On unramified abelian extensions of number fields arising from multiplication of elliptic curves

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(5) . . . . . Abstract We introduce a way to construct number fields with class numbers divisible by a power of a prime number. In particular, we start with an elliptic curve E defined over a number field k such that E[ln ] ⊂ E(k), where l is an odd prime number, and show a way to construct infinitely many quadratic extensions of k with class numbers divisible by l2n .. Keywords: Divisibility of Class Numbers, Isogeny of Elliptic Curves. 1. Introduction. (i) L/K is an abelian extension of exponent m.. Let k be an algebraic number field of finite degree, m ≥ 2 an integer, and let E be an elliptic curve defined over k such that E[m] ⊂ E(k). Let P be a point on E, and let K = k(P ) (resp. L = k([m]−1 P )) be the field generated over k by the coordinates of P (resp. the points in [m]−1 P ). Here, [m] denotes the multiplication-by-m map on E, and E[m] its kernel (that is, the m-torsion subgroup of E). Then, as is well known (see, e.g., [4, Chapter VIII]), we have: ∗. (ii) L/K is unramified at a finite place p if E has good reduction at p and if p m. We also note that L/K is a Kummer extension, since the assumption E[m] ⊂ E(k) implies μm ⊂ k, where μm denotes the group of m-th roots of unity. Furthermore, we have [L : K] = m2 for “generic” P . In the present paper, we show the following theorem: Theorem 1.1 Let k be an algebraic number. Associate Professor at Faculty of Liberal Arts, Tohoku Gakuin University. . .

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(8) . . field of finite degree, ln a power of an add. 2. prime number, and let E be an elliptic curve. Let k be a field of characteristic 0,. defined over k such that E[ln ] ⊂ E(k). Let y 2 = x3 + ax + b. (a, b ∈ Ok , 4a3 + 27b2 = 0). be a Weierstrass equation for E with x(T ), y(T ) ∈ Ok. (T ∈ E[l] − {O}).. Let P be a point on E with x(P ) ∈ k, and let K = k(P ), L =. k([ln ]−1 P ).. Moreover, as-. Dual isogeny via V´ elu’s formulas. E : y 2 = x3 +ax+b. an elliptic curve defined over k, and let Γ be a finite subgroup of E of order l ≥ 2 which is ¯ stable under the action of Gal(k/k). Here, k¯ is an algebraic closure of k, and Gal stands for Galois group. Let. sume that c = x(P ) satisfies the following condition: For any finite place p of k such that E. (2.1). has bad reduction at p or p | l, min ordp (c3 + ac + b), ordp (3c2 + a) ≤ 0. Here, we replace the inequality above by ordp (c2 + a) ≤ 0 if p | 2, and by. (a, b ∈ k, 4a3 +27b2 = 0). E ∗ : Y 2 = X 3 + AX + B, λ : X = ξ(x), Y = η(x) y. be the equations for λ : E → E ∗ = E/Γ that are given by Vélu’s formulas [5] (for the formulas, see also [3, Section 2] or [6, Section 12.3]). Then Γ∗ = λ(E[l]) is a subgroup of E ∗ of order ¯ l which is stable under the action of Gal(k/k). Let. ordp (l2n c) ≤ 0. (2.2). if p | l. Then L/K is unramified at all finite. E : (y )2 = (x )3 + a x + b , λ∗ : x = ξ ∗ (X), y = η ∗ (X) Y. places. Here, Ok denotes the ring of integers. be the equations for λ∗ : E ∗ → E = E ∗ /Γ∗. of k, while ordp the normalized additive valu-. that are given by Vélu’s formulas. Note that. ation of k associated with p.. E is naturally identified with E/E[l]. We can. The field K = k. √. c3 + ac + b in the the-. orem above is a quadratic extension of k for generic c. Thus, using the theorem and varying c in k, we can construct infinitely many quadratic extensions of k with class numbers divisible by l2n , for L/K is also unramified at all infinite places. We will give a proof of the theorem in Sections 3 and 4, after studying. also apply Vélu’s formulas to E[l], and then obtain the same equations as for λ∗ ◦ λ : E → E. On the other hand, E/E[l] is isomorphic to E via the multiplication-by-l map. Thus there exists an isomorphism φ : E → E such that λ∗ ◦ λ = φ ◦ [l]. In fact: Proposition 2.1 We have. about the explicit form of dual isogenies in Section 2.. . a = l4 a,. b = l6 b.. .

