Endomorphism rings of supersingular elliptic curves over Fp.

Let p>3 be a fixed prime. For a supersingular elliptic curve E over Fp, a result of Ibukiyama tells us that End(E) is a maximal order O(q) (resp. O′(q)) in End(E)⊗Q indexed by a (non-unique) prime q satisfying q≡3mod8 and the quadratic residue (pq)=−1 if 1+π2∉End(E) (resp. 1+π2∈End(E)), where π=((x,y)↦(xp,yp) is the absolute Frobenius. Let qj denote the minimal q for E whose j-invariant j(E)=j and M(p) denote the maximum of qj for all supersingular j∈Fp. Firstly, we determine the neighborhood of the vertex [E] with j∉{0,1728} in the supersingular ℓ-isogeny graph if 1+π2∉End(E) and p>qjℓ2 or 1+π2∈End(E) and p>4qjℓ2: there are either ℓ−1 or ℓ+1 neighbors of [E], each of which connects to [E] by one edge and at most two of which are defined over Fp. We also give examples to illustrate that our bounds are tight. Next, under GRH, we obtain explicit upper and lower bounds for M(p), which were not studied in the literature as far as we know. To make the bounds useful, we estimate the number of supersingular elliptic curves with qjp except p=11 or 23 and M(p)更多