# Introduction to Arithmetic Geometry Fall 2013 Lecture #24 12/03/2013

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1 18.78 Introduction to Arithmetic Geometry Fall 013 Lecture #4 1/03/ Isogenies of elliptic curves Definition 4.1. Let E 1 /k and E /k be elliptic curves with distinguished rational points O 1 and O, respectively. An isogeny ϕ: E 1 E of elliptic curves is a surjective morphism that maps O 1 to O. As an example, the negation map that send P E(k) to its additive inverse is an isogeny from E to itself; as noted in Lecture 3, it is an automorphism, hence a surjective morphism, and it clearly fixes the identity element (the distinguished rational point O). Recall that a morphism of projective curves is either constant or surjective, so any nonconstant morphism that maps O 1 to O is automatically an isogeny. The composition of two isogenies is an isogeny, and the set of elliptic curves over a field k and the isogenies between them form a category; the identity morphism in this category is simply the identity map from an elliptic curve to itself, which is is clearly an isogeny. Given that the set of rational points on an elliptic curve form a group, it would seem natural to insist that, as morphisms in the category of elliptic curves, isogenies should preserve this group structure. But there is no need to put this requirement into the definition, it is necessarily satisfied. Theorem 4.. Let E 1 /k and E /k be elliptic curves and let ϕ: E 1 E be an isogeny defined over k. Then ϕ is a group homomorphism from E 1 (L) to E (L), for any algebraic extension L/k. Proof. This is essentially immediate (just consider the pushforward map on divisors), but let us spell out the details. By base extension to L, it suffices to consider the case L = k. For i = 1,, let O i be the 0 distinguished rational point of E i and let φi : E i (k) Pic k E i be the group isomorphism that sends P E i (k) to the divisor class [P O i ]. Let ϕ : Pic 0 k E 1 Pic 0 k E be the pushforward map on divisor classes of degree zero. For any P E 1 (k) we have ϕ ([P ]) = [ϕ(p )], since P and ϕ(p ) both have degree one, and ϕ (φ 1 (P )) = ϕ ([P O 1 ]) = [ϕ (P O 1 )] = [ϕ(p ) ϕ(o 1 )] = [ϕ(p ) O ] = φ(ϕ(p )). For any P, Q E 1 (k) with P Q = R we have P Q = R φ 1 (P ) + φ 1 (Q) = φ 1 (R) ϕ (φ1(p ) + φ(q)) = ϕ (φ!(r)) ϕ (φ 1(P )) + ϕ (φ1( Q)) = ϕ (φ1(r)) φ (ϕ(p )) + φ (ϕ(q)) = φ (ϕ(r)) ϕ(p ) ϕ(q) = ϕ(r), where denotes the group operation on both E 1 (k) and E (k). Now that we know that an isogeny ϕ: E 1 E is a group homomorphism, we can speak of its kernel ϕ 1 (O ). One can view ker ϕ as a set of closed points of E 1 /k, but it is more useful to view it as a subgroup of E 1 (k). 1 Andrew V. Sutherland

2 Definition 4.3. Let ϕ: E 1 E be an isogeny of elliptic curves over k. The kernel of ϕ, denoted ker ϕ is the kernel of the group homomorphism ϕ: E 1 (k) E (k). Recall the translation-by-q automorphism τ Q : E E that sends P to P Q. The induced map τ Q : k(e) k(e) is an automorphism of the function field k(e). Lemma 4.4. Let ϕ: E 1 E be an isogeny of elliptic curves. For each P ker ϕ, the automorphism τ fixes ϕ P (k(e )), and the map ker ϕ Aut(k(E 1 )/ϕ (k(e ))) defined by P τ P is an injective group homomorphism. Proof. Let P ker ϕ and let f k(e ). Then τ P (ϕ (f))(q) = (f ϕ τ P )(Q) = f(ϕ(p Q)) = f(ϕ(q)) = (f ϕ)(q) = ϕ (f)(q), since ϕ is a group homomorphism and P lies in its kernel. Thus τ P fixes ϕ (k(e )). For any P, Q ker ϕ and f k(e 1 ) we have τ P Q (f) = f τ P Q = f τ Q τ P = τ P (f τ Q ) = τ P (τ Q (f)), so τ P Q = τ P τ Q, and the map P τ P is a group homomorphism. It is clearly injective, since if P = Q then P Q = O and τ 1 P Q = τ P (τ Q ) is not the identity map (apply it to any nonconstant f k(e 1 )). Corollary 4.5. For any isogeny ϕ: E 1 E of elliptic curves, # ker ϕ divides deg ϕ. In particular, the kernel of an isogeny is finite. Proof. By definition, deg ϕ = [k(e 1 ) : ϕ (k(e ))], and we know from Galois theory that the order of the automorphism group of a finite extension divides the degree of the extension. Since ker ϕ injects into Aut(k(E 1 )/ϕ (k(e ))), its order must divide deg ϕ. Remark 4.6. In fact, the homomorphism in Lemma 4.4 is an isomorphism, and the corollary implies that when ϕ is separable we have # ker ϕ = deg ϕ; see [1, III.4.10]. 4. Torsion points on elliptic curves Definition 4.7. Let E/k be an elliptic curve and let n be a positive integer. multiplication-by-n map [n]: E(k) E(k) is the group homomorphism defined by The np = P P P. The points P E(k) for which np = O are called n-torsion points. They form a subgroup of E(k) denoted E[n]. If ϕ: E 1 E is an isogeny, then we know from Corollary 4.5 that n = deg ϕ is a multiple of the order of ker ϕ. It follows that every point in ker ϕ is an n-torsion point. By definition, [n] is a group homomorphism. We now show that [n] is an isogeny. Theorem 4.8. The multiplication-by-n map on an elliptic curve E/k is an isogeny.