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(12) . Moreover, the isomorphism φ : E → E is. Example 2.3 (i) If E is given by y 2 = x3 +ax. given by. (a = 0), then E[2] consists of √ O, (0, 0), (± −a, 0),. x = l2 x,. y = l3 y.. Proof Since E is isomorphic to E, we have x = u2 x ◦ [l],. x ,. y,. x ◦ [l], y ◦ [l] as (E[l]-invariant) functions on E. It follows from Vélu’s formulas that 2. x. In this case, λ : E → E ∗ and. order 2.. λ∗ : E ∗ → E are given by. y = u3 y ◦ [l]. for some u ∈ k − {0}. Here we regard. and hence Γ = {O, (0, 0)} is a subgroup of. xl + lower degree terms , = l2 −1 x + lower degree terms. E ∗ : Y 2 = X 3 − 4aX, x2 + a x2 − a y, , Y = x x2 thus Γ∗ = {O, (0, 0)}, and by λ: X=. E : (y )2 = (x )3 + 16ax , X 2 − 4a X 2 + 4a Y, , y = X X2 ˆ : E ∗ → E is given by respectively. Hence λ λ ∗ : x =. xm + lower degree terms y = m y, x + lower degree terms where m=. 3 2 2 (l 3 2 2l. x=. − 1) if l odd,. X 2 + 4a Y. 8X 2. then E[3] consists of √ √ √ O, (0, ± b ), (− 3 4b, ± −3b ), √ √ √ √ (−ω 3 4b, ± −3b ), (−ω 2 3 4b, ± −3b ), √ where ω = (−1 + −3 )/2, and hence Γ = √ {O, (0, ± b )} is a subgroup of order 3. In. 2. xl + lower degree terms , l2 xl2 −1 + lower degree terms xm + lower degree terms y + lower degree terms. y ◦ [l] =. y=. (ii) If E is given by y 2 = x3 + b (b = 0),. if l even.. We also have x ◦ [l] =. X 2 − 4a , 4X. l 3 xm. by the formulas on division polynomials (see,. this case, λ : E → E ∗ and λ∗ : E ∗ → E are. e.g., [6, Section 3.2]), and hence conclude u =. given by E ∗ : Y 2 = X 3 − 27b,. l. Namely we have x = l2 x ◦ [l],. y = l3 y ◦ [l],. which immediately imply the assertions of the . proposition.. ˆ : E∗ → E Corollary 2.2 The dual isogeny λ is given by x=. 1 ∗ ξ (X), l2. y=. 1 ∗ η (X) Y. l3. . .

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(14) . . x3 + 4b x3 − 8b , Y = y, 2 x x3 √ thus Γ∗ = {O, (0, ±3 −3b )}, and by λ: X=. E : (y )2 = (x )3 + 729b, X 3 − 108b X 3 + 216b , y = Y, 2 X X3 ˆ : E ∗ → E is given by respectively. Hence λ λ∗ : x =. x=. X 3 − 108b , 9X 2. y=. X 3 + 216b Y. 27X 3. . .

