The quaternionic bridge between elliptic curves and Hilbert modular forms

Author: Socrates, Jude Thaddeus U.

Year: 1993

Degree: Dissertation (Ph.D.)

Advisor: Ramakrishnan, Dinakar

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/eb59-wy92

Abstract

The main result of this thesis is a matching between an elliptic curve E over F = Q(√509) which has good reduction everywhere, and a normalized holomorphic Hilbert modular eigenform f for F of weight 2 and full level. The curve E is not F-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 base change of an elliptic modular form, and does not satisfy f = f ⊗ ε for any quadratic character 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 Section 1.1) in the everywhere unramified case.

The method we use exploits the isomorphism between the spaces of holomorphic Hilbert modular cusp forms and quaternionic cusp forms. The construction of f involves explicity constructing a maximal order O in the quaternion algebra B/F which ramified precisely at the finite primes. We determine the type number T_1 of 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 an order in B for quadratic fields Q (√m) with class number 1 and whose fundamental unit u has nor -1. (Q(√509) is such a field.) The θ-series gives a way to obtain representatives of left O-ideal classes and hence representatives of maximal orders of different type. The Hecke action on quaternionic cusp forms is given by the modified Brandt matrices B'(ξ), hence a set of simultaneous eigenvectors for these matrices corresponds 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 another elliptic 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 normalized eigenforms with rational eigenvalues.

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