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HIGHER WEIERSTRASS POINTS on X0(P) TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 355, Number 4, Pages 1521{1535 S 0002-9947(02)03204-X Article electronically published on November 20, 2002 HIGHER WEIERSTRASS POINTS ON X0(p) SCOTT AHLGREN AND MATTHEW PAPANIKOLAS Abstract. We study the arithmetic properties of higher Weierstrass points on modular curves X0(p)forprimesp.Inparticular,forr 2; 3; 4; 5 ,we 2f g obtain a relationship between the reductions modulo p of the collection of r-Weierstrass points on X0(p) and the supersingular locus in characteristic p. 1. Introduction and Statement of Results Suppose that X is a smooth projective algebraic curve over C of genus g 2. Let ≥ ΩX denote the sheaf of holomorphic 1-forms on X and let r be a positive integer. r 0 r Then let (X):=H (X; ΩX⊗ ) denote the space of holomorphic r-differentials on X and letH r dr(X):=dim ( (X)): C H It follows from the Riemann-Roch theorem that g if r =1; (1.1) dr(X)= (2r 1)(g 1) if r 2: ( − − ≥ ApointQ X is called an r-Weierstrass point if there exists a holomorphic non- zero r-differential2 ! on X with the property that ordQ ! dr(X): ≥ r Suppose now that Q X and that !1;:::;! is a basis for (X) such that 2 f dr(X)g H 0=ordQ !1 < ordQ !2 < < ordQ ! : ··· dr(X) Then we define the Weierstrass weight of Q with respect to r(X)as H dr(X) (1.2) wtr(Q):= (ordQ(!j) j +1): − j=1 X r This definition is independent of the choice of basis for (X). Moreover, wtr(Q) is positive if and only if Q is an r-Weierstrass point, andH we have (1.3) wtr(Q)=dr(X)(g 1)(2r 1+dr(X)): − − Q X X2 Therefore, for each r 1, the set of r-Weierstrass points is a finite, non-empty set of distinguished points on≥ X. We remark that when g<2therearenor-Weierstrass Received by the editors July 31, 2002 and, in revised form, September 19, 2002. 2000 Mathematics Subject Classification. Primary 11G18; Secondary 11F33, 14H55. Key words and phrases. Weierstrass points, modular curves. The first author thanks the National Science Foundation for its support through grant DMS 01-34577. c 2002 American Mathematical Society 1521 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1522 SCOTT AHLGREN AND MATTHEW PAPANIKOLAS points for any r. For these and other general facts about Weierstrass points, one may consult, for instance, Chapter III of Farkas and Kra [F-K]. Higher Weierstrass points play an important role in the theory of algebraic curves. For example, one can use them to construct projective embeddings for moduli spaces of curves. Mumford has suggested that, on a curve of genus g 2, r-Weierstrass points are analogous to r-torsion points on an elliptic curve (see ≥A.I- II of [M], or [Si]). Higher Weierstrass points are also important in the arithmeticx of algebraic curves and their Jacobians; see for example Burnol [B]. In this paper we study the arithmetic of Weierstrass points on modular curves of level p for primes p. As usual, define the congruence subgroup Γ0(p)by ab Γ0(p):= SL2(Z):c 0(modp) ; cd2 ≡ and denote by H the upper half-plane of complex numbers. Define Y0(p):=Γ0(p) H; n and let the modular curve X0(p) be the compactification of Y0(p) obtained by adding cusps at 0 and .ThenX0(p) can be given the structure of a smooth 1 projective curve defined over Q (see, for example, 6.7 of [Sh]). The genus of X0(p) x grows linearly with p. To be precise, define the function m(k)fork N by 2 k=12 if k 2 (mod 12); (1.4) m(k):= b c 6≡ k=12 1ifk 2 (mod 12); (b c− ≡ then the genus gp of X0(p)isgivenby (1.5) gp = m(p +1): Properties of 1-Weierstrass points (or simply Weierstrass points) on modular curves have been studied by a number of authors. See, for example, papers by Atkin [At], Lehner and Newman [L-N], Ogg [O1], [O2], and Rohrlich [R1], [R2]. In more recent work [A-O], the first author and Ono showed that the collection of Weierstrass points on X0(p), when reduced modulo p, covers the supersingular 2 locus in characteristic p with multiplicity g gp. To describe this result precisely p − requires the introduction of some notation. Throughout we agree that q := e2πiz. Let 1 j(z):=q− + 744 + 196884q + ··· be the usual elliptic modular function on SL2(Z) (by a slight abuse of notation, we shall also speak of j(E) for elliptic curves E and of j(Q)forpointsQ Y0(p); for 2 the latter we take j(τ), where τ H is any point which corresponds to Q under the standard identification). The supersingular2 polynomial in characteristic p is given by (1.6) Sp(x):= (x j(E)); − E=Fp supersingularY where the product runs over Fp-isomorphism classes of elliptic curves. Then Sp(x) 2 Fp[x], and Sp(x) splits completely over Fp2 . Define the polynomial wt1(Q) Fp(x):= (x j(Q)) − Q X0(p) 2Y License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use HIGHER WEIERSTRASS POINTS 1523 (this definition makes sense, since it is known [O2] that the cusps of Γ0(p)arenot Weierstrass points). In [A-O] the following result is obtained. Theorem 1 ([A-O, Theorem 1]). If p is prime, then Fp(x) has p-integral rational coefficients and satisfies 2 g gp Fp(x) Sp(x) p− (mod p): ≡ The goal in the present paper is to investigate similar phenomena related to higher Weierstrass points on X0(p). Here there are complications which are not present in the previous case. For example, the cusps can be (and often are) Weier- strass points. Moreover, the p-integrality (or non-p-integrality) of the analogue of the polynomials Fp(x) becomes an issue. In the case of 1-Weierstrass points, the 1 bijection between the space (X0(p)) and the space of weight-two cusp forms on H Γ0(p) plays a crucial role; the absence of such a bijection in the current situation is to blame for these complications. If p is prime and r 2, then we define the polynomial ≥ (1.7) F (r)(x):= (x j(Q))wtr(Q) : p − Q Y0(p) 2Y By (1.1) and (1.3), and using the fact that 0 and are interchanged by the Atkin- Lehner involution, we see that 1 (r) 2 2 deg(F (x)) = (2r 1) (gp 1) gp 2 wtr( ): p − − − · 1 Further, we define the polynomial Sp∗(x) Fp[x]by 2 (1.8) S∗(x):= (x j(E)): p − E=Fp supersingularY isom. class j(E)=0;1728 6 Then we have the simple relationship S∗(x)ifp 1 (mod 12), p ≡ x Sp∗(x)ifp 5 (mod 12), (1.9) Sp(x)=8 · ≡ >(x 1728) Sp∗(x)ifp 7 (mod 12), <> − · ≡ x(x 1728) S∗(x)ifp 11 (mod 12). − · p ≡ > :> We obtain an analogue of Theorem 1 for r-Weierstrass points (2 r 5) on the r ≤ ≤ curves X0(p) under the assumption that (X0(p)) is good at p. This assumption H (r) is necessary to ensure that the polynomial Fp (x)hasp-integral coefficients, and is described completely in the next section. Although there are some spaces which fail to be good, computations suggest that this condition is almost always satisfied. 2 For example, the only space (X0(p)) which fails to be good with p<800 is 2 H (X0(373)). H License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1524 SCOTT AHLGREN AND MATTHEW PAPANIKOLAS Suppose that r 2andthatp is a prime with p 23 (this ensures that gp 2). Then we define the≥ polynomial ≥ ≥ (1.10) fp;r(x) 1ifp 1 (mod 12); 1 2 ≡ (2r 1) (gp 1)(gp 2) xd 3 − − − e if p 5 (mod 12); := 8 1 2 ≡ (2r 1) (gp 1)(gp 2) >(x 1728) 2 − − − if p 7 (mod 12); > 1− 2 1 2 ≡ < (2r 1) (gp 1)(gp 2) (2r 1) (gp 1)(gp 2) xd 3 − − − e(x 1728) 2 − − − if p 11 (mod 12): − ≡ > With:> this notation we can state our main result. Theorem 2. Suppose that p 23 is prime and that 2 r 5. Suppose that r ≥(r) ≤ ≤ (X0(p)) is good at p.ThenFp (x) has rational p-integral coefficients, and there H exists a polynomial H(x) Fp[x] such that 2 2 (r) (2r 1) (gp 1)(gp 2) F (x) H(x) fp;r(x) S∗(x) − − − (mod p): p ≡ · · p Examples. When p = 53, we have gp =4andwt2( )=0.Wehave 1 2 S53(x)=x(x +3)(x +7)(x +50x + 39) = x S∗ (x) · 53 and (2) 54 18 F (x) S53(x) x H1(x) (mod 53); 53 ≡ · · where H1(x) is the square of a polynomial of degree 18. When p = 61, we have gp =4andwt2( )=2.Wehave 1 2 S61(x)=(x + 11)(x + 20)(x + 52)(x +38x + 24) = S61∗ (x) and (2) 54 F (x) S61(x) H2(x) (mod 61); 61 ≡ · where H2(x) is the square of a polynomial of degree 25. (r) Remark 1. In view of (1.9) and (1.10), Theorem 2 shows that Fp (x) is divisible by a large power of the full supersingular polynomial Sp(x).
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