(15) . A chain of isogenies. Proof We first note that the equations (2.1). Let the notation and the assumptions be. for λ : E → E ∗ are derived from X =x+ x ◦ τT − x(T ) ,. the same as in the previous section. We define. T ∈Γ−{O}. isogenies λi : Ei → Ei∗ ,. E. Hence, regarding k(E) ⊃ k(E ∗ ) ⊂. Ei∗ : Yi2 = Xi3 + l4i A Xi + l6i B,. k(Ei ) ⊃ k(Ei∗ ). and by. λ∗i : xi+1. y ◦ τT − y(T ) .. Here τT denotes the translation-by-T map on. Ei : yi2 = x3i + l4i a xi + l6i b,. λi : Xi = l2i ξ. . T ∈Γ−{O}. λ∗i : Ei∗ → Ei+1. (i = 0, 1, 2, . . .) by (3.1). Y =y+. x i l2i. , Yi = η. x i l2i. by the commutative diagram yi ,. λ E −−−−→ ⏐ ⏐ φi λi Ei −−−− →. X X i i = l2i ξ ∗ 2i , yi+1 = η ∗ 2i Yi . l l. We note that Ei+1 is nothing but Ei with the. E∗ ⏐ ⏐ ∗ φ i Ei∗. (see Proposition 3.1), we have. notation in the previous section. We also de-. Xi = l2i X = l2i x +. fine isomorphisms φ i : E → Ei ,. ⊂. 3. . φ∗i : E ∗ → Ei∗. . x ◦ τT − x(T ). . T ∈Γ−{O}. by. = xi +. φi : xi = l2i x, yi = l3i y, φ∗i. : Xi =. l2i X,. Yi =. . xi ◦ τφi T − xi (φi T ) ,. T ∈Γ−{O}. l3i Y.. that is. Then:. Xi = xi +. . xi ◦ τTi − xi (Ti ) .. Ti ∈φi (Γ)−{O}. Proposition 3.1 We have λi ◦ φi = φ∗i ◦ λ,. We also have. ˆ λ∗i ◦ φ∗i = φi+1 ◦ λ.. Yi = yi +. . yi ◦ τTi − yi (Ti ). . Ti ∈φi (Γ)−{O}. Proof Immediate from Proposition 2.1 and. in the same manner. Thus we have shown the. . assertion for λi : Ei → Ei∗ . Since the equa-. Corollary 2.2.. tions (2.2) for λ∗ : E ∗ → E are derived from X ◦ τU − X(U ) , x = X +. Corollary 3.2 The equations for λi : Ei → Ei∗ and λ∗i : Ei∗ → Ei+1 described above are. U ∈Γ∗ −{O}. the ones that are given by Vélu’s formulas ap-. y. plied to φi (Γ) and φ∗i (Γ∗ ), respectively.. =Y +. . Y ◦ τU − Y (U ) ,. U ∈Γ∗ −{O}. 4. . .

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(19) . we can show the assertion for λ∗i : Ei∗ → Ei+1 . in a similar fashion.. 4. and by Kj∗ = k((λ∗n−1 ◦λn−1 ◦· · ·◦λn−j+1 ◦λ∗n−j )−1 Pn ),. Field extensions. respectively. Then we have. Let k be an algebraic number field of finite. Kn ⊃ Kn∗ ⊃ · · · ⊃ K2 ⊃ K2∗ ⊃ K1 ⊃ K1∗ ⊃ K0. degree, l an odd prime number, n a positive integer, and let E : y 2 = x3 +ax+b (a, b ∈ Ok , 4a3 +27b2 = 0) be an elliptic curve defined over k such that E[ln ] ⊂ E(k). We assume that the Weierstrass equation above is taken so that the condition x(T ), y(T ) ∈ Ok. (T ∈ E[l] − {O}). is satisfied. Taking a subgroup Γ of E of order l, we define λ : E → E ∗ , λ∗ : E ∗ → E , etc. in the same manner as in Sections 2 and 3. Then A, B ∈ Ok and X(U ), Y (U ) ∈ Ok. (U ∈ Γ∗ − {O}).. K, Kn = L. We also have: Proposition 4.1 Let i be an integer such that 1 ≤ i ≤ n − 1, and let j = n − i. Then: (i) For any point Pi on Ei with (λ∗n−1 ◦ λn−1 ◦ · · · ◦ λ∗i ◦ λi )Pi = Pn , we have Kj = k(Pi ) and an injective homomorphism Gal(Kj /Kj∗ ) σ

