1. Vélu s formulae GENERALIZATION OF VÉLU S FORMULAE FOR ISOGENIES BETWEEN ELLIPTIC CURVES. Josep M. Miret, Ramiro Moreno and Anna Rio

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1 Publ. Mat. 2007, Proceedings of the Primeras Jornadas de Teoría de Números. GENERALIZATION OF VÉLU S FORMULAE FOR ISOGENIES BETWEEN ELLIPTIC CURVES Josep M. Miret, Ramiro Moreno and Anna Rio Abstract Given an elliptic curve E and a finite subgroup G, Vélu s formulae concern to a separable isogeny I G : E E with kernel G. In particular, for a point P E these formulae express the first elementary symmetric polynomial on the abscissas of the points in the set P +G as the difference between the abscissa of I G P and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstraß coefficients of E as polynomials in the coefficients of E and two additional parameters: w 0 = t and w 1 = w. We generalize this by defining parameters w n for all n 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P +G. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group. 1. Vélu s formulae An elliptic curve E over a field K is a smooth projective curve over K of genus one with a distinguished K-rational point. If E/K is an elliptic curve then it has a plane model given by an affine Weierstraß equation y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 such that the coefficients a i K and the distinguished point is the point at infinity O = 0 : 1 : 0 EK Mathematics Subject Classification. 11G05. Key words. Elliptic curve, isogeny, rational subgroup. The first and second authors were supported in part by grant TIC The third author was supported in part by grants BFM C02-01 and 2005SGR

2 148 J. M. Miret, R. Moreno, A. Rio As usual see [3], we define the following quantities associated with the above equation b 2 = a a 2, b 4 = a 1 a 3 + 2a 4, b 6 = a a 6, b 8 = b 2 b 6 b 2 4 /4, = 9b 2 b 4 b 6 b 2 2 b 8 8b b2 6, j = b2 2 24b 4 3 /. The equation defines an elliptic curve if and only if is nonzero. And such equations define elliptic curves which are isomorphic over K if and only if they give the same quantity j. An elliptic curve E/K has a natural structure of a commutative algebraic group with the distinguished K-rational point as the identity element. Using the above Weierstraß model, the addition law is as follows. Let P = x 1, y 1, Q = x 2, y 2. If x 1 = x 2 and y 1 + y 2 + a 1 x 2 + a 3 = 0, then P 1 + P 2 = O. Otherwise, let y 2 y 1 if x 1 x 2, x 2 x 1 λ = ν = 3x a 2x 1 + a 4 a 1 y 1 2y 1 + a 1 x 1 + a 3 otherwise y 1 x 2 y 2 x 1 x 2 x 1 if x 1 x 2, x a 4x 1 + 2a 6 a 3 y 1 2y 1 + a 1 x 1 + a 3 otherwise. Then, P + Q = λ 2 + a 1 λ a 2 x 1 x 2, λ + a 1 x 3 ν a 3. An isogeny between elliptic curves E 1 and E 2 over K is a morphism I : E 1 E 2 such that IO E1 = O E2. Its equations are of the form Ix, y = Rx, Sx, y where Rx, Sx, y are rational functions, Rx Kx and Sx, y Kx, y. An isogeny is a group morphism, and if it is non constant, namely I O E2, then it is surjective finite morphism. It induces a field immersion I : KE2 KE 1 between the corresponding function fields. The degree of the isogeny is the degree of the field extension and if this extension is separable then it is Galois with Galois group isomorphic to ker I. Given an elliptic curve E and a finite subgroup G there exists a unique, up to K-isomorphism, elliptic curve E and a separable isogeny I G : E E defined up to AutE such that ker I G = G see [3, III.4.12]. This elliptic curve is denoted by the quotient E/G. If E is defined over K

