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10 Sep 2003: L(x,v) =1. 2gx(v,v) θx(v). Recall that the action of the Lagrangian L over an absolutely continuous curve γ :[a,b] M is defined by. ... 7. Appendix B: Proof of Proposition 2.4. Our main reference here is [36].

17 Jul 2003: Some examples of 5 and 7 descent for elliptic curves over Q 175. ... To treat the dual isogeny φ̂, the rough idea. is that the involution η swaps over the sets of primes A and B.

19 Feb 2003: where the maximum is taken over all pairs (u,u+) of conjugate functions. ... 15. where, again, the intersection is taken over all pairs (u,u+) of conjugate func-tions.