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 (equivalently 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 ∈ Ok, valp(I(f)) := max {νp(Ii(f)) for all i = 0, . . . , n} 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 ∈ Ok which divides p | 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 m|d 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 ∈ Ok which divides d p | 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 ∈ 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 ∈ 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 ∈ Z is a prime such that p 6= 2, 3, and p| 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 ∈ Ok as possible. It is a topic of interest to explore the stability of such curves. Adjusting the method of reduction a) would give a method of obtaining a stable os semistable form for any binary form. Theoretically, both reductions can be performed for all superelliptic curves, providing that we explicitly know the generators of the ring of invariants Rd. For non-superelliptic curves this is a much more difficult problem which requires the full arsenal of GIT. An explicit description of the moduli point is not known in general. Moreover, there is no known algorithm to determine the equation of the curve starting with the moduli point even in the case a planar curves. Superellip- tic curves are a natural extension of the theory of hyperelliptic curves as explored in [13] and [20]. Extending this approach beyond superelliptic curves seems very difficult at the moment.

Aknowledgments: I would like to thank Mike Fried for helpful comments and discussions. 4 TANUSH SHASKA

2. Preliminaries 2.1. Superelliptic curves. A a genus g ≥ 2 smooth, irreducible, algebraic curve X defined over an algebraically closed field k is called a superelliptic curve of level n if there exist an element τ ∈ Aut (X ) of order n such that τ is central and the quotient X / hτi has genus zero; see [13] for details. Superelliptic curves n Qd have affine equation X : y = f(x) = i=1(x − αi). Denote by σ : X → X the superelliptic automorphism, i.e. σ(x, y) → (x, ξny), where ξn is a primitive n-th root of unity. Notice that σ fixes 0 and the point 1 1 at infinity in Py. The natural projection π : X → Px = X / hσi is called the superelliptic projection. It has deg π = n and π(x, y) = x. This cover is branched at exactly at the roots α1, . . . , αd of f(x). Then the affine equation of X m Qd is X : z = i=1(x − αi). Denote the projective equation of X by m d−m d d−1 d−1 d (1) z y = f(x, y) = adx + ad−1x y + ··· + a1xy + a0y defined over a field k. Hence, f(x, y) is a binary form of degree deg f = d. Let k[x, y] be the polynomial ring in two variables and Vd(k) denote the (d+1)- dimensional subspace of k[x, y] consisting of homogeneous polynomials f(x, y) of degree d. Elements in Vd are called binary forms of degree d. The general linear group GL2(k) acts as a group of automorphisms on k[x, y]. SL2(k) leaves a bilinear form (unique up to scalar multiples) on Vd invariant. Consider a0, a1, ... , ad as parameters (coordinate functions on Vd). Then the coordinate ring of Vd can be identified with k[a0, ..., ad]. For I ∈ k[a0, ..., ad] and M M M M ∈ GL2(k), define I ∈ k[a0, . . . , ad] as I (f) := I(f ), for all f ∈ Vd. Then MN M N I = (I ) and we have an action of GL2(k) on k[a0, . . . , ad]. A homogeneous polynomial I ∈ k[a0, . . . , ad, x, y] is called a covariant of index M s d s if I (f) = λ I(f), where λ = det(M) . The homogeneous degree in a0, . . . , ad is called the degree of I, and the homogeneous degree in x, y is called the order of I. A covariant of order zero is called invariant. An invariant is a SL2(k)-invariant on Vd. From Hilbert’s theorem the ring of invariants Rd of degree d binary forms is finitely generated. Let I0,...,In be the generators of Rd with degrees q0, . . . , qn respectively. Usually we assume an ordering q0 < q1 < . . . < qn of degrees. Denote the ordered tuple of invariants by I := (I0,...,In). Over the algebraic closure we have d Y x xi f(x, y) = (y1x − x1Y ) ··· (ydx − xdy) = det , y yi i=1 1 where (xi, yi) are the homogenous coordinates of the roots in P . For M ∈ GL2(k), M denote by δ = det M and by f the action of M on f. Each root (xi, yi) goes to x x x x x M i and i → M i . Hence, we have yi y yi y yi d Y x x d f M (x, y) = det M · i = (det M) f(x, y). y yi i=1 d Hence, all the coefficients ai, i = 0, . . . , d are multiplied by δ . Since an invariant of degree s is a homogenous polynomial of degree s in terms of ai,

X α0 αd Is = a0 . . . ad REDUCTION OF SUPERELLIPTIC RIEMANN SURFACES 5

M ds where αi = 0, . . . , d and α0 + ··· + αd = s, then I(f ) = δ Is(f). However, it is more diﬃcult if we want to determine f M (x, y) when f(x, y) is given as in Eq. (1). α β Consider f(x, y) as in Eq. (1), M = ∈ GL (k), and f M (x, y) := γ δ 2 f (αx + βy, γx + δy). We would like to determine the invariants of f M (x, y) in terms of the invariants of f(x, y). The following result is fundamental to our approach.

