1 Elliptic Curves Over Finite Fields

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1 1 Elliptic Curves Over Fiite Fields 1.1 Itroductio Defiitio 1.1. Elliptic curves ca be defied over ay field K; the formal defiitio of a elliptic curve is a osigular (o cusps, self-itersectios, or isolated poits projective algebraic curve over K with geus 1 with a give poit defied over K. If the characteristic of K is either 2 or 3, the every elliptic curve over K ca be writte i the form y 2 = x 3 px q where p,q K such that the RHS does ot have ay double roots. If the characteristic of K is 3, the the most geeral equatio is of the form such that RHS has distict roots. y 2 = 4x 3 + b 2 x 2 + 2b 4 x + b 6 I characteristic 2, the most geeral equatio is of the form provided that the variety it defies is o-sigular. y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 Defiitio 1.2. A ratioal poit is a poit whose coordiates lie i K. W deote the set of ratioal poits as E(K ad this set forms a group. 1.2 Isogeies Defiitio 1.3. A o-costat morphism φ : E 1 E 2 betwee elliptic curves such that φ(o = O is called a isogey. Let φ : E 1 E 2 be a isogey. The there is a uique isogey ˆφ : E 2 E 1 satisfyig ˆφ φ = [degφ]. ˆφ is called the dual of φ. Remark. I the above statemet, [] deotes the isogey which adds times. If E is defied over F q ad π q,e : E E is the Frobeius morphism (x,y (x q,y q, the E(F q = ker(1 π q,e. As we oted, a isogey has fiite kerel. Lemma 1.1. Let E 1 ad E 2 be isogeous elliptic curves defied over F q. The #E 1 (F q = #E 2 (F q. Proof. Ay isogey φ : E 1 E 2 commutes with the Frobeius map o E 1 ad E 2. Now, φ is surjective. So we have y E 2 (F q π q,e2 (φ(x = φ(x x ker((1 π q,e2 φ. Now, each φ 1 (y has deg sep φ elemets. Thus, #E 2 (F q = ker((1 π q,e2 φ/deg sep φ = #ker(φ(1 π q,e2 /deg sep φ = #ker(φ(1 π q,e1 /deg sep φ = deg sep (1 φ q,e1 = #E 1 (F q. Recall the followig useful facts o degrees ad dual maps (i φ + ψ = ˆφ + ˆψ (ii [ ˆ] = [] (iii deg[] = 2 (iv deg ˆφ = degφ (v ˆφ = φ (vi deg( φ = deg(φ (vii d(φ,ψ := deg(φ + ψ degφ degψ is symmetric, biliear o Hom(E 1,E 2, where E i is a elliptic curve. (viii degφ > 0 for ay isogey φ

2 1.3 Riema Hypothesis for Elliptic Curves For a elliptic curve E defied over a fiite field F q, the most obvious parameter is the umber of poits i E(F q. Theorem 1.1. (Riema hypothesis for elliptic curves (Hasse, 1934 Let E be a elliptic curve defied over F q. The #E(F q 1 q 2q /2 1 Proof. Choose a Weierstrass equatio with coefficiets i F q. Sice Gal(F q /F q is topologically geerated by x π q x q, a poit P of E(F q lies i E(F q if ad oly if π q,e (P = P. Thus P E(F q if ad oly if πq,e (P = P, i.e., E(F q = ker(1 πq,e. Now 1 π q,e is a separable morphism (sice its differetial is the idetity. Thus, #E(F q = deg(1 πq,e. We oted that, for ay two elliptic curves over a field, the fuctio d : Hom(E 1,E 2 Hom(E 1,E 2 Z, (φ,ψ deg(φ + ψ degφ degψ is a positive defiite biliear form. By the Cauchy-Schwarz iequality, we get deg(1 πq,e deg1 degπq,e 2 deg1degπq,e i.e. #E(F q 1 q 2q /2 1.4 The Weil Cojectures Let K = F q. If V is a projective variety, we wat to keep accout of #V (K. We ca do this by usig the zeta fuctio of V ad it is defied as the formal power series Note that Z(V /K 1 : T = exp ( #V (K T =1 1 #V (K = ( 1! dt logz(v /K 1,T T =0 The reaso for defiig the zeta fuctio i this maer is that the series 1 #V (K T ratioal fuctio of T. Let V be ay smooth projective variety of dimesio, defied over K 1 = F q. The: Ratioality cojecture d ofte looks like the log of a Fuctioal cojecture There exists a iteger χ such that Factorizatio There exists a factorizatio ( Z V /K 1 ; Z = Z(V /K 1 ;T Q(T 1 q T = ±q χ/2 T χ Z(V /K 1 ;T P 1 (T P 3 (T P 2 1 (T P 0 (T P 2 (T P 2 2 (T P 2 (T

