Endomorphism rings of supersingular elliptic curves over Fp

Let p > 3 be a fixed prime. For a supersingular elliptic curve E over F-p, a result of Ibukiyama tells us that End(E) is a maximal order O(q) (resp. O'(q)) in End(E) circle times Q indexed by a (non-unique) prime q satisfying q 3 mod 8 and the quadratic residue (p/q) = -1 if 1+pi/2 is not an element of End(E) (resp. 1+pi/2 is an element of End(E)), where pi = ((x,y) bar right arrow (x(p) ,y(p)) is the absolute Frobenius. Let q(j) denote the minimal q for E whose j-invariant j(E) = j and M(p) denote the maximum of q(j) for all supersingular j is an element of F-p. Firstly, we determine the neighborhood of the vertex [E] with j is not an element of {0, 1728} in the supersingular l-isogeny graph if 1+pi/2 is not an element of End(E) and p > q(j)l(2) or 1+pi/2 is an element of End(E) and p > 4q(j)l(2): 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 F-p. 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 q(j) < c root P for c = 4 or 1/2. In the appendix, we compute M(p) for all p < 2000 numerically. Our data show that M(p) > root P except p = 11 or 23 and M(p) < p log(2)p for all p. (C) 2019 Elsevier Inc. All rights reserved.