3 Generalization Vélu s Formulae for Isogenies 149 and G is Gal K/K invariant, then the curve E/G and the isogeny I G are defined over K. For example, if Q EK is a point of order m and G = Q then I : E E/ Q is a degree m isogeny defined over K. If E is an elliptic curve over K, when we look at the local ring K[E] O and take the completion at its maximal ideal, we get a power series ring in one variable, say K[[z]], for some uniformizer z at O. We can take z = x/y and then the expression of the Weierstraß coordinate x as formal Laurent power series in z is xz = 1 z 2 α 1 1 z α 2 α 3 z α 4 z 2 α 5 z 3 α 6 z 4, where α 1 = a 1, α 2 = a 2, α 3 = a 3, α 4 = a 1 a 3 +a 4, α 5 = a 2 a 3 +a 2 1 a 3+a 1 a 4 and α 6 = a 2 1a 4 + +a 6. We have y = x/z = z 3 +α 1 z 2 +α 2 z 1 + α 3 + α 4 z +, and the function field KE is isomorphic to Kx, y. For a finite subgroup G of E, Vélu defines XP = xp + xp + Q xq = x + Q G {O} T PG {O} tt x xt + ut x xt 2 where PG is a system of representatives of the orbits of G under the action of {±1}, ut = 4xT 3 +b 2 xt 2 +2b 4 xt+b 6, tt = 6xT 2 + b 2 xt + b 4 if 2T O and tt = 3xT 2 + 2a 2 xt + a 4 a 1 yt if 2T =O. A function Y P is defined analogously: Y P = yp + Q G {O} yp + Q yq. The functions X, Y KE are invariant under the action of G by translation, the function field KE/G is isomorphic to KX, Y and the isogeny I G is identified with the transformation x, y X, Y see [4, 2]. From the second equality for XP we get on one hand the expression of X as rational function of x, which gives the equation of the isogeny I G : x, y X = Rx, Y

4 150 J. M. Miret, R. Moreno, A. Rio and on the other hand the expression of X as formal Laurent power series in z Xz = 1 z 2 α 1 z α 2 α 3 z α 4 w 0 z 2 α 5 w 0 α 1 z 3 α 6 w 0 α α 2 w 1 z 4 α 7 w 0 α α 1 α 2 + α 3 2w 1 α 1 z 5, which determines Weierstrass coefficients for E/G a 1 = a 1, a 2 = a 2, a 3 = a 3, a 4 = a 4 5w 0, a 6 = a 6 b 2 w 0 7w 1, where w 0 = T PG tt and w 1 = T PG ut + xttt. Looking over all these formulae, we see that in order to make explicit computations concerning the isogeny, we have to operate with the abscissas of the points in the kernel G. As we mentioned above, if this group is Gal K/K invariant, then the curve E/G and the isogeny I G are defined over K. But the computations to obtain equations, for the curve or the isogeny, may involve non K-rational abscissas. The Galois invariance of the subgroup G is equivalent to the fact that the polynomial ψ G x = x xq Q G {O} has coefficients in K see [2], namely that the elementary symmetric polynomials in the abscissas of G {O} lie in K. The efficiency of computations would then be improved if we can perform them in terms of these symmetric polynomials. The above equality XP = Q G can be rewritten in the form 2. Generalization xp + Q S 1 = X + S 1, Q G {O} xq showing that the first elementary symmetric polynomial in the abscissas of P + G is expressed as the sum of the abscissa of I G P and the first elementary symmetrical polynomial in the abscissas of the points in G {O}.

5 Generalization Vélu s Formulae for Isogenies 151 Our goal is to obtain analogous formulas for S r, the r-th elementary symmetric polynomial in the abscissas of P + G and for S r, the r-th power sum of these abscissas, for all r such that 1 r deg I G = G Symmetric polynomials. The first goal is achieved if we are able to determine the coefficients of the polynomial x xp + Q = x G X + S 1 x G 1 + Q G attached to the preimage of the point I G P E. But this polynomial is obtained from the equality tt X = Rx = xp + x xt + ut x xt 2, T PG {O} and becomes of the form R 1 x + x Xψ G x when we express the rational function Rx as x + R 1 x/ψ G x. Theorem 1. Let E be an elliptic curve and G a nontrivial finite subgroup of E. If P is a point of E, then x xp +Q=x G X + S 1 x G 1 Q G G + r=2 r 2 1 S r r 1 X+S r + 1 i w i S r i 2 x G r where X is the abscissa of the isogenous point I G P, S j is the j-th elementary symmetric polynomial in the abscissas of the points in G {O} with S G = 0 and w i = ttxt i + utixt i 1, T PG {O} with PG a system of representatives of the orbits of G under the action of {±1} and 6xT 2 + b 2 xt + b 4, if T G \ E[2], tt = 6xT 2 + b 2 xt + b 4, if T G E[2], 2 ut = 4xT 3 + b 2 xt 2 + 2b 4 xt + b 6, for all T G {O}, where E[2] denotes the 2-torsion subgroup of E.