Proposition 1. Let f ∈ Vd(k), M ∈ GL2(k), and I0,...In be the generators of Rd with degrees q0, . . . , qn respectively. Then for each i = 0, . . . , n

M qi (2) Ii f (x, y) = λ Ii(f)

d where λ = (det M) 2 . Proof. Let J be a covariant of degree deg J = s, order ord (J), and weight wt (J). Recall that the weight of a covariant is the integer r such that J (f (αx + βy, γx + δy)) = (αδ − βγ)rJ(f(x, y)). We know that deg J + 2wt (J) = (deg f + 2wt f) ord(J), see [17, Prop. 2.29]. Let I be an invariant of order ord (I) := s. Then deg I = 0, d wt (f) = 0 and we have 2wt (I) = ds. So the weight of any invariant is 2 s. This completes the proof. Remark 1. The above result was stated in [13, Prop. 14], but the proof ac- cidentally cut oﬀ. It was probably known to classical invariant theorists, but we couldn’t ﬁnd it stated anywhere in the literature.

2.2. Proj Rd as a weighted projective space. Since all I0,...,Ii,...,In are homogenous polynomials then Rd is a graded ring and Proj Rd is a weighted n+1 projective space. Let w := (q0, . . . , qn) ∈ Z be the fixed ordered tuple of positive integers called weights. Consider the action of k? = k \{0} on An+1(k) as follows q0 qn ∗ λ ? (x0, . . . , xn) = (λ x0, . . . , λ xn), for λ ∈ k . The quotient of this action is n called a weighted projective space and denoted by WPw(k). It is the projective variety P roj (k[x0, ..., xn]) associated to the graded ring k[x0, . . . , xn] where the n variable xi has degree qi for i = 0, . . . , n. We will denote a point p ∈ WPw(k) by p = [x0 : x1 : ··· : xn]. Since the isomorphism class of a superelliptic curve X : zmyd−m = f(x, y) is determined by the equivalence class of binary form f(x, y) we denote the set of invariants of X by I(X ) := I(f). Corollary 1. Let X be as in Eq. (1). The k¯-isomorphism class of X is n determined by the weighted moduli point p := [I(f)] ∈ WPw(k). 2.3. Heights. If we want to create a list of all isomorphism classes of superel- n liptic curves we have to create a database of points in WPw(k). Obviously, we n would prefer to take for each point p ∈ WPw(k) its smallest representative and also want to order such points. We struggled with both such tasks in [2], until n we were able to define a height function on WPw(k), which solves both problems. This is due to a Northcott like theorem for this weighted height, which says that there are only finitely many points for a bounded height. Also the representation of n the point p ∈ WPw(k) with smallest coordinates correspond precisely to the tuple 6 TANUSH SHASKA

(x0, . . . , xn) such that its weighted greatest common divisor is = 1. These ideas were developed in detail in [1]. Fix the following notation: k is a number field, Ok is ring of integers of k, 0 Mk is a complete set of absolute values of k, Mk is the set of all non-archimedian ∞ places in Mk, Mk is the set of archimedian places, and X /k is a smooth projective superelliptic curve defined over k. For a place ν ∈ Mk, the corresponding absolute value is denoted by |·|ν , normalized with respect to k such that the product formula Q holds and the Weil height for x ∈ k is H(x) = ν max{1, |x|ν }. n For P = [x0, . . . , xn] ∈ P (k) the multiplicative height of P is defined as follows H (P ) := Q max |x |nv ,..., |x |nv . The height H (P ) is well defined, k v∈Mk 0 v n v k in other words it does not depend on the choice of homogenous coordinates of P . Moreover, Hk(P ) ≥ 1. [L:k] If L/k is a finite extension, then HL(P ) = Hk(P ) . Hence, we can define the height on Pn(Q), which is called the absolute (multiplicative) height and n ¯ 1/[k: ] is the function H : P (Q) → [1, ∞), such that H(P ) = Hk(P ) Q . The height is n invariant under Galois conjugation. In other words, if P ∈ P (Q) and σ ∈ GQ, then H(P σ) = H(P ). The following are the two main results in the theory of heights on projective spaces; see [9] or [11].

Theorem 1 (Northcott). Let c0 and d0 be constants. Then the set n {P ∈ P (Q): H(P ) ≤ c0 and [Q(P ): Q] ≤ d0} n has ﬁnitely many points. In particular, {P ∈ P (k): Hk(P ) ≤ c0} is a ﬁnite set. n Theorem 2 (Kronecker’s theorem). Let P = [x0, . . . , xn] ∈ P (k). Fix any i0 with xi0 6= 0. Then H(P ) = 1 if and only if the ratio xj/xi0 is a root of unity or zero for every 0 ≤ j ≤ n. Amazingly, the above theory can be extended to weight projective spaces with necessary adjustments. We give a quick recapture here; see [1, 15] for details. n+1 Let x = (x0, . . . xn) ∈ Z be a tuple of integers, not all equal to zero. A n+1 weighted integer tuple is a tuple x = (x0, . . . , xn) ∈ Z such that to each coordinate xi is assigned the weight qi and w := (q0, . . . , qn) is called the set of weights. We multiply weighted tuples by scalars λ ∈ Q via

q0 qn λ ? (x0, . . . , xn) = (λ x0, . . . , λ xn) n+1 For an ordered tuple of integers x = (x0, . . . , xn) ∈ Z , whose coordinates are not all zero, the weighted greatest common divisor with respect to the set qi of weights w is the largest integer d such that d | xi, for all i = 0, . . . , n. For n+1 a weighted tuple x = (x0, . . . , xn) ∈ Ok the weighted greatest common divisor is given by nj ν (x ) k j ν (x ) ko Y min p 0 ,..., p n wgcd(x) = p q0 qn