(20) −→ Piσ − Pi ∈ φi (Γ). (ii) For any point Pi∗ on Ei∗ with (λ∗n−1 ◦ λn−1 ◦ · · · ◦ λi+1 ◦ λ∗i )Pi∗ = Pn ,. Consequently, we have (4.1) xi (Ti ), yi (Ti ) ∈ Ok. and Kj = k([lj ]−1 P ). In particular, K0 =. (Ti ∈ φi (Γ)−{O}). and (4.2) Xi (Ui ), Yi (Ui ) ∈ Ok (Ui ∈. φ∗i (Γ∗ )−{O}).. Let P be a point on E with x(P ) ∈ k, and let K = k(P ), L = k([ln ]−1 P ). It follows from the assumption E[ln ] ⊂ E(k) that L/K is an abelian extension of exponent ln . Putting Pn = φn P , which is a point on En , we define intermediate fields Kj (0 ≤ j ≤ n) and Kj∗ (1 ≤ j ≤ n) of L/K by Kj = k((λ∗n−1 ◦ λn−1 ◦ · · · ◦ λ∗n−j ◦ λn−j )−1 Pn ). . .

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(22) . . we have Kj∗ = k(Pi∗ ) and an injective homomorphism Gal(Kj∗ /Kj−1 ) σ

(23) −→ (Pi∗ )σ − Pi∗ ∈ φ∗i (Γ∗ ). Proof Immediate from Ker(λ∗n−1 ◦ λn−1 ◦ · · · ◦ λ∗i ◦ λi ) = (λ∗i−1 ◦ λi−1 ◦ · · · ◦ λ∗0 ◦ λ0 )E0 [ln ], which is a subgroup of Ei (k), and Ker(λ∗n−1 ◦ λn−1 ◦ · · · ◦ λi+1 ◦ λ∗i ) = (λi ◦ λ∗i−1 ◦ · · · ◦ λ∗0 ◦ λ0 )E0 [ln ], which is a subgroup of Ei∗ (k).. . . .

(24) . . Now, we assume that c = x(P ) satisfies the condition. i ≤ n − 1) and Pi∗ ∈ E∗i (L; P) (0 ≤ i ≤ n − 1) such that. (4.3) min ordp (c3 +ac+b), ordp (3c2 +a) ≤ 0. λ. λ∗. λ∗n−3. λn−2. 0 0 ∗ P0

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(31) −→ Pn .. for any finite place p of k such that E has bad reduction at p or p | l. Here, we replace the. These points might depend on the place P.. inequality above by. However, the fields. (4.3). ∗ (0 ≤ i ≤ n − 1) k(Pi ) = Kn−i , k(Pi∗ ) = Kn−i. ordp (c2 + a) ≤ 0. do not depend on the choice of such points. if p | 2, and by. (see Proposition 4.1). Hence we can show that (4.3). . 2n. ordp (l c) ≤ 0. if p | l. Then we have: Proposition 4.2 The extensions Kj /Kj∗ and Kj∗ /Kj−1 are unramified at all finite places.. Kj /Kj∗ and Kj∗ /Kj−1 are unramified at P in a similar fashion to the argument in [3, Section 5], in which we use Proposition 4.1 again (note that an extension of degree 1 is unrami. fied).. Proposition 4.2 immediately implies that Proof We fix a finite place P of L such that. L/K is unramified at all finite places, which is. E has bad reduction at P or P | l, for L/K is. the assertion of Theorem 1.1.. unramified at other finite places, and put. 5. Some examples.

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(33) i )ns (κ)}, Ei (L; P) = {Q ∈ Ei (L) ; Q

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(35) ∗ )ns (κ)}. E∗i (L; P) = {Q∗ ∈ Ei∗ (L) ; Q i Here,

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(41) −→ Q i (κ denotes the residue field of P) are the reduc-. We close the present paper with giving some examples of biquadratic number fields, √ which contain −3, with class numbers divis√ ible by 9. Let k = Q( −3 ), l = 3, n = 1, and let E : y 2 = x3 + 16. Then, by using Magma [1], we have. tion modulo P maps with respect to the equations (3.1), and the symbol “ns” means non-. x ◦ [3] =. singular points. Then the assumption (4.3) (or. rank E(Q) = rank Ek/Q (Q) = 0,. (4.3) or (4.3) ) implies Pn ∈ En (L; P). Therefore, by Corollary 3.2, (4.1), (4.2) and by [3, Theorem 4.5], there exist Pi ∈ Ei (L; P) (0 ≤. . x9 − 1536x6 + 12288x3 + 262144 , 9x2 (x3 + 64)2. where Ek/Q : −3y 2 = x3 + 16. .