6 152 J. M. Miret, R. Moreno, A. Rio Therefore, if 1 r G, for the r-th elementary symmetric polynomial in the abscissas of the points in P + G we have r 2 S r = S r 1 X + S r + 1 i w i S r i 2. Proof: As we said above, we begin with the expression of the abscissa of I G P as rational function of the abscissa of P: X = x + T PG {O} tt x xt + ut x xt 2 = Rx = x + R 1x ψ G x. Note that the formulae for tt given in the theorem agree with the definition previously given. Taking into account that xq = x Q for all Q G \ E[2] and ut = 0 for all T E[2], we have x xt x xt 2 = x xq T PG E[2]\{O} T PG\E[2] and this is the polynomial ψ G x = As for the numerator, R 1 x = = T PG {O} T PG {O} + T PG {O} G 1 r=0 Q G {O} 1 r S r x G r 1. tt ψ Gx x xt + ut ψ G x x xt 2 G 2 tt r=0 G 3 ut 1 r S r T, tx G r 2 r=0 1 r S r T, ux G r 3 where we have denoted S r T, t the r-th elementary symmetric polynomial in the abscissas of the points in G {O, T } and S r T, u the r-th elementary symmetric polynomial in the abscissas of the points in G {O, ± T }. These polynomials can be expressed in terms of the S r using the following recurrences S r T, t = S r xts r 1 T, t S r T, u = S r 2xTS r 1 T, u xt 2 S r 2 T, u,

7 Generalization Vélu s Formulae for Isogenies 153 with S 0 T, = 1 and S 1 t, = 0. We have S r T, t = S r T, u = r 1 i S r i xt i, r 1 i i + 1S r i xt i and R 1 x = = = G 2 1 r r=0 T G 3 + r=0 G 2 1 r T 1 r r=0 T G 2 tts r T, t uts r T, u r 1 r 1 i S r i r=0 x G r 2 x G r 3 tts r T, t uts r 1 T, u x G r 2 ttxt i +utixt x i 1 G r 2. T On the other hand, for 0 r G the coefficient of x G r in x Xψ G x is 1 r S r +XS r 1. Now we have computed the coefficients of the polynomial R 1 x + x Xψ G x and the result follows. We call the w i = T ttxt i + utixt i 1

8 154 J. M. Miret, R. Moreno, A. Rio generalized Vélu parameters. They appear when we consider further terms of X as formal Laurent power series in z: XP = 1 z 2 α 1 z α 2 α 3 z α 4 w 0 z 2 α 5 w 0 α 1 z 3 α 6 w 0 α α 2 w 1 z 4 α 7 w 0 α α 1 α 2 + α 3 2w 1 α 1 z 5 α 8 w 0 α α2 1 α α 1α 3 + α 4 w 1 3α α 2 w 2 z 6 α 9 w 0 α α3 1 α 2+ 3α 2 1 α 3+ 2α 2 α 3 + 3α 1 α α 1α 4 + α 5 2w 1 2α α 1 α 2 + α 3 3α 1 w 2 z 7 More precisely, the power series XP xp = A j z j has coefficients A j = n 1< <n s h 1n 1+ +h sn s+2i=j 2 j 2 h h s + i! h 1!...h s! i! α h1 n 1... α hs n s w i. Remark 1. The theorem gives the first equation of the isogeny in terms of the elementary symmetric polynomials in the abscissas of the points in G {O} and the generalized Vélu parameters. X = x + R 1x ψ G x x G S 1 x G 1 G + 1 r S r + r 2 1 i w i S r i 2 x G r r=2 =. ψ G x For the efficiency of computations we would like to obtain these parameters w i without computing the abscissas of the points in G. We seek an expression for them which is directly computable from the symmetric polynomials S j. Plugging the formulae for tt and ut in the formula giving the w i we get an alternative expression w i = 2i + 3S i+2 + i + 1b 2 2 S i+1 + 2i + 1b 4 S i + ib Si 1 where S j indicates the j-th power sum of the abscissas of the points in G {O}. Now, the Newton formulae allow us to obtain these power sums, and the w i, from the symmetric polynomials S j.