p∈Ok n where νp is the valuation corresponding to the prime p. We call a point p ∈ WPw(k) a normalized point if the weighted greatest common divisor of its coordinates is n 1 1. For any point p ∈ WPw(k), there exists its normalization given by q = wgcd(p) ?p. Moreover, this normalization is unique up to a multiplication by a q-root of unity, where q = gcd(q0, . . . , qn), see [1]. The absolute weighted greatest common qi divisor of x = (x0, . . . , xn) is the largest real number d such that d ∈ Z and qi d | xi, for all i = 0, . . . n. We denote it by wgcd(x0, . . . , xn). REDUCTION OF SUPERELLIPTIC RIEMANN SURFACES 7

n Let p = [x0, . . . , xn] ∈ WP (k). Without any loss of generality we can assume that p is normalized. The weighted multiplicative height of p is

nv nv Y q0 qn (3) hk(p) := max |x0|v ,..., |xn|v

v∈Mk

The height hk(p) is well defined, in other words it does not depend on the choice of coordinates of p and hk(p) ≥ 1; see [1]. Denote by K = k(wgcd(p)). Then, over K, the weighted greatest common divisor is the same as the absolute greatest common divisor, wgcd K (p) = wgcd K (p). Moreover, [K : k] < ∞ and we have the following. Proposition 2. If p is normalized in K, then n o nν /qi (4) hK (p) = h∞(p) = max |xi|∞ . 0≤i≤n [L:K] Moreover, if L/K is a finite extension, then hL(p) = hK (p) . n Using Prop.2, we can define the height on WP (Q). The height of a point n on WP (Q) is called the absolute (multiplicative) weighted height and is ˜ n ¯ ˜ 1/[K:Q] the function h : WP (Q) → [1, ∞) such that h(p) = hK (p) , where p ∈ n n WP (K), for any K which contains Q(wgcd(p)). Moreover, for p ∈ WP (Q) and σ σ ∈ GQ we have h(p ) = h(p). The field of definition of p is defined as Q(p) := q0 qn q q x0 ,..., 1,..., xn For any point p ∈ n ( ), we have [ (p): ] ≤ Q xi xi WPw Q Q Q q · [Q(φ(p)) : Q]. The following result is analogue to Northcott’s theorem for weighted projective spaces.

Theorem 3 (Theorem [1]). Let c0, d0 ∈ R. Then the set {p ∈ n ( ): h (p) ≤ c and [ (p): ] ≤ d } WPw Q Q 0 Q Q 0 contains only finitely many points. n Hence, {p ∈ WPw(Q): h¯ (p) ≤ c0} is a finite set for any constant c0. For any Q n number field K, the set {p ∈ WPw(K): Q(p) ⊂ K and hK (p) ≤ c0}, is a finite set. The next result is the analogue of Kronecker’s theorem.

Theorem 4 (Theorem [1]). Fix any i with xi 6= 0. Then h(p) = 1 if the ratio qj xj/ξi , where ξi is the qi-th root of unity of xi, is a root of unity or zero for every 0 ≤ j ≤ n and j 6= i. Now we have the following two problems in terms of curves. m d−m Problem 1. Given a curve X : z y = d(x, y) deﬁned over Ok, determine 0 0 m d−m X , k-isomorphic to X , such that deﬁned over Ok, say X : z y = g(x, y), such n that p := [I(g)] ∈ WPw(k) has minimal height over k.. n Problem 2. Determine a twist Y of X such that p := [I(Y)] ∈ WPw(k) has minimal height over the algebraic closure k¯.

3. Reduction of the moduli point The reduction of superelliptic curves consists of two steps. On the first step we perform the necessary coordinate changes so we have a minimal weighted moduli point, and the second step is to minimize the coefficients of the equation of the curve. Both of these steps are possible due to the fact that a superelliptic curve 8 TANUSH SHASKA can be written as a curve with projective equation ynzd−n = f(x, z), such that f is a binary form of degree deg f = d with nonzero discriminant. The first step is based on the concept of weighted heights in weighted projective spaces as defined in [1]. Applications of this method were used in [2,3,6,15]. The second step is well known by work of Hermite for quadratics and extended by Julia for higher degree forms. For more recent work on this type of reduction see work of Cremona, Stoll, and Beshaj [5, 19, 24].

3.1. Minimal integral models of binary forms. We say that a binary form f(x, y) has a integral minimal model over k if it is integral (i.e. f ∈ Ok[x, y]) and valp(I(f)) is minimal for every prime p ∈ Ok. n Let f ∈ Ok and x := I(f) ∈ WPw(Ok) its corresponding weighted moduli point. We deﬁne the weighted valuation of the tuple x = (x0, . . . xn) at the prime p ∈ Ok as

j qi valp(x) := max j | p divides xi for all i = 0, . . . n , Then we have the following.

Proposition 3. A binary form f ∈ Vd is a minimal model over Ok if for every prime p ∈ Ok such that p | wgcd(I(f)) the following holds d val (I(f)) < q , for all i = 0, . . . , n. p 2 i Moreover, for every integral binary form f its minimal model exist.

Proof. Let x = I(f). From Eq. (2) we know that for any M ∈ SL2(Ok), d M 2 I(f ) = (det M) I(f). Hence, for every prime p ∈ Ok which divides wgcd(x) we d j must ”multiply” x by the maximum exponent j such that p 2 divides wgcd(x). 1 0 For a given binary form f we pick M = λ , where λ is the weighted 0 1 j k j k dq0 dqn greatest common divisor of I(f) with respect the weights 2 ,..., 2 . The x transformation x → λ gives the minimal model of f over Ok. This completes the proof.