3 Proof assuming char(k) = : The case n = 1 is clear, and for n = the map P P P is a rational map, hence a morphism (by Theorem 18.6, a rational map from a smooth projective curve is a morphism), since it can be defined in terms of rational functions of the coordinates of P via the algebraic formulas for the group operation on E(k). More generally, given any morphism φ: E E, plugging the coordinate functions of φ into the formulas for the group law yields a morphism that sends P to φ(p ) P. It follows by induction that [n] is a morphism, and it clearly fixes the identity element O. It remains to show that [n] is surjective. For this it suffices to show that it does not map every point to O, since a morphism of smooth projective curves is either surjective or constant (by Corollary 18.7). We have already seen that there are exactly 4 points in E(k) that are fixed by the negation map, three of which have order (in short Weierstrass form, these are the point at infinity and the 3 points whose y-coordinate is zero). For n odd, [n] cannot map a point of order to O, so [n] is surjective for n odd. For n = k m with m odd we may write [n] = [] [] [m]. We already know that [m] is surjective, so it suffices to show that [] is. But [] cannot map any of the infinitely many points in E(k) that are not one of the 4 points fixed by the negation map to O, so [] must be surjective. Remark 4.9. Note that in characteristic there are not four -torsion points, in fact there may be none. But one can modify the proof above to use 3-torsion points instead. Corollary Let E/k be an elliptic curve. For any positive integer n, the number of n-torsion points in E(k) is finite. Remark In fact one can show that the number of n-torsion points divides n, and for n not divisible by char(k), is equal to n. 4.3 Torsion points on elliptic curves over Q Let E be an elliptic curve Q, which we may assume is given by a short Weierstrass equation E : y = x 3 + a 4 x + a 6, with a 4, a 6 Q. Let d be the LCM of the denominators of a 4 and a 6. After multiplying both sides by d 6 and replacing y by d 3n y and x by d n x, we may assume a 4, a 6 Z. Since E is non-singular, we must have = (E) := 16(4a a 6 ) = 0.1 For each prime p the equation for E also defines an elliptic curve over Q p. For the sake of simplicity we will focus our attention on primes p that do not divide, but everything we do below can be extended to arbitrary p (as we will indicate as we go along). Let E 0 denote the elliptic curve over Q p obtained by base extension from Q to Q p. Let E/F p denote the curve over F p obtained by reducing the equation for E modulo p. Here we are assuming 0 mod p so that the reduced equation has no singular points, meaning that E is an elliptic curve. We say that E has good reduction at p when this holds. The reduction map E 0 (Q p ) E(F p ) is a group homomorphism, and we define E 1 (Q p ) to be its kernel; these are the points that reduce to (0 : 1 : 0) modulo p. In fact, E 1 (Q p ) can be defined as the kernel of the reduction map regardless of whether E has good reduction at p or not and one can show that the points in E 1 (Q p ) still form a group. The points in E 1 (Q p ) are precisely the points in E 0 (Q p ) that can be represented as (x : y : z), with v p (x), v p (z) > 0 and v p (y) = 0; equivalently, the points with v p (x/y) > 0 1 The leading factor of 16 appears for technical reasons that we won t explain here, but it is useful to have a factor of in because a short Weierstrass equation always has singular points in characteristic. 3