9 Generalization Vélu s Formulae for Isogenies Power sums. The formula S 1 = X + S 1 admits yet another generalization, since S 1 is also the first power sum in the abscissas of the points in P + G. We denote S r the r-th power sum of these abscissas, for all r such that 1 r G, and S r the r-th power sum of the abscissas of the points in G {O}. Proposition 1. Let E/K be an elliptic curve and G a nontrivial finite subgroup of E. If x, X denote Weierstraß coordinates of points P, I G P, respectively, and z is a uniformizer at O, then for all r such that 2 r G there exist elements β 0,r, β 1,r,..., β r 2,r K such that r f r z = xz r Xz r β i 2,r Xz r i belongs to the maximal ideal of K[[z]]. Namely, as an element of the function field of E, the function f r vanishes at the infinite point O. Proof: We denote v the valuation of Kz. Since vx = 2, all nonzero rational functions in x have valuations in the ideal 2Z. Therefore, all nonzero polynomial expressions in x and X have valuation in this ideal. Since xz r = 1 z 2r rα 1 z 2r 1 rα 2 r 2 α 2 1 i=2 z 2r 2 rα 3 2 r 2 α1 α 2 + r 3 α 3 1 z 2r 3 rα 4 r 2 α α 1 α r 3 α 2 1 α 2 r 4 α Oz 2r 5, z 2r 4 Xz r = 1 z 2r rα 1 z 2r 1 rα 2 r 2 α 2 1 z 2r 2 rα 3 2 r 2 α1 α 2 + r 3 α 3 1 z 2r 3 rα 4 w 0 r 2 α α 1 α r 3 α 2 1 α 2 r 4 α Oz 2r 5, z 2r 4 we have that vx r X r 2r 4 with equality if w 0 0. Since X r 2 has valuation 2r 4, there exists β 0,r K concretely, β 0,r = rw 0 such that vx r X r β 0,r X r 2 2r 5. If x r X r β 0,r X r 2 = 0, we take all the remaining β s equal to zero and we have

10 156 J. M. Miret, R. Moreno, A. Rio finished. If not, vx r X r β 0,r X r 2 is even and 2r 6. We continue with X r 3 and iterate the process until we get vf r z > 0. For r fixed, the computation of β 0,r, β 1,r,...,β r 2,r amounts to solving an homogenous linear system. In this way, we obtain β 0,r = rw 0, β 1,r = rw 1, β 2,r = r w 2 + r 3 w0 2 2!, β 3,r = r w 3 + r 4w 0 w 1, β 4,r = r w 4 + r 5 w1 2 r 4r 5 + r 5w 0 w 2 w0 3, 2! 3! r 5r 6 β 5,r = r w 5 + r 6w 0 w 3 + r 6w 1 w 2 w 2 2! 0w 1. Proposition 2. With the hypothesis and notations of the previous proposition, for all r such that 2 r G, let S r be the r-th power sum of the abscissas of the points in G {O} and S r the r-th power sum of the abscissas of the points in P + G. Then, r S r S r = X r + β i 2,r X r i. Proof: Let us denote ϕ 1 = xp r + Q G {O} i=2 ϕ 2 = XP r + β i 2,r XP r i. xp + Q r xq r = S r S r Both functions belong to the function field of the isogenous curve E = E/G. In KE lies also its difference, which can be written in the following way: ϕ 1 ϕ 2 = f r + xp + Q r xq r. Q G {O} It is a function without poles, therefore constant, which vanishes at the infinite point. We conclude that ϕ 1 ϕ 2 = 0 as we claimed.

