenforms for F.
Applying these algorithms to Q(√509), we prove that there are exactly three normalized eigenforms which have rational eigenvalues for all the Hecke operators.We show that for one of these eigenforms f, a_ρ(f) ≠ a_ρ(f^σ) for certain primes ρ,proving that f does not come form base change. We also note that there is anotherelliptic curve E'/Q(√509) which is isogenous to its Galois conjugate and hence not isogenous to either E or E^σ. We show that a_ρ(E') = a_ρ(f') ∀ρ ε Ʃ, where f'is the third normalized eigenform that we found above. This is compatible with the expectation that all three non-isogenous elliptic curves correspond to normalizedeigenforms with rational eigenvalues.