3 with P 0 (T = 1 T, P 2 (T = 1 q T, each P i (T Z[T ] ad where b i = degp i = degp 2 i Riema hypothesis Each root of P i (T satisfies α = q i/2 ( ( 1 1 bi P i q = P 2 i (T T T q i/2 The cojecture was prove i its etirety by the efforts of Weil, Dwork, M. Arti, Grothedieck, Lubki, Delige, Laumo. But the first case for elliptic curves was solved by Hasse i 1934 before the cojectures were formulated i this geerality by Weil i Weil poited out that if oe had a suitable cohomology theory for abstract varieties aalogous to the usual cohomology for varieties over C, the stadard properties of the cohomology would imply all the cojectures. For istace, the fuctioal equatio would follow from Poicaré duality property. Such a cohomology is the étale cohomology. 1.5 Tate Modules ad the Weil Pairig Let E be a elliptic curve defied over F q. Suppose l is a prime ot dividig q. We kow that the l divisio poits of E, i.e., E[l ] d = ker[l ] is Z/l Z/l. The iverse limit of the groups E[l ] with respect to the maps E[l +1 ] [l] E[l ] is the Tate module T l (E = lime[l ]. Sice each E[l ] is aturally a Z/l module, it ca be checked that T l (E is a Z l (= limz/l module. It is a free Z l module of rak 2. Ay isogey φ : E 1 E 2 iduces a Z l module homomorphism φ l : T l (E 1 T l (E 2. I particular, we have a represetatio Ed(E M 2 (Z l,φ φ l if l q. Note that Ed(E Ed(T l (E is ijective because if φ l = 0 the φ is 0 o E[l ] for large, i.e., φ = O. Fially, let us recall the Weil pairig. This is a o-degeerate, biliear, alteratig pairig It has the importat property that e(φx,y = e(x, ˆφy. e : T l (E T l (E T l (µ d = lim µ l = Zl Remark. For ay geeral curves C,D, ad a ocostat morphism φ : C D, recall that φ : Div(D Div(C is a homomorphism defied by (P Q φ 1 (P e φ (Q(Q where e φ (Q is the ramificatio idex at Q. For C = D a elliptic curve, all the e φ (Q = deg isep φ. For a geeral C ad D, ord P ( f φ = e φ (Qord φ(p ( f for every ocostat ratioal fuctio o D. 1.6 Weil Cojectures for Elliptic Curves Lemma 1.2. Let φ Ed(E ad l q be a prime. The, detφ l = degφ traceφ l = 1 + degφ deg(1 φ I particular, detφ l,traceφ l are idepedet of l, ad are itegers. Proof. Let (v 1,v 2 be a Z l basis of T l (E ad write ( a b φ l = c d