4 (note that v p (x/y) does not depend on how the coordinates are scaled). For all positive integers n we thus define E n (Q p ) = { (x : y : z) E 0 } (Q p ) : v p (x/y) n, and note that this agrees with our previous definition of E 1 ( Q p ). Lemma 4.1. For n > 0, each of the sets E n+1 (Q p ) is an index p subgroup of E n (Q p ). Proof. Containment is clear from the definition, but we need to show that the sets E n (Q p ) are actually groups. For O = (0 : 1 : 0) we have v p (x/y) =, so O E n (Q p ) for all n. Any affine point P E n (Q p ) E n+1 (Q p ) has v p (x/y) = n, and and after dividing through by z can be written as (x : y : 1) with v p (y) < 0. Since a 4, a 6 Z p, the equation y = x 3 +a 4 x+a 6 implies 3v p (x) = v p (y), so n = v p (x/y) = v p (x) v p (y) = v p (y)/3, and therefore v p (y) = 3n and v p (x) = n. After multiplying through by p 3n we can write P = (pnx 3 0 : y 0 : p n ) with x 0, y 0 Z p. We then have p 3n y 3 0 = p n x a 4 p 7n x 0 + a 9 6p n 4n 6n y0 = x a4p x 0 + a 6 p. After reducing mod p we obtain an affine point (x 0 : y 0 : 1) whose coordinates are all nonzero and which lies on the singular variety C 0 /F p defined by y z = x 3, which also contains the reduction of O = (0 : 1 : 0). If we consider the image of the group law on E 0 (Q p ) on C 0 (F p ), we still have an operation defined by the rule that three colinear points sum to zero. We claim that this makes the set S of nonsingular points in C 0 (F p ) into a group of order p. To show this, we first determine S. We have ( / x)(y z x 3 ) = 3x, ( / y)(y z x 3 ) = yz, ( / z)(y z x 3 ) = y. It follows that a point in C 0 (F p ) is singular if and only if its y-coordinate is zero. In particular all of the reductions of points in E n (Q p ) are non-singular, for any n 1. Every non-singular point in C 0 (F p ) can be written as (x : 1 : x 3 ), and this gives a bijection from F 3 p to S defined by x (x : 1 : x ). Thus the set S has order p and it contains the identity element. It is clearly closed under negation, and we now show it is closed under addition. If P and Q are two elements of S not both equal to (0 : 1 : 0), then at least one of them has non-zero z-coordinate and the line L defined by P and Q can be written in the form z = ax + by. Plugging this into the curve equation gives y (ax + by) = x 3, and it is then clear that the third point R in C 0 L must have nonzero y-coordinate, since y 0 = 0 x 0 = 0 z 0 = 0 for any (x 0 : y 0 : z 0 ) C 0 L. Since P and Q are both in C 0 (F p ), so is R, thus R lies in S, as does its negation, which is P Q. Therefore the reduction map E n (Q p) C 0(F p ) defines a group homomorphism from E n (Q p ) to S, and its kernel is E n+1 (Q p ), an index p subgroup of E n (Q p ). 4

5 Definition The infinite chain of groups is called the p-adic filtration of E/Q. E 0 (Q p ) E 1 (Q p ) E (Q p ) Theorem Let E/Q be an elliptic curve and let p be a prime not dividing (E). The p-adic filtration of E satisfies (1) E 0 (Q p )/E 1 (Q p ) E(F p ); () E n (Q p )/E n+1 (Q p ) E(F p ) Z/pZ for all n > 0; (3) n E n = {O}. Proof. The group E 1 (Q p ) is, by definition, the kernel of the reduction map from E 0 (Q p ) to E(F p ). To prove (1) we just need to show that the reduction map is surjective. Let P = (a 1 : a : a 3 ) be a point in E(F p ) with the a i Z/pZ. The point P is nonsingular, so at least one of the partial derivatives of the curve equation f(x 1, x, x 3 ) = 0 for E does not vanish. Without loss of generality, suppose f/ x 1 does not vanish at P. If we pick an arbitrary point P ˆ = (â 1 : â : â 3 ) with coefficients â i Z p such that â i a i mod p, we can apply Hensel s to the polynomial g(t) = f(t, â, â 3 ) using a 1 Z/pZ as our initial solution, which satisfies g (a i ) = 0. This yields a point in E 0 (Q p ) that reduces to P, thus the reduction map is surjective, which proves (1). Property () follows from the lemma above. For (3), note that E 1 (Q p ) contains only points with nonzero y-coordinate, and the only such point with v p (x/y) = is O; every other other point (x : y : z) E 1 (Q p ) lies in E n (Q p ) E n+1 (Q p ), where n = v p (x/y). Remark Theorem 4.14 can be extended to all primes p by replacing E(F p ) in (1) with the set S of non-singular points on the reduction of E modulo p. As in the proof of Lemma 4.1, one can show that S always contains O and is closed under the group operation, but there are now three different group structures that can arise: 1. A cyclic group of order p isomorphic to the additive group of F p ; in this case E is said to as additive reduction at p.. A cyclic group of order p 1 isomorphic to the multiplicative group of F p ; in this case E is said to have split multiplicative reduction at p. 3. A cyclic group of order p + 1 isomorphic to the subgroup of the multiplicative group of a quadratic extension of F p consisting of elements of norm one; in this case E is said to have non-split multiplicative reduction at p. Parts () and (3) of the theorem remain true for all primes p (as we will now assume). Corollary Suppose P = (x : y : 1) is an affine point in E 0 (Q p ) with finite order prime to p. Then x, y Z p. Proof. Suppose not. Then both x and y must have negative p-adic valuations in order to satisfiy y = x 3 + a 4 x + a 6 with a 4, a 6 Z p, and we must have v p (x) = 3v p (y), so v p (x/y) > 0. Let n = v p (x/y). Then P E n (Q p ) E n+1 (Q p ), and the image of P in E n (Q p )/E n+1 (Q p ) Z/pZ is not zero, hence it has order p. The order m of P is prime to p, so the image of mp in E n (Q p )/E n+1 (Q p ) is also nonzero. Thus mp E n+1 (Q p ), but this is a contradiction, because mp = O E n+1 (Q p ). 5