2 qi q Remark 2. If a prime p ∈ Ok divides wgcd(x) then p divides xi, so p i qi divides xi . Taking qt = min(q0, . . . , qn), we have a lower bound for the weighted 2 valuation of the point I(f) = (x0, . . . , xn), that is valp(I(f)) ≥ qt . Notice that it is possible to ﬁnd a twist of f with ”smaller” invariants. In this case the new binary form is not in the same SL2(Ok)-orbit as f. For example, the 1 1 transformation (x, y) → 2 x, 2 y , will give us the form with smallest invariants, λ d λ d but not necessarily k-isomorphic to f. It is worth noting that for a binary form f given in its minimal model, the point I(f) is not necessarily normalized as in the sense of [1].

n Corollary 2. If f(x, y) ∈ Ok[x, y] is a binary form such that I(f) ∈ WPw(k) is normalized over k, then f is a minimal model over Ok. We see an example for binary sextics. REDUCTION OF SUPERELLIPTIC RIEMANN SURFACES 9

Example 1. Let be given the sextic f(x, y) = 7776x6 +31104x5y+40176x4y2 +25056x3y3 +8382x2y4 +1470xy5 +107y6 Notice that the polynomial has content 1, so there is no obvious substitution here to simplify sextic. The moduli point is p = [J2 : J4 : J6 : J10], where

15 5 12 9 16 13 J2 = 2 · 3 ,J4 = −2 · 3 · 101 · 233,J6 = 2 · 3 · 29 · 37 · 8837, 26 21 J10 = 2 · 3 · 11 · 23 · 547 · 1445831

1 Recall that the transformation (x, y) → p x, y will change the representation of the point p via 1 1 1 1 1 ? [J : J : J : J ] = J : J : J : J p3 2 4 6 10 p6 2 p12 4 p18 6 p30 10

6 12 18 30 So we are looking for prime factors p such that p |J2, p |J4, p |J6, and p |J10. Such candidates for p have to be divisors of wgcd(p) = 22 · 32. Obviously neither p = 2 or p = 3 will work. Thus, f(x, y) is in its minimal model over Ok. Corollary 3. The transformation of f(x, y) by the matrix

 1  εd 2 0 (wgcd(I(f))) d M =  1  0 εd 2 (wgcd(I(f))) d where εd is a d-primitive root of unity, will always give a minimal set of invariants. 3.2. Reduction a): Moduli points with minimal weighted height. Let n X be as in Eq. (1) and p = [I(f)] ∈ WPw(k). Let us assume that for a prime p ∈ Ok, x we have νp (wgcd(p)) = α. If we use the transformation x → pβ x, for β ≤ α, then from Eq. (2) the set of invariants will become 1 ? I(f). To ensure that d β p 2 βd the moduli point p is still with integer coeﬃcients we must pick β such that p 2 divides pνp(xi) for i = 0, . . . , n. Hence, we must pick β as the maximum integer 2 such that β ≤ d νp(xi), for all i = 0, . . . , n. This is the same β as in Prop.3. The 1 0 transformation (x, y) → x , y , has corresponding matrix M = pβ with pβ 0 1 d/2 1 1 det M = pβ . Hence, from Eq. (2) the moduli point p changes as p → pβ ? p, which is still an integer tuple. We do this for all primes p dividing wgcd(p). Notice n that the new point is not necessarily normalized in WPw(k) since β is not necessarily equal to α. This motivates the following deﬁnition.

Definition 1. Let X be a superelliptic curve deﬁned over an integer ring Ok n and p ∈ WPw(Ok) its corresponding weighted moduli point. We say that X has a minimal model over Ok if for every prime p ∈ Ok the valuation of the tuple at p

valp(p) := max {νp(xi) for all i = 0, . . . n} , is minimal, where νp(xi) is the valuation of xi at the prime p. 10 TANUSH SHASKA

Theorem 5. Minimal models of superelliptic curves exist. An equation X : m d−m z y = f(x, y) is a minimal model over Ok, if for every prime p ∈ Ok which divides p | wgcd (I(f)), the valuation valp of I(f) at p satisﬁes d (5) val (I(f)) < q , p 2 i for all i = 0, . . . , n. Moreover, then for λ = wgcd(I(f)) with respect the weights j dq k j dq k d 0 n x m ( 2 ,..., 2 ) the transformation (x, y, z) → λ , y, λ z gives the minimal model of X over Ok. If m|d then this isomorphism is deﬁned over k.

Proof. Let X be a superelliptic curve given by Eq. (1) over Ok and p = I(f) ∈ n n WPw(Ok) with weights w = (q0, . . . , qn). Then p ∈ WPw(Ok) and from Prop.3 1 0 exists M ∈ SL (O ) such that M = λ and λ as in the theorem’s hypothesis. 2 k 0 1 By Prop.3 we have that Eq. (9) holds. Let us see how the equation of the curve X changes when we apply the transformation by M. We have x xd xd−1 x zmyd−m = f , y = a + a y + ··· + a yd−1 + a yd λ d λd d−1 λd−1 1 λ 0 Hence, 0 d m d−m d d−1 d−1 d−1 d d (6) X : λ z y = adx + λad−1x y + ··· + λ a1xy + λ a0y M 1 This equation has coefficients in Ok. Its weighted moduli point is I(f ) = d ? λ 2 I(f), which satisfies Eq. (9). It is a twist of the curve X since λd is not necessary d a m-th power in Ok. The isomorphism of the curves over the field k λ m is given