11 Generalization Vélu s Formulae for Isogenies Newton formulae. At this point we have obtained formulae for symmetric polynomials involving the w i and formulae for the power sums involving the β k,r. The classical link between them are the Newton formulae for r G S r = 1 r 1 1 r+i 1 S r i S i, r which will provide some kind of link between w s and β s, and hopefully an efficient way to compute the β s. Direct computations that we have already shown give the initial values β 0,r = rw 0 and β 1,r = rw 1. This is all we need if G is 2 or 3. For G 4 we would have to compute β k,r for 4 r G and 2 k r 2. Proposition 3. With the hypothesis and notations of the previous propositions, assume that G 4. For 4 r G and 2 k < r 2 we have k 2 β k,r = β k,r 1 w k w i β k i 2,r i 2. For the last one, we have r 4 β r 2,r = rw r 2 w j β r j 4,r j 2. Example 1. If G =5, we have β 0,2 = 2w 0, β 0,3 = 3w 0, β 1,3 = 3w 1, β 0,4 = 4w 0 and β 1,4 = 4w 1. Now, using the second formula, β 2,4 = 4w 2 w 0 β 0,2 = 4w 2 + 2w 2 0. We also know β 0,5 = 5w 0, β 1,5 = 5w 1, and from the first formula we compute β 2,5 = β 2,4 w 2 w 0 β 0,3 = 5w 2 + 5w 2 0. Finally, β 3,5 = 5w 3 w 0 β 1,3 + w 1 β 0,2 = 5w 3 + 5w 0 w 1 and we have the whole family of β s. Proof of Proposition 3: We plug the above formulae for symmetric polynomials and power sums into the Newton formula to obtain an equality F r X = 0 which is polynomial in X. Since X is transcendental over K, the coefficients must be zero. The degree of F r is a priori r, but we see that the coefficient of X r is S 0 S 0 and the coefficient of X r 1 is S 1 + S 1. On the other hand, the coefficients of X r 2 and X r 3 are β 0,r β 0,r 1 + w 0 = 0 and β 1,r β 1,r 1 + w 1 = 0, respectively.

12 158 J. M. Miret, R. Moreno, A. Rio We have then F r X = r 2 where k=2 F k,r X r 2 k, with k 2 F k,r = 1 j S j γ k j,r j, for 2 k r 4, k 2 r 1 F r 3,r = 1 j S j γ k j,r j 1 j S r j 1 S j, F r 2,r = r 1 r j S r j δ j + j=4 r 1 j S r j S j r 2 j + w r j 2 1 i S j i S i, k 2 γ k,r = β k,r β k,r 1 + w k + w i β k i 2,r i 2, for 2 k r 1, j 4 δ j = β j 2,j + jw j 2 + w i β j i 4,j i 2, for 4 j r. Using the Newton formulae, now considering that the variables are the abscissas of the points in G {O}, we see that all the summations containing power sums vanish and k 2 F k,r = k 1 j S j γ k j,r j, for 2 k r 3, F r 2,r = r 1 r j S r j δ j. j=4 Let us begin with F r 2,r = 0. If r = 4, it gives that δ 4 = 0 namely, that β 2,4 = 4w 2 w 0 β 0,2. Let us assume that r > 4 and δ j = 0 for all j between 4 and r 1. Then, r 1 0 = F r 2,r = 1 r j S r j δ j + δ r j=4

