ON SUBFIELDS OF THE MODULAR FUNCTION FIELDS K. SRILAKSHMI Abstract. Let XΓ0 (N ) be the modular curve which is the compact Riemann surface Γ0 (N ) \ H∗ . For primes p, we determine the essential elliptic subfields of the function fields of the genus 2 modular curves X0 (p), and we study the decomposition property of the Jacbian of X0 (p).
1. Introduction a b ∈ SL2 (Z)/c ≡ 0 (mod N ) , The congruence subgroups Γ0 (N ) = c d a b Γ(N ) = ∈ SL2 (Z)/a ≡ d ≡ 1 (mod N ), b ≡ c ≡ 0 (mod N ) , act on the completed upper c d half plane H∗ = H ∪ P1 (Q) by linear fractional transformations, and the quotient X(N ) and X0 (N ) respectively are compact Riemann surfaces which are modular curves. The image points of P1 (Q) in X(N ) and X0 (N ) are known as cusps. The non cusps of the modular curve X(N ) parametrize elliptic curves together with a basis of the N -torsion group of the elliptic curve and the non cusps of the modular curves X0 (N ) parametrizes elliptic curves together with the structure of a cyclic isogeny, or equivalently, a cyclic subgroup of order N of the N -torsion of the elliptic curve. These modular curves also have canonical models over Q. The subfields of the function fields of projective curves have been studied by many authors in various contexts. The bounds for the number of subfields (essential elliptic subfields) can be found in [8]. There is also a deep study of automorphism group of function fields of curves. In this paper, for primes p, we investigate the essential elliptic subfields of the function field of the genus 2 modular curves X0 (p). We determine these subfields by computing a set of generators of the corresponding modular function field. 2. Notation and preliminaries Let C and C 0 be nonsingular projective algebraic curves over C. Let gF denote the genus of function field F of C. If f : C → C 0 is a nonconstant morphism over C, then there is a pullback f ∗ : C(C 0 ) → C(C). Moreover, f ∗ is an injective map of function fields (See [10]). Note that [C(C) : f ∗ (C(C 0 ))] = deg(f ). Definition 2.1. A subfield F of C(C) is called essential if gF is positive and it is not properly contained 0 in any other subfield F of C(C) of the same genus. Definition 2.2. A subfield F of C(C) is called elliptic if gF = 1. Genus of X0 (p) = b(p + 1)/12c -1 if p ≡ 1 (mod 12). Otherwise it equals to b(p + 1)/12c. Thus genus of X0 (p) = 2 only for p = 23, 29, 31, 37. Let S be the set {23, 29, 31, 37}. The modular curves X0 (p), p ∈ S, are hyperelliptic curves defined over Q. Explicit equations of canonical models of these hyperelliptic curves are given by (cf. [14]) X0 (23) : y 2 = x6 − 8x5 + 2x4 + 2x3 − 11x2 + 10x − 7 X0 (29) : y 2 = x6 − 4x5 − 12x4 + 2x3 + 8x2 + 8x − 7 Date: April 12th, 2013. 1
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X0 (31) : y 2 = x6 − 8x5 + 6x4 + 18x3 − 11x2 − 14x − 3 X0 (37) : y 2 = x6 + 8x5 − 20x4 + 28x3 − 24x2 + 12x − 4
Let Fp be the function field C(X0 (p)) of the modular curve X0 (p). If F is an essential elliptic subfield of Fp , then F ⊆ Fp and gF = 1. Thus there is an elliptic curve E whose function field is F and there is a morphism f : X0 (p) → E. Let J0 (N ) be the jacobian of the modular curve X0 (N ), EQ = End(J0 (N )) ⊗ Q, it’s endomorphism algebra tensored with Q and EC = EQ ⊗ C. Let TQ denote the algebra generated over Q by Tn for all n ≥ 1, TC = TQ ⊗ C. Let T0Q denote the algebra generated over Q by Tn , where (n, N ) = 1 and T0C = T0Q ⊗ C. Let NFp (n)ess = ]{F |[Fp : F ] = n, gF = 1, F is an essential subfield of Fp } and NFp = ]{F |gF = 1, F is an essential subfield of Fp }. When the genus of Fp is 2, NFp is 0 or 2 or ∞ (thm. 3.2). For fixed n, NFp (n)ess is finite. There is a bound for NFp (n)ess interms of n, genus of Fp and r = dimC (End(J0 (p))). For the genus 2 modular curve X0 (p), this bound turns out to be independent of n (lemma 3.3). In fact, NFp = 0 or 2, for p ∈ S. In Section 4, we determine the essential elliptic subfields. By Poincare’s complete irreducibility theorem, J0 (p) is isogeneous over Q to product of simple abelian varieties. In Section 5, we list this decomposition of J0 (p) for primes p ≤ 500.