d x m 0 by (x, y, z) → λ , y, λ z . If m|d then this isomorphism is deﬁned over k and X has equation 0 m d−m d d−1 d−1 d−1 d d X : z y = adx + λad−1x y + ··· + λ a1xy + λ a0y Then we have the following. Corollary 4. There exists a curve X 0 given in Eq. (10) isomorphic to X over d the ﬁeld K := k wgcd(p) m with minimal SL2(Ok)-invariants. Moreover, if m|d then X and X 0 are k-isomorphic. A simple observation from the above is that in the case of hyperelliptic curves we have m = 2 and d = 2g + 2. Hence, the curves X and X 0 would always be isomorphic over k. So we have the following.

Corollary 5. Given a hyperelliptic curve deﬁned over a ring of integers Ok. 0 There exists a curve X k-isomorphic to X with minimal SL2(Ok)-invariants. Thm.5 above provides an algorithm which is described next. Given a superelliptic curve X deﬁned over Ok, we denote the corresponding point in the weighted moduli space by p = [x0 : ··· : xn]. Algorithm: Computing an equation of the curve with minimal moduli point.

m d−m Input: A curve X : z y = f(x, y), deg(f) = d and f ∈ Ok[x, y]. d m Output: A curve Y : λ z = g(x, y), deﬁned over Ok and k-isomorphic to X such REDUCTION OF SUPERELLIPTIC RIEMANN SURFACES 11 that valp I(g) is minimal for each prime p ∈ Ok.

Step 1: Compute the generating set I := [Iq0 ,...,Iqn ] for the ring of invariants Rn. n Step 2: Compute the moduli point p ∈ WPw(k) for X by evaluating I(f). Step 3: Computing λ the weighted greatest common divisor wgcd(I(f)) ∈ Ok j k j k dq0 dqn with respect the weights 2 ,..., 2 1 0 Step 4: Compute f M , where M := λ . We have g(x) := λd · f x . 0 1 λ Step 5: Return the curve X 0 with equation λdzmyd−m = g(x, y).

Let us see the cases for g = 1, 2. 3.2.1. Elliptic curves. Technically elliptic curves are not superelliptic curves, but the method will work the same. Let E be an elliptic curve with Weierstrass equation (7) E : y2 = f(x) = ax4 + bx3 + cx + d. Invariants of f(x) are deﬁned as 2 2 2 3 I2 = 12ae − 3bd + c ,I3 = 72ace + 9bcd − 27ad − 27eb − 2c

The corresponding weighted moduli space is WP(2,3)(k). Isomorphism classes of elliptic curves over k correspond to points in WP(2,3)(k). Corollary 6. The equation in Eq. (7) is a minimal models if for every prime p ∈ Z, p 6= 2, 3 which divides p| wgcd(I2,I3), the valuation valp(I2,I3) satisﬁes valp(I2,I3) < 2qi, for qi = 2, 3.

α β Hence, we can reduce any prime p 6= 2, 3 such that p |I2 and p |I3 when α ≥ 4 and β ≥ 6. The above result was proved in [7] using a case by case analysis; see [7, Lem. 3]. Lemma 4 and Lemma 5 in [7] describe the cases when p = 2 and p = 3 respectively. Then we have the followin; see [7, Theorem 1]. Theorem 6 (Birch, Swinnerton-Dyer). If E is an elliptic curve, and U is a non-trivial 2-covering of E then U can be represented by a curve y2 = g(x), where g(x) is reduced. The reduced g(x) of a given E are ﬁnite in number and computable. 3.2.2. Genus 2 curves. Let X be a genus 2 curve with equation z2y4 = f(x, y) as in Example1. By applying the transformation x (x, y, z) → , y, 63 · z 6 we get the equation (8) z2 = x6 + 24x5 + 186x4 + 696x3 + 1397x2 + 1470x + 642. Computing the moduli point of this curve we get p = [211 · 3 : −24 · 3 · 101 · 233 : 24 · 3 · 29 · 37 · 8837 : 26 · 3 · 11 · 23 · 547 · 1445831], 3 which is obviously normalized in WPw(Q) since wgcd(p) = 1. Hence, the Eq. (8) is a minimal model. 12 TANUSH SHASKA

3.3. Minimal integral twists. Let k be a field of characteristic zero and Gal(k/k¯ ) the Galois group of k/k¯ . Let X a genus g ≥ 2 smooth, projective algebraic curve defined over k. We denote by Aut (X ) the automorphism group of X over ¯ the algebraic closure k. By Aut k(X ) is denoted the subgroup of automorphisms of X defined over k.A twist of X over k is a smooth projective curve X 0 defined over k which is isomorphic to X over k¯. We will identify two twists which are isomorphic over k. The set of all twists of X , modulo k-isomorphism, is denoted by Twist (X /k). Let X and X 0 be twists of each other over k. Hence, there is an isomorphism φ : X 0 → X defined over k¯. For any σ ∈ Gal(k/k¯ ) there exists the induced map φσ : X 0 → X . To measure the failure of φ being defined over k one considers the map ξ : Gal(k/k¯ ) → Aut (X ) such that ξ(σ) = φσφ−1 for any σ ∈ Gal(k/k¯ ). The following is the main result on twists; see [23, Th. 2.2. pg. 285]

Proposition 4. The following are true: i) ξ is a 1-cocycle ii) The cohomology class {ξ} is determined by the k-isomorphism class of X 0 and is independent of φ. Hence, there is a natural map

θ : Twist (X /k) → Hom1 Gal(k/k¯ ), Aut (X )

iii) The map θ is a bijection.

