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/n_{1} ⊕Z/n_{2} with
n≥1 and n_{1}, n_{2} ≥1integers, n_{1}|n_{2}.

2. Let E be an elliptic curve over a finite field F^{q} 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 µ_{n} ⊂Fq.
(c) Let a=q+ 1−#E(Fq). Deduce thata ≡2(modn).

(d) Show that q=n^{2}+ 1 orq=n^{2}±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 α_{n} of E[n].

ii. Let ^{a b}_{c d}

the matrix of α_{n} in the base {T_{1}, T_{2}}. Show that
deg α≡det(α_{n})(mod n)

(one could express ζ^{deg 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(rα_{n}+sβ_{n})−r^{2}detα_{n}−s^{2}detβ_{n}=rs(det(α_{n}+β_{n})−detα_{n}−detβ_{n})
(one can start by showing that det(α_{n} +β_{n}) − detα_{n} − detβ_{n} =
T race(α_{n}β_{n}^{∗}), where β_{n}^{∗} is the adjoint matrix : if β_{n} = ^{x y}_{z t}

, then
β_{n}^{∗} = _{−z x}^{t} ^{−y}

).

ii. Deduce that

deg rα+sβ =r^{2}deg α+s^{2}deg β +rs(deg(α+β)−deg α−deg β).

4. (a) Let E be an elliptic curve defined over a finite field Fq, q = p^{r}, and let
a_{q}=q+1−#E(Fq). As before, we denoteφ_{q}the Frobenius morphism on
E and for any integermprime toq one denote (φ_{q})_{m} the endomorphism
induced by φ_{q} on E(Fq)[m]. Show that

det(φ_{q})_{m} ≡q(mod m) and T race(φ_{q})_{m} ≡a_{q}(modm)

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

(b) Deduce that the endomorphism φ^{2}_{q} − a_{q}φ_{q} + q is identically zero on
E(Fq)[m].

(c) Show that the kernel of the map φ^{2}_{q} −a_{q}φ_{q} +q is infinite; deduce that
the polynomial g(x) = x^{2}−a_{q}x+q annihilatesφ_{q}.

(d) Assume that b is an integer such that the polynomial x^{2} −bx+q an-
nihilates φ_{q}. Deduce that (a_{q} −b) annihilates E(Fq) and finally that
a_{q}=b.

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

g_{n}(x) = x^{2n}−(α^{n}+β^{n})x^{n}+q^{n}.
Show that g(x)divides g_{n}(x) for all n. Deduce that

(φ^{n}_{q})^{2}−(α^{n}+β^{n})φ^{n}_{q} +q^{n}= 0.

(f) Deduce that E(Fq^{n}) has cardinality q^{n}+ 1−(α^{n}+β^{n}).

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

Z(E/F^{q}, T) =exp(

∞

X

n=1

#E(F^{q}^{n})T^{n}
n ).

Show that Z(E/Fq, T) is a rational function
1−aqT +qT^{2}
(1−T)(1−qT).

2

Additional exercise (DO NOT SUBMIT WITH THE HOMEWORK):

LetE be an elliptic curvey^{2} =x^{3}+ax+b defined over a fieldk,char(k)6= 2,3.

One defines the division polynomials ψ_{m}(x, y) in a recursive way : ψ_{0} = 0, φ_{1} = 1,
ψ_{2} = 2y

ψ3 = 3x^{4}+ 6ax^{2}+ 12bx−a^{2}

ψ_{4} = 4y(x^{6}+ 5ax^{4} + 20bx^{3}−5a^{2}x^{2}−4abx−8b^{2}−a^{3})
ψ_{2m+1} =ψ_{m+2}ψ_{m}^{3} −ψm−1ψ_{m+1}^{3} ,m ≥2

ψ2m = [ψm(ψm+2ψ_{m−1}^{2} −ψm−2ψ^{2}_{m+1})]/2y, m≥3.

1. Show that ψ_{n} is a polynomial in x, y^{2} if n is odd and that yψ_{n} is polynomial
inx, y^{2}, if n is even.

2. One defines φ_{m} =xψ^{2}_{m}−ψ_{m+1}ψm−1

ω_{m} = [ψ_{m+2}ψ_{m−1}^{2} −ψm−2ψ_{m+1}^{2} ]/4y.Show that φ_{n}is a polynomial inx, y^{2}, that
ω_{n} is a polynomial in x, y^{2} if n is odd, and that yω_{n} is a polynomial in x, y^{2}
if n is even.

3. By the previous question, on can define the polynomials φ_{n}(x) and ψ_{n}^{2}(x) by
replacing y^{2} by x^{3} +ax+b in the polynomials φ_{n}(x, y) and ψ^{2}_{n}(x, y). Show
that φ_{n}(x) is the sum of x^{n}^{2} and the terms of lower degree, and that ψ_{n}(x)^{2}
is the sum of n^{2}x^{n}^{2}^{−1} and the terms of lower degree.

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

nP = ( φ_{n}(x)

ψ_{n}(x)^{2},ω_{n}(x, y)
ψ_{n}(x)^{3} )

5. Show that the polynomialsφ_{n}(x)and ψ_{n}(x)^{2} are relatively prime. Deduce the
the multiplication by n map is of degree n^{2}.

3