3. Subfields of function fields of X0 (p) Lemma 3.1. dim(EndC (J0 (p))) is 2, where p ∈ S. Proof. Suppose N = p is prime. We know that TQ = T0Q = EQ . and TC = T0C = EC ( cf. [13], [9] ). Thus dim(EC ) = dim(EndC (J0 (p))) = dim(T0C ). By [1], dim(T0C ) = dim(S2 (Γ0 (p))) and there is a basis for S2 ((Γ0 (p)) of newforms of level p, where S2 (Γ0 (p)) is the space of cuspforms of weight 2 on Γ0 (p). The lemma follows from dim(S2 (Γ0 (p))) = genus of X0 (p) = 2 for p ∈ S. Theorem 3.2. The number of essential elliptic subfields of Fp , NFp , is 0 or 2 or ∞, where p ∈ S. Proof. The number of essential elliptic subfields of Fp corresponds to the number of elliptic factors of J0 (p). Dimension of J0 (p) = genus of X0 (p) = 2. So, by Poincare’s complete irreducibility theorem, either J0 (p) is simple or isogeneous to a product of two elliptic curve E and E 0 . If E 0 is E, then the embedding of E into E x E via P → (mP, nP ) for some positive integers m, n will lead infinitely many elliptic factors of J0 (p). Lemma 3.3. For any fixed n, NFp (n)ess ≤ 2, where p ∈ S. Proof. By thm. 4 of [8], NFp (n)ess ≤ (c2 n + 1)r−1 − (c2 n − 1)r−1 , where r = dimC (End(J0 (p))), p c2 = (2(g − 1)/g). When genus g = 2, c2 = 1. It follows from lemma 3.1 that r = 2. Thus NFp (n)ess ≤ (n + 1) − (n − 1) = 2. In fact, if p ∈ S, then NFp is finite ( 0 or 2 ) by the following proposition.
ON SUBFIELDS OF THE MODULAR FUNCTION FIELDS
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Proposition 3.4. There is no essential elliptic subfield of Fp , for p ∈ S \{37}. There are two essential elliptic subfields of F37 . Proof. Let F be the essential elliptic subfield of Fp . Then, F ⊆ Fp and gF = 1. Hence there is an elliptic curve E such that the function field of E is F and there is a push forward morphism f : X0 (p) → E. The conductor of E should be p (since conductor of E is the smallest integer n such that there exists a mapping Φ: X0 (n) → E). Moreover, essentiality of F insists that E is optimal, f doesn’t factor through any other elliptic curve. There is no elliptic curve of conductor 23, 29, 31 by thm. 3 of [2]. It follows that there is no essential elliptic subfield of Fp , for p ∈ S \ {37}. There are two optimal elliptic curves of conductor 37. Thus there are two essential elliptic subfields of Fp for p = 37. Remark 3.5. The function field of the modular curve X(11) has infintely many essential elliptic subfields. Note that gX(11) = 26. Proof. Klein’s equations for X(11): xm + x−m = 0, xa+b xa−b xc+d xc−d + xa+c xa−c xd+b xd−b + xa+d xa−d xb+c xb−c = 0, where a, b, c, d and m ∈ Z/11Z. √ Hecke proved that the Jacobian of X(11) is isogeneous over Q( −11) to a product of 26 elliptic curves. The decomposition of J(11) contains two copies of J(X1 (11)), the Jacobian of X1 (11) which is of dimension 1. Thus as in the proof of theorem 3.2 the function field of X(11) has infinitely many essential elliptic subfields. 