As noted by Silverman in [23, Remark 2.3, pg. 285], Hom1 Gal(k/k¯ ), Aut (X ) is not necessarily a group since Aut (X ) is not necessarily Abelian. For nonabelian Galois cohomology we refer to [18]. Fix an integer n ≥ 2. Let k be a number ﬁeld which contains all primitive n-th roots of unity ξn and Ok its ring of integers. Consider a smooth superelliptic curve n X deﬁned over Ok with equation y = f(x) where f is a separable polynomial in k. The multiplicative group of n-th roots of unity will be denoted by µn. Then, µn is embedded in Aut (X ) in the obvious way. Hence, Aut (X ) is an extension of the group µn. The list of isomorphism classes of such groups is determined; see for example [10]. Thus, determining the set Twist (X /k) is equivalent to determining Hom1 Gal(k/k¯ ), Aut (X ) for each possible group Aut (X ).

Lemma 1. Let X be as in Eq. (1). Any twist X 0 of X has equation λyn = f(x), 1/n 0 for some λ ∈ k such that√ λ ∈ k. Moreover, the isomorphism φ : X −→ X is given by (x, y) −→ (x, n λ y).

Proof. Two curves X and X 0, with equations yn = f(x) and vn = g(u), where deg f = deg g = s, are isomorphic over k¯, if and only if

au + b λ v a b x = , y = , where ∈ GL (k), λ ∈ k∗ cu + d (cu + d)s/n c d 2

Let f(x) be given by

s−1 s f(x) = a0 + a1x + ··· + as−1x + asx , REDUCTION OF SUPERELLIPTIC RIEMANN SURFACES 13 then au + b au + b au + bs 1 f = a + a + ··· + a = g(u), cu + d 0 1 cu + d s cu + d (cu + d)s where g(u) is of degree s in u. hence, the equation of the curve becomes (cu + s n n n d) y = g(u). Replacing y as above we get λ v = g(u). Theorem 7. Minimal twists of minimal models of superelliptic curves exist. m d−m An equation X : z y = f(x, y) is a minimal twist over Ok, if for every prime p ∈ Ok which divides p | wgcd (I(f)), the valuation valp of I(f) at p satisﬁes d (9) val (I(f)) < q , p 2 i for all i = 0, . . . , n. Moreover, then for λ = wgcd(I(f)) with respect the weights j dq k j dq k d 0 n x m ( 2 ,..., 2 )the transformation (x, y, z) → λ , y, λ z gives the minimal model of X over Ok. If m|d then this isomorphism is deﬁned over k.

This equation has coeﬃcients in Ok. Its weighted moduli point is

M 1 I(f ) = d ? I(f), λ 2 which satisﬁes Eq. (9). It is a twist of the curve X since λd is not necessary a m-th d power in Ok. The isomorphism of the curves over the ﬁeld k λ m is given by

d x m 0 (x, y, z) → λ , y, λ z . If m|d then this isomorphism is deﬁned over k and X has equation 0 m d−m d d−1 d−1 d−1 d d X : z y = adx + λad−1x y + ··· + λ a1xy + λ a0y This completes the proof.

4. Reduction b): Reduction of coeﬃcients In this section we will focus on the reduction of coeﬃcients of a binary form and equivalently a superelliptic curve. The method has an interesting story on its own culminating with Julia’s thesis in [12]. First we go over some of the basic properties of heights of polynomials and binary forms. Standard references on the theory of heights are [9] and [11]. 14 TANUSH SHASKA

4.1. Heights of polynomials. A non-homogenous polynomial with n variables will be denoted as f(x , . . . , x ) = P a xi1 ··· xin , where all 1 n i=(i1,...,in)∈I i 1 n ≥0 ai ∈ K, I ⊂ Z , and I is finite. Let deg f denote the total degree of f. We will use lexicographic ordering to order the terms in a given polynomial, and x1 > x2 > ··· > xn. The (affine) multiplicative height of f is defined as n o n o A Y nv HK (f) = max 1, |f|v , where |f|v := max |aj|v j v∈MK is called the Gauss norm for any absolute value v. Hence, the affine height of a polynomial is defined to be the height of its coefficients taken as affine coordinates. The (projective) multiplicative height of a polynomial is the height of its coefficients taken as coordinates in the projective space. Thus, HK (f) = Q |f|nv . v∈MK v The (projective) absolute multiplicative height is defined as follows H : n 1/[k: ] A P (Q) → [1, ∞) such that H(f) = Hk(f) Q , and in the same way h(f), H (f), hA(f). Lemma 2. i) Let f ∈ k[x, y]. Then there are only finitely many polynomials g ∈ k[x, y] such that Hk(g) ≤ Hk(f). ii) Let f(x0, . . . , xn) and g(y0, . . . , yn) be polynomials in different variables. Then, H(f · g) = H(f) · H(g). The following lemma is true for the product of a finite number of polynomials.

Lemma 3 (Gauss’s lemma). Let k be a number ﬁeld and f, g ∈ k[x1, . . . , xn]. If v is not Archimedean, then |fg|v = |f|v|g|v. The proof can be found in [9, pg. 22]. Gauss’s lemma applies to all non- Archimedean absolute values but the Archimedean case is more complicated. 4.1.1. Homogenous polynomials. Next we focus on homogenous polynomials. The following gives a bound for the homogenous polynomial evaluated at a point.