13 Generalization Vélu s Formulae for Isogenies 159 and we get δ r = 0, that is, r 4 β r 2,r + rw r 2 + w j β r j 4,r j 2 = 0. Now we proceed with F k,r = 0 and use induction on k. If k = 2, it gives 0 = F 2,r = γ 2,r = β 2,r β 2,r 1 + w 2 + w 0 β 0,r 2 for all r 5. Let us assume k > 2 and γ h,s = 0 for all h between 2 and k 1 and all s h + 3. If r 3 k, then k 2 0 = F k,r = γ k,r + 1 j S j γ k j,r j j=1 and the induction hypothesis gives γ k,r = 0, namely and we are done. k 2 β k,r = β k,r 1 w k w i β k i 2,r i 2 The recurrences given in the above proposition show that β k,r Q[w 0, w 1,..., w k ] and provide an efficient way to compute them. For the sake of completeness we give also their explicit expression as polynomials in the generalized Vélu parameters: k 2 β k,r = r 1 j+1 r k + j 2 B j k, j where B j k = n 1< <n s h 1+ +h s=j+1 h 1n 1+ +h sn s=k 2j 3. Example j! h 1!... h s! wh1 n 1... w hs n s. We take the example from Vélu s paper, namely the elliptic curve E/Q defined by the equation E/Q : y 2 + xy + y = x 3 x 2 3x + 3, and the subgroup G of order 7 generated by the point Q = 1, 0. Its elements are the infinite point O and the points Q = 1, 0, 2Q = 1, 2, 3Q = 3, 6, 4Q = 3, 2, 5Q = 1, 2, 6Q = 1, 2.

14 160 J. M. Miret, R. Moreno, A. Rio First of all we compute the symmetric polynomials in the abscissas of these points and the generalized Vélu parameters. Since in this case all the points in G are rational, these parameters can be easily computed from the defining formula. i S i w i Using the formula of Theorem 1, we compute the symmetric polynomials in the abscissas of the points in P + G: i S i 1 X X X X X X X 178 where X is the abscissa of the isogenous point I G P. In order to compute the power sums, we need the elements β k,r, that we obtain from the recurrences of the last proposition: r\k

15 Generalization Vélu s Formulae for Isogenies 161 Finally, using the formula of Proposition 2 we can get the power sums of the abscissas of the points in P + G: i S i 1 X X X 3 126X X 4 168X 2 792X X 5 210X 3 990X X X 6 252X X X X X 7 294X X X X X The elliptic curve of this example is the curve 26b1 in Cremona s tables see [1]. The isogenous curve, given by Vélu, is 1 E /Q : y 2 + xy + y = x 3 x 2 213x Its 7-division polynomial has irreducible factor x x 2 94x The polynomial ψ G x = x x 2 94x = x x x x x x has roots the abscissas of the nontrivial points of an order 7 subgroup G of E. Its coefficients provide the new S i, the elementary symmetric polynomials in these six abscissas. Now that we deal with non rational points, to compute the generalized Vélu s parameters we make use of the formula in terms of power sums.

16 162 J. M. Miret, R. Moreno, A. Rio In the following table we provide the values of S i, S i and w i which we need to compute the equation of the isogeny I G. i S i S i w i / / / / / / / / / / / / / /7 The isogenous curve, obtained from w 0 and w 1, is E /Q : y 2 + xy + y = x 3 x x and the abscissa of I G x, y, computed with the formula in the remark above, is 49x x x x x x x x x x x x x The isomorphism σ = u, r, s, t = 7, 12, 3, 177 see [3] transforms E into the original elliptic curve E and σ I G is the dual isogeny ÎG. Acknowledgement. We are grateful to the referee for her/his helpful comments to the manuscript. References [1] J. E. Cremona, Elliptic Curve Data, Available at [2] J. González, On the division polynomials of elliptic curves, Contributions to the algorithmic study of problems of arithmetic moduli, Spanish, Rev. R. Acad. Cienc. Exactas Fís. Nat. Esp , [3] J. H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics 106, Springer-Verlag, New York, [4] J. Vélu, Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris Sér. A-B , A238 A241.

17 Generalization Vélu s Formulae for Isogenies 163 Josep M. Miret and Ramiro Moreno: Departament de Matemàtica Universitat de Lleida Lleida Spain address: miret@matematica.udl.es address: ramiro@matematica.udl.es Anna Rio: Departament de Matemàtica Aplicada II Universitat Politècnica de Catalunya Barcelona Spain address: ana.rio@upc.edu