4. Function field of X0 (37) In this section, we determine the modular functins x, y which are generators of C(X0 (37)) and it’s two essential elliptic subfields interms of x and y. We also list rank, torsion subgroups of E1 , E2 and J0 (37). Theorem 4.1. (i) There are only two essential elliptic subfields of C(X0 (37)), and the function field C(X0 (37)) is the compositum of those two subfields. (ii) Suppose p is prime. Then J0 (p) is isogeneous over Q to a product of elliptic curves for only p = 37. P k The functions Gk : H → C maps τ 7→ (c,d) 1/(cτ + d) , where (c, d) ∈ Z2 − (0, 0) are modular forms of weight k on Γ0 (N ). We denote g2 = 64G2 and g3 = 140G6 , discriminant function 4 = g23 − 27g32 and jN (τ ) = j(N τ ) for τ ∈ H, where the modular function j: H → C maps τ 7→ 1728 3 g2 (τ ) /4(τ ). From [5], Section 7.5, C(X0 (N )) = C(j, jN ). We tried to write down the subfields C(E1 ) and C(E2 ) in terms of the generators j, jN of C(X0 (N )). But the polynomial ΦN (x, y) over Q satisfied by j, jN is difficult to treat. For any prime p, Φp (x, y) ≡ xp+1 + y p+1 − xp y p − xy
(mod p).
For the smallest prime p = 2, Φ2 (x, y) is equal to x3 + y 3 − x2 y 2 + 24 .3.31.xy(x + y) − 24 .34 .53 (x2 + y 2 ) + 34 .53 .4027.xy + 28 .37 .56 (x + y) − 212 .39 .59 . Since it is difficult to handle the coefficients of Φ37 (x, y), we compute a basis of cusp forms S2 (Γ0 (37)), then we use this basis to compute a suitabe set of generators x, y of C(X0 (37)). Then C(E1 ) and C(E2 ) is obtained interms of x and y. We use the commands CuspForms(Gamma0(37),2).i; (i = 1,2) of the package Magma, we find that the q-expansions of a basis s1 and s2 of S2 (Γ0 (37)) are s1 = q + q 3 − 2q 4 − q 7 − 2q 9 + 3q 11 + O(q 12 ), s2 = q 2 + 2q 3 − 2q 4 + q 5 − 3q 6 − 4q 9 − 2q 10 + O(q 11 ),
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where q = e2iπz . Let h1 be the modular function s1 /s2 on Γ0 (37). Its q-expansion is h1 = q −1 + q − 2q 2 + 4q 3 − 4q 4 + 5q 5 − 6q 6 + 9q 7 − 12q 8 + 13q 9 − 18q 10 + O(q 11 ).
Let h2 be the differential of x with respect to z divided by the cuspform s2 . Note dx/dz = (2iπ)qdx/dq. Upto a scalar multiple h2 = q1/s2 dx/dq, its q-expansion is h2 = −q −3 − q −1 − 4 + 9q − 24q 2 + 44q 3 − 88q 4 + 163q 5 − 288q 6 + O(q 7 ). Note that [C(X0 (37)) : C(Ei )] = 2 = degree of the optimal maps fi : X0 (37) → Ei (i = 1, 2). Let x = h1 − 2/h1 , y = 4h2 /x3 . Since x and y satisfy the equation of the hyper elliptic curve X0 (37): y 2 = x6 + 8x5 − 20x4 + 28x3 − 24x2 + 12x − 4, x and y are generators for C(X0 (37)). Let x1 = (2y − x4 − 6x2 − 15)/(x2 − 1), y1 = x1 (6 − 2x2 ) − 3x2 − 21/(x2 − 1). The set {x1 , y1 } is a generating set of C(E1 ), since x1 and y1 satisfy the equation of the elliptic curve E1 : y12 + y1 = x31 − x1 . Let x2 = (37 − 5x2 )/4x2 , y2 = (37yx−3 − 4)/8. The set {x2 , y2 } is a generating set of C(E2 ), since x2 and y2 satisfy the equation of the elliptic curve E2 : y22 + y2 = x32 + x22 − 23x2 − 50. The proof of (i) follows from C(X0 (37)) (= C(x, y)) is the compositum of C(E1 ) (= C(x1 , y1 ) = C(x2 , y)) and C(E2 ) (= C(x2 , y2 ) = C(x2 , xy)). Proof of (ii) Let p be prime. Suppose J0 (p) is isogeneous over Q to a product of elliptic curves, then the genus of X0 (p) ≤ 26 ([4]). Genus of X0 (p) = bp + 1/12c -1 if p ≡ 1 (mod 12). Otherwise it equals to bp + 1/12c. Thus p ≤ 323. We conclude from the following tables that p = 37 is the only prime such that J0 (p) is isogeneous over Q to a product of elliptic curves. We denote E37a , E37b for the elliptic curves E1 and E2 . Rank of J0 (37) = 1, torsion subgroup is Z/3Z. Rank of E37a = 1, torsion subgroup is trivial, a point of infinite order on E37a = [0:0:1]. Rank of E37b = 0, torsion subgroup is Z/3Z, a generator of the torsion subgroup of E37b = [8:18:1]. From the following table, J0 (37) is isogeneous over Q to E37a × E37b . Remark 4.2. The modular curve X0 (N ) is hyperelliptic for exactly nineteen values of N . The prime p = 37 is the unique case where there is an exceptional hyperelliptic involution, not the Atkin-Lehner involution. Proof. See thm. 1, [12].
5. Appendix
We list d = dim(J0 (p)) = genus(X0 (p)) and Decomposition of J0 (p) for primes p ≤ 323 by using sage. In the following tables, we use the notation E for elliptic curve, A for abeian variety and i in Ai means dimension A.
ON SUBFIELDS OF THE MODULAR FUNCTION FIELDS
p 11 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251
d Decomposition 1 J0 (11) 1 J0 (17) 1 J0 (19) 2 J0 (23) 2 J0 (29) 2 J0 (31) 2 E37a × E37b 3 J0 (41) 3 E43a × A243b 4 J0 (47) 4 E53a × A353b 5 J0 (59) 4 E61a × A361b 5 E67a × A267b × A267c 6 A371a × A371b 5 E73a × A273b × A273c 6 E79a × A579b 7 E83a × A683b 7 E89a × E89b × A589c 7 A397a × A497b 8 E101 × A7101b 8 A2103a × A6103b 9 A2107a × A7107b 8 E109a × A3109b × A4109c 9 E113a × A2113b × A3113c × A3113d 10 A3127a × A7127b 11 E131a × A10 131b 11 A4137a × A7137b 11 E139a × A3139b × A7139c 12 A3149a × A9149b 12 A3151a × A3151b × A6151c 12 A5157a × A7157b 13 E163a × A5163b × A7163c 14 A2167a × A12 167b 14 A4173a × A10 173b 15 E179a × A3179b × A11 179c 14 A5181a × A9181b 16 A2191a × A14 191b 2 15 A193a × A5193b × A8193c 16 E197a × A5197b × A10 197c 16 A2199a × A4199b × A10 199c 17 A2211a × A3211b × A3211c × A9211d 18 A2223a × A4223b × A12 223c 2 19 A227a × A2227b × A2227c × A3227d × A10 227e 18 E229a × A6229b × A11 229c 19 E233a × A7233b × A11 233c 20 A3239a × A17 239b 19 A7241a × A12 241b 21 A4251a × A17 251b
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p 257 263 269 271 277 281 283 293 307 311 313 317
d Decomposition 21 A7257a × A14 257b 22 A5263a × A17 263b 22 E269a × A5269b × A16 269c 22 A6271a × A16 271b 22 E277a × A3277b × A9277c × A9277d 23 A7281a × A16 281b 23 A9283a × A14 283b 24 A8293a × A16 293b 25 E307a × E307b × E307c × E307d × A2307e × A9307f × A10 307g 26 A4311a × A22 311b 12 25 A2313a × A11 313b × A313c 11 15 26 A317a × A317b 6. Acknowledgements
The author is greatful to Prof. Nagaraj for his helpful comments. The author was supported by a postdoctoral fellowship at the Institute of Mathematical Sciences, Chennai.
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The Institute of Mathematical Sciences, IV Cross Road, CIT Campus, Taramani, Chennai 600 113, Tamil Nadu, India. E-mail address:
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