6 Lemma Suppose P 1, P, P 3 are colinear points in E n (Q p ), for some n > 0, with P i = (x i : 1 : z i ). Then v p (x 1 + x + x 3 ) 5n. Proof. We have already seen that for P E n (Q ) we have x n 3n i p i p Z p and z i p Z p. Fixing y = 1, if P 1 = P then the equation of the line through P 1 and P in the x-z plane has the form z = αx + β with α = (z z 1 )/(x x 1 ). Using the curve equation z = x 3 + a 3 xz + z 3 (with y = 1), we can rewrite α as z z 1 α = x x 1 z z 1 1 a4z a 6(z = + z 1z + z1 ) x x 1 1 a 4 z a 6(z + z 1 z + z1 ) (z 3 a 6 z = ) (z 1 a 4 x 1 z1 a 6z1 3) a 4x 1 z (x 1 x )(1 a4z a 6 (z + z 1z + z1 )) (x 3 + a xz ) x a 4 x 1 z = (x 1 x )(1 a 4 z a 6(z + z 1 z + z1 )) (x x 1 )(x + x 1x + x1 = + a 4z ) (x1 x )(1 a4z a 6(z + z 1z + z 1 )) x + x 1x + x 1 + a 4z =. 1 a z a (z + z z + z ) The key point is that the denominator of α is then a p-adic unit. It follows that α p n Z p, and then β = z 1 αx 1 p 3n Z p. Substituting z = αx + β into the curve equation gives αx + β = x 3 + a 4 x(αx + β) + b(αx + β) 3. We know that x 1, x, x 3 are the three roots of the cubic defined by the equation above, thus x 1 + x + x 3 is equal to the coefficient of the quadratic term divided by the coefficient of the cubic term. Therefore The case P 1 = P is similar. x 1 + x + x 3 = a 4 αβ + 3a 6 α β p 5n 1 + a 4 α + a 6 β 3 Z p. Corollary The group E 1 (Q p ) is torsion-free. Proof. By the previous corollary we only need to consider the case of a point of order p. So suppose pp = 0 for some P E n (Q p ) E n+1 (Q p ). Consider the map E n (Q p ) p n Z p /p 5n Z p that sends P := (x : 1 : z) to the reduction of x in p n Z p /p 5n Z p. By the lemma above, this is a homomorphism of abelian groups, so it sends pp to the reduction of px p n+1 Z p p n+ Z p modulo p 5n Z p, which is not zero, a contradiction. Corollary Let E/Q be an elliptic curve and let p be a prime of good reduction. The torsion subgroup of E(Q) injects into E(F p ). in particular, the torsion subgroup is finite. Proof. The group E(Q) is isomorphic to a subgroup of E 0 (Q p ) and E(F p ) = E 0 (Q p )/E 1 (Q p ). But E 1 (Q p ) is torison free, so the torsion subgroup of E(Q) injects into E(F p ). 6

