The quaternionic bridge between elliptic curves and Hilbert modular forms

• Main Author: Socrates, Jude Thaddeus U.
• Format: Others
• Language: en
• Published: 1993
• Online Access: https://thesis.library.caltech.edu/7376/2/Socrates_jtu_1993.pdf; Socrates, Jude Thaddeus U. (1993) The quaternionic bridge between elliptic curves and Hilbert modular forms. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/eb59-wy92. https://resolver.caltech.edu/CaltechTHESIS:01082013-084908017 <https://resolver.caltech.edu/CaltechTHESIS:01082013-084908017>

<p>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.</p> <p>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.</p> <p>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.</p> <p>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.</p>