4 with respect to this basis. We ow use the Weil pairig e which is biliear ad alteratig Sice e is odegeerate, we have that degφ = degφ l. Fially, e(v 1,v 2 degφ = e(deg(φv 1,v 2 = e(( ˆφ l (φ l v 1,v 2 = e(φ l v 1,φ l v 2 = e(av 1 + cv 2,bv 1 + dv 2 = e(v 1,v 2 ad bc = e(v 1,v 2 detφ l traceφ l = 1 + detφ l det(id φ l = 1 + degφ deg(1 φ To prove the Weil cojectures for E, we have to compute #E(K, where K = F q. Now #E(K = deg(1 φ where φ = π q,e is the Frobeius isogey. A cosequece of the lemma is the fact that the characteristic polyomial of φ l has coefficiets i Z whe l charf q. Write det(id T φ l = (T α(t β for α,β C. Moreover, for all m Q, we get m det( Id φ l = 1 2 det(mid φ l = deg(m φ 1 2 > 0 This implies α = β. Note by triagularizig, that detid T φ l = (T α (T β, we get Theorem 1.2. For all 1, #E(K = 1 α α + q where α = q 1/2. I particular, Z(E/K 1 ;T = 1 at + qt 2 (1 T (1 qt where a Z ad 1 at + qt 2 = (1 αt (1 αt. Further, Z(E/K 1 ; 1 qt = Z(E/K 1;T. Proof. We have that ad α = q. Hece #E(K = deg(1 φ = det(1 φ l = 1 α α + q logz(e/k 1 ;T = (1 α α + q T 1 = T (αt (αt (1 αt (1 αt = log (1 T (1 qt (qt + The fuctioal equatio is obvious from the expressio. The factorizatio Z = P 1 P 0 P 2 is with P 1 (T = 1 at + qt 2, so ( P 1 1 qt = P 1(T ( 1/T q 2

5 Remark. Puttig ζ E/Fq (s = Z(E/K 1 ;q s, oe has ζ E/Fq (s = 1 aq s + q 1 2s (1 q s (1 q 1 s = ζ E/F q (1 s Note that the Riema hypothesis for Z(E/K 1 ;T is equivalet to the fact that the zeros of ζ E/Fq (s are o the lie Re(s = Supersigularity Supersigular curves are a special class of elliptic curves which arise aturally. Oe of the most useful properties they have, as we shall prove, is that their defiitio forces them to be defies over a small fiite field ad, over ay field, there are oly fiitely may elliptic curves isogeous to a supersigular oe. Before defiig supersigularity, let us recall that a elliptic curve E is said to have complex multiplicatio if Ed(E Z. Let us recall the followig result o Ed(E Propositio 1.1. (i Ed(E has o zero divisors. (ii Ed(E is torsio free. (iii Ed(E is either Z, or a order i a imagiary quadratic field, or a order i a quaterio divisio algebra over Q. Defiitio 1.4. A elliptic curve E defied over a field of characteristic p > 0 is said to be supersigular if E[p] = O. The followig characterizatio of supersigular elliptic curves is very useful Propositio 1.2. Let K be a perfect field of characteristic p > 0. The the followig statemets are equivalet: (a E is sigular (b [p] : E E is purely iseparable ad j(e F p 2 (c E[p r ] = O for some r 1 (d E[p r ] = O for all r 1 (e Ed K (E is a order i a quaterio divisio algebra over Q. Remark. By the above propositio, up to isomorphism, there are oly fiitely may elliptic curves isogeous to a supersigular curve. For p = 2, Y 2 +Y = X 3 is the uique supersigular curve. For p > 2, we have the followig theorem: 1.8 Structure of E(F q Theorem 1.3. A group G of order N = q + 1 m is isomorphic to E(F q for some elliptic curve E over F q if oe of the followig holds: (i (q,m = 1, m 2 q, ad G = Z/A Z/B where B (A,m 2 (ii q is a square, m = ±2 q, ad G = (Z/A 2 where A = q 1 (iii q is a square, p 1 mod 3, m = ± q, ad G is cyclic (iv q is ot a square, p = 2 or 3, m = ± pq, ad G is cyclic (v q is ot a square, p 3 mod 4, m = 0, ad G is cyclic or q is a square, p 1 mod 4, m = 0, ad G is cyclic (vi q is ot a square, p 3 mod 4, m = 0, ad G is either cyclic or G = Z/M Z/2 where M = q+1 2