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MATH-GA 2420.006 : Homework 2; due by Thursday February 18 morning (before 10am), late submission implies -50% of this homework

grade; send the solutions to pirutka@cims.nyu.edu

1. Let E be an elliptic curve over a finite field Fq. Show that the group E(Fq) is either a cyclic group Z/n for some n ≥ 1, or the group Z/n1 ⊕Z/n2 with n≥1 and n1, n2 ≥1integers, n1|n2.

2. Let E be an elliptic curve over a finite field Fq of characteristic p. Assume E(Fq) = Z/n⊕Z/n.

(a) Show that (n, p) = 1.

(b) Show that E(Fq)[n]⊂E(Fq). Deduce that µn ⊂Fq. (c) Let a=q+ 1−#E(Fq). Deduce thata ≡2(modn).

(d) Show that q=n2+ 1 orq=n2±n±1 orq= (n±1)2. 3. (a) Let α be an endomorphism ofE and (n, char.k) = 1.

i. Show that α induces an endomorphism αn of E[n].

ii. Let a bc d

the matrix of αn in the base {T1, T2}. Show that deg α≡det(αn)(mod n)

(one could express ζdeg α in terms of a, b, c, d.)

(b) Let α, β be two endomorphisms of E and r, stwo integers.

i. Show that

det(rαn+sβn)−r2detαn−s2detβn=rs(det(αnn)−detαn−detβn) (one can start by showing that det(αnn) − detαn − detβn = T race(αnβn), where βn is the adjoint matrix : if βn = x yz t

, then βn = −z xt −y

).

ii. Deduce that

deg rα+sβ =r2deg α+s2deg β +rs(deg(α+β)−deg α−deg β).

4. (a) Let E be an elliptic curve defined over a finite field Fq, q = pr, and let aq=q+1−#E(Fq). As before, we denoteφqthe Frobenius morphism on E and for any integermprime toq one denote (φq)m the endomorphism induced by φq on E(Fq)[m]. Show that

det(φq)m ≡q(mod m) and T race(φq)m ≡aq(modm)

(One could use that #Ker(φq−1) =deg(φq−1) =q+ 1−aq, see the proof of Hasse theorem. Also use the formulas for det and Trace from the previous exercise)

(2)

(b) Deduce that the endomorphism φ2q − aqφq + q is identically zero on E(Fq)[m].

(c) Show that the kernel of the map φ2q −aqφq +q is infinite; deduce that the polynomial g(x) = x2−aqx+q annihilatesφq.

(d) Assume that b is an integer such that the polynomial x2 −bx+q an- nihilates φq. Deduce that (aq −b) annihilates E(Fq) and finally that aq=b.

(e) Let α, β be the roots of the polynomial g(x) and let gn(x) be the poly- nomial

gn(x) = x2n−(αnn)xn+qn. Show that g(x)divides gn(x) for all n. Deduce that

nq)2−(αnnnq +qn= 0.

(f) Deduce that E(Fqn) has cardinality qn+ 1−(αnn).

(g) We define the sets function of the curve E by

Z(E/Fq, T) =exp(

X

n=1

#E(Fqn)Tn n ).

Show that Z(E/Fq, T) is a rational function 1−aqT +qT2 (1−T)(1−qT).

2

(3)

Additional exercise (DO NOT SUBMIT WITH THE HOMEWORK):

LetE be an elliptic curvey2 =x3+ax+b defined over a fieldk,char(k)6= 2,3.

One defines the division polynomials ψm(x, y) in a recursive way : ψ0 = 0, φ1 = 1, ψ2 = 2y

ψ3 = 3x4+ 6ax2+ 12bx−a2

ψ4 = 4y(x6+ 5ax4 + 20bx3−5a2x2−4abx−8b2−a3) ψ2m+1m+2ψm3 −ψm−1ψm+13 ,m ≥2

ψ2m = [ψmm+2ψm−12 −ψm−2ψ2m+1)]/2y, m≥3.

1. Show that ψn is a polynomial in x, y2 if n is odd and that yψn is polynomial inx, y2, if n is even.

2. One defines φm =xψ2m−ψm+1ψm−1

ωm = [ψm+2ψm−12 −ψm−2ψm+12 ]/4y.Show that φnis a polynomial inx, y2, that ωn is a polynomial in x, y2 if n is odd, and that yωn is a polynomial in x, y2 if n is even.

3. By the previous question, on can define the polynomials φn(x) and ψn2(x) by replacing y2 by x3 +ax+b in the polynomials φn(x, y) and ψ2n(x, y). Show that φn(x) is the sum of xn2 and the terms of lower degree, and that ψn(x)2 is the sum of n2xn2−1 and the terms of lower degree.

4. Show that for P = (x, y) a point ofE, one has

nP = ( φn(x)

ψn(x)2n(x, y) ψn(x)3 )

5. Show that the polynomialsφn(x)and ψn(x)2 are relatively prime. Deduce the the multiplication by n map is of degree n2.

3

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