Monday, 3 October 2011

nt.number theory - elliptic curve with j-invariant T

The idea for (2) is the following: the modular curve Y(elln) classifying elliptic curves
over mathbbC together with an isomorphism (mathbbZ/elln)2congE[elln]
identifying the standard symplectic pairing on the left (i.e. langle(a1,a2),(b1,b2)rangle=e2pii(a1b2a2b1)/elln) with the Weil pairing on the right,
is irreducible. (It is isomorphic to mathcalH/Gamma(elln), where
mathcalH is the complex upper half-plane and Gamma(elln) is the congruence
subgroup of full level elln.)



(3) follows from (2) and the irreducibility of cyclotomic polynomials over mathbbQ.

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