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Title: The Jacobian of modular curves associated to Cartan subgroups
Author: Chen, Imin
ISNI:       0000 0001 3532 9402
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 1996
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The mod p representation associated to an elliptic curve is called split/non-split dihedral if its image lies in the normaliser of a split/non-split Cartan subgroup of GL2(Fp). Let X+split(p) and X+non-split(p) denote the modular curves which classify elliptic curves with dihedral split and non-split mod p representation, respectively. We call such curves (split/non-split) Cartan modular curves. It is well known that X+split(p) is isomorphic to the curve X+0(p2). On the other hand, the curve X+non-split(p) is distinctly different from any of the classical modular curves. Despite this apparent disparity, it is shown in this thesis that the jacobian of X+non-split(p) is isogenous to the new part of the jacobian of X+0(p2). The method of proof uses the Selberg trace formula. An explicit formula for the trace of Hecke operators is derived for both split and non-split Cartan modular curves. Comparing these two trace formulae, one obtains a trace relation, which in combination with the Eichler-Shimura relations allows us to conclude that the L-series of the two abelian varieties in question are the same, up to finitely many L-factors. The result then follows by Faltings' isogeny theorem.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Pure mathematics