Reduction of Superelliptic Riemann Surfaces

Contemporary Mathematics Reduction of superelliptic Riemann surfaces Tanush Shaska Abstract. For a superelliptic curve X , defined over Q, let p denote the corresponding moduli point in the weighted moduli space. We describe a method how to determine a minimal integral model of X such that: i) the corresponding moduli point p has minimal weighted height, ii) the equation of the curve has minimal coefficients. Part i) is accomplished by reduction of the moduli point which is equivalent with obtaining a representation of the moduli point p with minimal weighted height, as defined in [1], and part ii) by the classical reduction of the binary forms. 1. Introduction Let k be an algebraic number field and Ok its ring of integers. The isomorphism class of a smooth, irreducible algebraic curve X defined over Ok is determined by its set of invariants which are homogenous polynomials in terms of the coefficients of X . When X is a superelliptic curve, as defined in [13], these invariants are generators of the invariant ring of binary forms of fixed degree. Such invariant rings are theoretically well understood and their generators have been known explicitly for degree d ≤ 8 due to work of classical invariant theorists such as Clebsch, Bolza, Gordan, van Gall, et al. In the last two decades computational invariant theory has been revived and have been attempts to compute such generators for higher degree binary forms. Given a binary form f(x; y), there are two main reduction problems (equiv- alently for superelliptic curves). Determine g(x; y), SL2(k)-equivalent to f(x; y), such that g(x; y) has minimal: a) invariants b) coefficients Explaining what minimal means will be the main focus of this paper. Completing of each task we will call reduction type a) and reduction type b). Reduction a) described in this paper, to the best of our knowledge is new and has not appeared in the literature before, apart from elliptic curves in the work of Birch and Swinnerton-Dyer as explained below (cf. Section 3.2.1). Reduction b) is explained in details in Section4 and is based on work of G. Julia [ 12]. For the purposes of this paper we will focus on the case when k = Q, even though the majority of results follow for any number field k. Key words and phrases. Superelliptic curve and Minimal invariants and weighted height. 1 2 TANUSH SHASKA 1.1. Reduction a): Minimality of the moduli point. Let X be a superelliptic curve defined over a field k given by an affine equation zn = f(x) with coefficients in Ok. There are two questions at hand: i) determine an equation of 0 a curve X , defined over Ok and k-isomorphic to X , with smallest invariants, ii) " ¯ determine the equation of a curve X , defined over Ok and k-isomorphic to X with smallest invariants, in other words the twist of X with smallest invariants. The isomorphism class of X correspond to the equivalence class of a binary form f(x; y). Thus, the isomorphism classes of superelliptic curves are determined by the set of generators of the ring of invariants Rd of degree d ≥ 2 binary forms, in other words by SL2(Ok)-invariants. By Hilbert's basis theorem Rd is finitely generated. Let I0, ::: , In be the generators of Rd such that the homogenous degree of Ii is qi for each i = 0; : : : ; n. We denote by I := (I0;:::;In) the tuple of invariants and by I(f) = (I0(f);:::;In(f)) such invariants evaluated at the binary form f. The corresponding set of invariants I(f) is a point in the weighted projective space n WPw(k) with weights w = (q0; : : : ; qn); see [15] or [1] for details. The weighted moduli point corresponding to X is p = [I(f)]. We denote the weighted greatest common divisor of the weighted tuple I(f) by wgcd(I(f)) and by wgcd(I(f)) its absolute weighted greatest common divisor; see [1]. The curve X is said to have a minimal model over Ok when it has minimal height in the weighted moduli n space WPw(k). In other words, when the weighted valuation of the tuple I(f) for each prime p 2 Ok, valp(I(f)) := max fνp(Ii(f)) for all i = 0; : : : ; ng is minimal, where νp(Ii(f)) is the valuation at p of Ii(f). The weighted moduli point p = [I(f)] is called normalized when wgcd(I(f)) is 1. A given curve X which has normalized moduli point (cf. ??) is already in the minimal model. Our main result is Thm.5, which says that the minimal models of