Lemma 4. Let k be a number ﬁeld, f ∈ k[x0, . . . , xn] a homogenous polynomial n+1 of degree d, and α = (α0, . . . , αn) ∈ k . Then the following hold: d n+d i) |f(α)|v ≤ |c(d, n)|v · maxj |αj|v · |f|v, where |c(d, n)|v is d if v is non-Archimedean and 1 otherwise. d ii) H(f(α)) ≤ c0 · H(α) · H(f). The following is an immediate corollary. Corollary 7. Let k be a number ﬁeld, f ∈ k[x, z] a homogenous polynomial Pd i d−i 2 of degree d as f(x, z) = i=1 aix z and α = (α0, α1) ∈ k . Then, H(f(α)) ≤ min d + 1, 2d+1 · H(α)d · H(f).

We will use Lemma4 to bound the height of the SL 2(k) invariants on Vd.

Theorem 8. Let M ∈ GL2(k), f ∈ Vd(k), and H(f) the absolute height of f. Then, H(f M ) ≤ 2n · (n + 1) · H(M)n · H(f) Next we follow a diﬀerent approach. Theorem 9. Let k be an algebraic number ﬁeld, and f, f¯ as above. The following are true: REDUCTION OF SUPERELLIPTIC RIEMANN SURFACES 15

i) For any valuation v ∈ Mk we have

¯ d d d n o |f|v ≤ 2v · c(d)v · |u|v · |w|v · max |bi|v 0≤i≤d ii) The height of the form is bounded as follows

H(f¯) ≤ (d + 1) · 2d · ud · wd · H(f)

Let f(x, y) be a binary form and Orb(f) its GL2(k)-orbit in Vd. Let H(f) be the height of f as defined in the previous section. Note that there are only finitely many f 0 ∈ Orb(f) such that H(f 0) ≤ H(f). Define the height of the binary form f(x, y) as follows n o H(˜ f) := min H(f 0)|f 0 ∈ Orb(f), H(f 0) ≤ H(f) .

We want to consider the following problem. For every f let f 0 be the binary form 0 0 such that f ∈ Orb(f) and H(˜ f) = H(f ). Determine a matrix M ∈ GL2(k) such that f 0 = f M .

4.2. Quadratic forms. A quadratic form over R is a function Q : Rn → R that has the form Q(x) = xT Ax where A is a symmetric n × n matrix called the matrix of the quadratic form. Let F , G be quadratic forms and AF , AG their corresponding matrices. Then, F ∼ G if and only if AF is similar to AG. Let Q(x, y) = ax2 + bxy + cy2 be a binary quadratic in R[x, y]. We will use the following notation to represent the binary quadratic, Q(x, y) = [a, b, c]. The discriminant of Q is ∆ = b2 − 4ac and Q(x, y) is positive definite if a > 0 and + ∆ < 0. Denote the set of positive definite binary quadratics with V2 (R). Let + SL2(R) acts as usual on the set of positive definite binary quadratic forms V2 (R). + Consider the following map ξ : V2 (R) → H2, which is called the zero map: √ −b + ∆ (11) [a, b, c] → ξ(Q) = . 2a It is a bijection since given z = x+iy, we can find a, b, c such that Q(x, y) is positive definite given as [1, −2x, x2 + y2]. The map ξ gives us a one to one correspondence between positive definite quadratic forms and points in H2. Let Γ be the modular + group acting on H2 and on V2 as described above. Lemma 5. The map ξ is a Γ-equivariant map (i.e. ξ(QM ) = M −1ξ(Q)).

+ Define Q ∈ V2 (R) to be reduced if ξ(Q) ∈ F. Theorem 10. The following are true: + i) A quadratic form Q ∈ V2 (R) is reduced if and only if |b| ≤ a ≤ c. ii) Let Q be reduced with discriminant ∆ = −D. Then, b ≤ pD/3. iii) The number of reduced forms of a fixed discriminant ∆ = −D is finite. + iv) Every Q ∈ V2 (R) is equivalent to a reduced form of the same discriminant.

Theorem 11. Let f(x, y) = ax2 + bxy + cy2 be reduced. Then, H([f]) = c. Moreover, if f is reduced, then f has minimal height H(f) in its Γ-orbit. 16 TANUSH SHASKA

4.3. Reduction of higher degree binary forms in their Γ-orbit. Let f ∈ Vn(k) and ΓOK := SL2(OK )/{±I}. To every binary form f it is associated a positive deﬁnite quadratic Jf called the Julia quadratic. In [12] is proved that this is a covariant of the degree n binary forms and we can develop reduction theory using this quadratic. A degree n binary form is called reduced when ξ(Jf ) is in the fundamental domain of the action of the modular group Γ on H2. Let f ∈ Vn as in Eq. (1) the real roots of f(x, y) be αi, for 1 ≤ i ≤ r and the ¯ pair of complex roots βj, βj for 1 ≤ j ≤ s, where r + 2s = n. Then r s Y Y ¯ (12) f(x, 1) = (x − αi) · (x − βi)(x − βi). i=1 i=1 The ordered pair (r, s) of numbers r and s is called the signature of f. We associate to f the two quadratic forms Tr(x, 1) and Ss(x, 1) of degree r and s respectively given by the formulas r s X 2 2 X 2 ¯ (13) Tr(x, 1) = ti (x − αi) , and Ss(x, 1) = 2uj (x − βj)(x − βj), i=1 j=1 where ti, uj are to be determined. Then, r ! r ! r ! X 2 2 X 2 X 2 2 Tr(x, 1) = ti x − 2 ti αi x + ti αi i=1 i−1 i−1 (14)  s   s   s  X 2 2 X 2 X 2 2 Ss(x, 1) = 2  uj  x − 4  uj Re(βj) x + 2  uj · ||βj||  . j=1 j=1 j=1