7 Now that we know that each elliptic curve over Q has a finite number of rational torsion points, one might ask whether there is a uniform upper bound that applies to every elliptic curve over Q. It s not a priori clear that this should be the case; one might suppose that by varying the elliptic curve we could get an arbitrarily large number of rational torsion points. But this is not the case; an elliptic curve over Q can have at most 16 rational points of finite order. This follows from a celebrated theorem of Mazur that tells us exactly which rational torsion subgroups can (and do) arise for elliptic curves defined over Q. Theorem 4.0 (Mazur). Let E/Q be an elliptic curve. The torsion subgroup of E(Q) is isomorphic to one of the fifteen subgroups listed below: Z/nZ (n = 1,, 3,..., 9, 10, 1), Z/Z Z/nZ (n = 1,, 3, 4). The proof of this theorem is well beyond the scope of this course. However, as a further refinement of our results above, we can prove the Nagell-Lutz Theorem. Theorem 4.1 (Nagell-Lutz). Let P = (x 0 : y 0 : 1) be an affine point of finite order on the elliptic curve y = x 3 + a 4 x + a 6 over Q, with a 4, a 6 Z. Then x 0, y 0 Z, and if y 0 = 0 then y0 divides 4a a 6. Proof. For any prime p, if v p (x 0 ) < 0 then v p (y 0 ) = 3v p (x 0 ) and v p (x 0 /y 0 ) > 0. It follows that P E 1 (Q), but then P cannot be a torsion point. So v p (x) 0 for all primes p. Thus x 0 is an integer, and so is y0 = x3 0 + a 4x 0 + a 6, and therefore y. If P has order then y 0 = 0; otherwise, the x-coordinate of P is an integer equal to λ x 0, where λ = (3x 0 + a 4)/(y 0 ) is the slope of the tangent at P. Thus 4y0 and therefore y0 divides λ = (3x a 4 ), as well as x 0 + a 4 x 0 + a 6. We now note that (3x 0 + 4a 4 )(3x 0 + a 4 ) = 7x a 4 x a 4x 0 + 4a 3 4 (3x 0 + 4a 4 )(3x 0 + a4) = 7(x a 4 x 0 ) + 4a a 6 + 4a 3 4 mod y0, since (x a 4x 0 ) a 6 mod y 0, thus y 0 divides 4a a 6. The Nagell-Lutz theorem gives an effective method for enumerating all of the torsion points in E(Q) that is quite practical when the coefficients a 4 and a 6 are small. By factoring D = 4a a 6, one can determine all the squares y 0 that divide D. By considering each of these, along with y 0 = 0, one then checks whether there exists an integral solution x 0 to y0 = x3 0 + ax 0 + a 6 (note that such an x 0 must be a divisor of a 6 y0 ). This yields a list of candidate torsion points P = (x 0 : y 0 : 1) that are all points in E(Q), but do not necessarily all have finite order. To determine which do, one computes multiples np for increasing values of n (by adding the point P at each step, using the group law on E), checking at each step whether np = O. If at any stage it is found that the affine coordinates of np are not integers then np, and therefore P, cannot be a torsion point, and in any case we know from Mazur s theorem that if np = O for any n 1 then P is not a torsion point; alternatively, we also know that n must divide #E(F p ), where p is the least prime that does not divide (E). However, this method is not practical in general, both because it requires us to factor D, and because D might have a very large number of square divisors (if D is, say, the product There was recently a graduate seminar at Harvard devoted entirely to the proof of Mazur s theorem; see chaoli/mazurtorsionseminar.html for notes and references. 7

8 of the squares of the first 100 primes, then we have 100 values of y 0 to consider). But Cororllary 4.19 gives us a much more efficient alternative that can be implemented to run in quasi-linear time (roughly proportional to the number of bits it takes to represent a 4 and a 6 on a computer). We first determine the least odd prime p that does not divide D (we don t need to factor D to do this), and then exhaustively compute the set E(F p), which clearly has cardinality at most p (in fact, at most p p). We then use Hensel s lemma to lift these points to Z by computing p-adic approximations of their coordinates to sufficiently high precision; the fact that y0 must divide D gives us an upper bound on x 0 and y 0. This yields a small set of candidate points P = (x 0 : y 0 : 1) with x 0, y 0 Z that satisfy the curve equation modulo some power of p. For each candidate P we then check whether P E(Q) by plugging its coordinates into the curve equation (it need not be satisfied, P could be the reduction of a Q p -point that doe not lie in E(Q)). If P E(Q), we then check that it is actually a torsion point by computing np = E(Q), where n is the order of the reduction of P in E(F p ). References [1] J. H. Silverman, The arithmetic of elliptic curves, Springer,

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