superelliptic curves exist. Moreover, an equation X : zmyd−m = f(x; y) is a minimal model over Ok, if for every prime p 2 Ok which divides p j wgcd (I(f)), the valuation d valp of I(f) at p satisfies valp(I(f)) < qi, for all i = 0; : : : ; n. Moreover, for 2j k j k dq0 dqn λ = wgcd(I(f)) with respect the weights 2 ;:::; 2 the transformation d x m (x; y; z) ! λ ; y; λ z gives the minimal model of X over Ok. If mjd then this isomorphism is defined over k. Such minimal integral models can be found not only up to k-isomorphisms, but also over its algebraic closure k¯. We call them minimal twists. In Thm.7 we prove that minimal twists of superelliptic curves exist. An equation X : zmyd−m = f(x; y) is a minimal twist over Ok, if for every prime p 2 Ok which divides d p j wgcd (I(f)), the valuation valp of I(f) at p satisfies valp(I(f)) < 2 qi, for all i = 0; : : : ; n. The result in Thm.5 makes it possible to create a database of superelliptic curves defined over Ok, by storing only the curves with minimal weighted moduli point. The result in Thm.7 does this also for all their twists as well. Our motivation of exploring such reduction came from computational issues for genus 2 curves; see [2]. The idea of weighted heights and weighted greatest common divisors were developed in [1] and the moduli reduction was first developed in [13]. n 1.2. Reduction b): Minimality of coefficients. Let p 2 WPw(Q) be a reduced point via reduction a) and X the corresponding superelliptic curve with equation zn = f(x) over some minimal field of definition k. Determining such equation is part of the math folklore for elliptic curves. For genus 2 such equation is determined in [16] for curves with automorphism group of order 2, in [21, Lemma 3] REDUCTION OF SUPERELLIPTIC RIEMANN SURFACES 3 where the group is the dihedral group D4 or D6, and in [22, Theorem 3] for all groups of order > 2. Recently in [14] a universal equation for all genus 2 curves is given. One of the main concerns when obtaining such equations is to have coefficients as small as possible. This comes down to what historically is referred to as reduction of binary forms and goes back to Hermite for quadratic forms and Julia [12] for cubic forms. In [24] such reduction was considered again for cubics and quartics. We briefly explain below. Let f(x; y) be a degree d ≥ 2 binary form with real coefficients. Then its Julia quadratic Jf is defined as in Eq. (15). It is a positive definite quadratic and therefore has one root in the upper-half complex plane H2, say αf . Since J(f) is SL2(Z)-covariant, then bringing αf to the fundamental domain F by a matrix M M M 2 SL2(Z), induces an action f ! f on binary forms. The form f is called reduction of f. A detailed description of reduction b) is intended in [20]. 1.3. Birch and Swinnerton-Dyer computations. After the first draft of this paper was written we discovered that a reduction combining both methods described above had been used in the seminal paper of Birch and Swinnerton-Dyer in [7] and [8] for computations with elliptic curves. While the case of the elliptic curves is simpler, it is also the only case that is fully understood, since only for cubics and quartics we have precise results of the Julia reduction as described in the work of Cremona and Stoll in [24] and further explored by Beshaj in [4] and [5] as remarked above. The initial computations in [7] start with reduction a). Let f(x; y) = ax4+bx3+ 2 cx +dx+e, where a; b; c; d; e are rational integers, and I4;I6 its invariants. If p 2 Z is a prime such that p 6= 2; 3, and pj wgcd 4;6(I4;I6), there there a quartic integral −4 −6 binary form g(x; y), SL2(Z) equivalent to f(x; y) with invariants (p I4; p I6). Birch and Swinnerton-Dyer go to a case by case analysis to prove this result; see [7, Lemma 3]. However, this is an immediate consequence of the reduction a) described in Section 3.2. Lemma 4 and Lemma 5 in [7] give the reduction for p = 2 and 3 respectively. In Section 3.2.1 we give a quick recap of Birch and Sinnerton-Dyer spectacular results. The minimal integral models that we determine in this paper give the "nicest" equation of the paper over a global field, since the corresponding invariants are non-zero for as many primes p 2 Ok as possible. It is a topic of interest to explore the stability of such curves.

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