For a binary form f(x, 1) of signature (r, s) the quadratic form Qf is deﬁned as

(15) Qf (x, 1) = Tr(x, 1) + Ss(x, 1)

The discriminant of Qf is a degree 4 homogenous polynomial in t1, . . . tr, u1, . . . , us. We would like to pick values for t1, . . . tr, u1, . . . , us such that this discriminant is square free and minimal. Then we can use the reduction theory of quadratics (with square free, minimal discriminant) to determine the reduced form for Qf . For quadratics T and S in Eq. (13) we deﬁne 2 2 a0 · ∆T a0 · ∆S (16) θT = 2 2 , θS = 4 4 t1 ··· tr u1 ··· us The case of cubic and quartic forms were discussed in detail in [12]. Notice that Tr and Ss are given recursively as 2 2 4 2 2 2 Tr = Tr−1 + tr(x − αr)) ,Ss = Ss−1 + us x − 2asx + (as + bs)

Proposition 5. Let f ∈ Vn,R with signature (r, s) and equation as in Eq. (12). Then Qf is a positive deﬁnite quadratic form with discriminant Df given by the formula X 2 2 2 2 (17) Df =∆(Tr) + ∆(Ss) − 8 ti uj (αi − aj) + bj i,j

We deﬁne the θ0 of a binary form as follows 2 n/2 a0 · |Df | (18) θ0(f) := Qr 2 Qs 4 . i=1 ti j=1 uj REDUCTION OF SUPERELLIPTIC RIEMANN SURFACES 17

Consider θ0(t1, . . . , tr, u1, . . . , us) as a multivariable function in the variables t1, . . . , tr, u1, . . . , us. We would like to pick such variables such that Qf is a reduced quadratic, hence Df is minimal. This is equivalent with θ0(t1, . . . , tr, u1, . . . , us) obtaining a minimal value.

r+s Proposition 6. The function θ0 : R → R obtains a minimum at a unique point (t¯1,..., t¯r, u¯1,..., u¯s). Proof. Julia in his thesis [12] proves existence and Stoll and Cremona prove uniqueness in [24]. Choosing (t¯1,..., t¯r, u¯1,..., u¯s) that make θ0 minimal gives a unique positive deﬁnite quadratic Qf (x, y). We call this unique quadratic Qf (x, y) for such a choice of (t¯1,..., t¯r, u¯1,..., u¯s) the Julia’s quadratic of f(x, y), denote it by Jf (x, y), and the quantity θf := θ0(t¯1,..., t¯r, u¯1,..., u¯s) the Julia invariant. From the previous remarks, this is well deﬁned.

Lemma 6. Consider GL2(C) acting on Vn,R. Then, θ is an invariant and J is a covariant of order 2.

4.3.1. Minimal height of a binary form in its SL2(OK )-orbit. Consider f a degree n binary form and K its minimal ﬁeld of deﬁnition. Let M ∈ SL2(OK ) be M ¯ M a matrix such that f is reduced, i.e. ξ(f ) ∈ FK where FK is the fundamental domain of SL2(OK ) acting on H3. A bound on the height of the reduced binary form with respect to Julia invariant is given below. Qn Lemma 7. Let f(x, 1) = a0 i=1(x − αi) be a reduced binary form where αi are the roots. Then, the height of this form can be bounded by Julia’s invariant as

n2 n(n−1) 4 2 n/2 1 4 1 H(f) ≤ c · θf , where c = n 3 n − 1 a0 Let f be a binary form and K its minimal field of definition. If f is reduced over K, then it has minimal height in its ΓK -orbit. Let f be a degree n binary form defined over K and Jf its Julia quadratic, Df its discriminant, and L = K(Df ). Then, [L : K] ≤ n. Let r be the class number of Jf over L and M1,...,Mr the matrices with Mj entries in SL2(OK ) that send Jf respectively to {J1,...,Jr}. The form f for some j = 1, . . . , r has minimal height over SL2(OK ). See [5] for a proof.

5. Concluding remarks The methods of this paper are new and give a new approach for bookkeeping of point in the moduli space of superelliptic curves, provided that we have explicit descriptions of the invariants. For example, for genus g = 2, since we know explicitly the Igusa arithmetic invariants J2,J4,J6,J10 we can explicitly list all points in 3 WP2(k) of a given weighted moduli height, including their twists. This makes it possible to study the arithmetic of the moduli space M2 and its rational points. A similar approach can be used for any moduli space of curves when the corresponding invariants are explicitly known. To our knowledge, the only time when these two types of reduction have been combined is in seminal work of Birch and Swinnerton- Dyer in [7,8] for their computation with elliptic curves in an attempt to verify their famous conjecture. 18 TANUSH SHASKA

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Department of Mathematics and Statistics, Oakland University, Rochester, MI. 48309, USA. E-mail address: [email protected]