Twists of Elliptic Curves Sample Clauses

Twists of Elliptic Curves. ‌ First we recall a theorem of Xxxx [Fre88], which gives a relationship between Selmer groups of quadratic twists of certain elliptic curves E and the ideal class group of the quadratic field associated with the twisting discriminant. For the quadratic fields Q(√D) appearing in Xxxx’x result, all of the primes of bad reduction of E factor (i.e. split, ramify, or remain inert) in OQ(√D) in a prescribed way depending on their reduction type. To state Xxxx’x conditions precisely, we must define Xxxx curves, which are a class of elliptic curves over local fields. The local fields we will refer to are simply the p-adic completions of Q, denoted by Qp. Xxxx curves admit a uniformizing map (which here means a complex analytic isomorphism of Riemann surfaces that also preserves the group structure) that is analogous to one admitted by all elliptic curves over C. We recall the analogous facts for elliptic curves over C in the next paragraph. Every elliptic curve over C admits a uniformization E(C) ∼= C/Λ, where Λ is an additive discrete subgroup of C which can be taken to be of the form Λ = Z+τ Z, with τ ∈ H. The map z → e2πiz applied to C/Λ yields the uniformization E(C) ∼= C/e2πiτZ, where C∗ is the multiplicative subgroup of C and e2πiτZ denotes the subgroup of C∗ generated by e2πiτ . Now let K be a local field which is complete with respect to a discrete valuation