2023/12/14 by Xiao, Guanju, Zhou, Zijian, Qu, Longjiang
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2312.08844
Let p>3 be a prime and E be a supersingular elliptic curve defined over \mathbbFp2. Let c be a prime with c < 3p/16 and G be a subgroup of E[c] of order c. The pair (E,G) is called a supersingular elliptic curve with level-c structure, and the endomorphism ring End(E,G) is isomorphic to an Eichler order with level c. We construct two kinds of Eichler orders Oc(q,r) and O'c(q,r') with level c. Interestingly, we prove that each Oc(q,r) or O'c(q,r') can represent a primitive reduced binary quadratic form with discriminant -16cp or -cp respectively. If a curve E is ℤ[√(-cp)]-oriented or ℤ[(1+√(-cp))/(2)]-oriented, then we prove that End(E,G) is isomorphic to Oc(q,r) or O'c(q,r') respectively. Due to the fact that ℤ[√(-cp)]-oriented isogenies between ℤ[√(-cp)]-oriented elliptic curves could be represented by quadratic forms, we show that these isogenies are reflected in the corresponding Eichler orders via the composition law for their corresponding quadratic forms.