Citation
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/eb59wy92. https://resolver.caltech.edu/CaltechTHESIS:01082013084908017
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 Fisogenous 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 nontrivial 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 θseries 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 θseries gives a way to obtain representatives of left Oideal 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 nonisogenous elliptic curves correspond to normalized eigenforms with rational eigenvalues.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  17 May 1993 
Record Number:  CaltechTHESIS:01082013084908017 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:01082013084908017 
DOI:  10.7907/eb59wy92 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  7376 
Collection:  CaltechTHESIS 
Deposited By:  Benjamin Perez 
Deposited On:  08 Jan 2013 23:26 
Last Modified:  16 Apr 2021 23:21 
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