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(45) . is the quadratic twist of E with respect to k/Q,. (a) ord3 (9c) ≤ 0.. and E(k)tors = E[3] consists of. (b) fc (x) is irreducible over Q. √ √ Then the class number of Q( −3 , c3 + 16 ). √ O, (0, ±4), (−4, ±4 −3 ), √ √ √ √ (2 + 2 −3 , ±4 −3 ), (2 − 2 −3 , ±4 −3 ).. is divisible by 9. Example 5.2 Using PARI/GP [2], we have. Therefore E(k) = E[3]. It is not hard to ob-. the following table. Here hK denotes the class √ √ number of K = Q( −3 , c3 + 16 ). We have. serve that E is isomorphic to (y )2 + y = (x )3 , which has discriminant −27.. 9 | hK , except for c = ±3/9, ±6/9, ±9/9 or c = −4/9. The values c = ±3/9, ±6/9 and. Thus E has. c = ±9/9 do not satisfy the condition (a),. good reduction at every finite place except √ p = ( −3 ). In this case, the condition on. while the values c = 9/9 and c = −4/9 do not satisfy the condition (b).. c = x(P ) in Theorem 1.1 becomes ordp (9c) ≤ 0.. c. hK. Moreover L = k([3]−1 P ) coincides with the. 1/9. 540. −1/9. 1296. 2/9. 108. −2/9. 432. 3/9. 96. −3/9. 21. 4/9. 54. −4/9. 3. 5/9. 189. −5/9. 432. splitting field of fc (x) = x9 − 1536x6 + 12288x3 + 262144 − 9cx2 (x3 + 64)2 √. c. hK. over K = k(P ) = k( c3 + 16 ), since y(P ) =. 6/9. 24. −6/9. 6. 0. Consequently, if ordp (9c) ≤ 0 and if fc (x). 7/9. 270. −7/9. 315. 8/9. 9. −8/9. 36. 9/9. 1. −9/9. 2. 10/9. 72. −10/9. 144. is irreducible over K, then L/K is an unramified abelian extension of degree 9, and hence the class number of K is divisible by 9. We note that these conditions imply [K : k] = 2, for the class number of k is 1.. References. In what follows, we shall consider the case √ √ where c ∈ Q. Then K = Q( −3 , c3 + 16 ). [1] W. Bosma, J. Cannon, and C. Play-. is a biquadratic field (that is, Gal(K/Q) is iso-. oust, The Magma algebra system I: The. morphic to the Klein 4-group) unless c = 0 or. user language, J. Symbolic Comput. 24. c = −4. Hence, if fc (x) is irreducible over Q,. (1997), 235–265.. so is over K. Thus Theorem 1.1 implies:. [2] The PARI Group, PARI/GP version. Corollary 5.1 Let c be a rational number sat-. 2.5.2, Univ. Bordeaux, 2012, http://. isfying the following two conditions:. pari.math.u-bordeaux.fr/. 7. . .

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(48) . . [3] A. Sato, On the class numbers of certain. [5] J. Vélu, Isogénies entre courbes ellip-. number fields obtained from points on el-. tiques, C. R. Acad. Sc. Paris 273 (1971),. liptic curves II, Osaka J. Math. 45 (2008),. 238–241.. 375–390.. [6] L.C. Washington, Elliptic Curves: Number Theory and Cryptography, 2nd ed.,. [4] J.H. Silverman, The Arithmetic of Ellip-. Discrete Mathematics and Its Applica-. tic Curves, Graduate Texts in Math. 106,. tions, Chapman & Hall/CRC, Boca Ra-. Springer-Verlag, New York, 1986.. ton, FL, 2008.. . .

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