Let E: y = f(x) be an elliptic curve defined over Fp. (1) Show that if (*) denotes the Legendre symbol, then #E(F)) =p+1

[answerhappygod]

Let E Y F X Be An Elliptic Curve Defined Over Fp 1 Show That If Denotes The Legendre Symbol Then E F P 1 1 (31.9 KiB) Viewed 29 times

Let E: y = f(x) be an elliptic curve defined over Fp. (1) Show that if (*) denotes the Legendre symbol, then #E(F)) =p+1+I (""). TEF (2) Let n e Zx, be such that płn. Show that if p = 3 (mod 4) and f(x) = 2(x2 – n?) € Fy, then #E(FM) = p+1. (3) Let n € Z>. Show that the elliptic curve E : y = x(22 – nº) defined over Q has complex multiplication and that ap(E) = 0 for every prime p in a set of density 1/2 Hint: Recall that for 2 € Fp, the Legendre symbol is defined as 0- if x = 0, if x € (FM) otherwise. For (2), use that if p = 3 (mod 4), then (+5) = -1, and for (3) apply Dirich- let's density theorem.