Theta functions and algebraic curves with automorphisms

Transcription

1 arxiv:.68v [math.ag 5 Oct Theta functions and algebraic curves with automorphisms T. Shaska and G.S. Wijesiri AbstractLet X be an irreducible smooth projective curve of genus g defined over the complex field C. Then there is a covering π : X P where P denotes the projective line. The problem of expressing branch points of the covering π in terms of the transcendentals (period matrix thetanulls e.g.) is classical. It goes back to Riemann Jacobi Picard and Rosenhein. Many mathematicians including Picard and Thomae have offered partial treatments for this problem. In this work we address the problem for cyclic curves of genus 3 and and find relations among theta functions for curves with automorphisms. We consider curves of genus g > admitting an automorphism σ such that X σ has genus zero and σ generates a normal subgroup of the automorphism group Aut(X ) of X. To characterize the locus of cyclic curves by analytic conditions on its Abelian coordinates in other words theta functions we use some classical formulas recent results of Hurwitz spaces and symbolic computations especially for genera and 3. For hyperelliptic curves we use Thomae s formula to invert the period map and discover relations among the classical thetanulls of cyclic curves. For non hyperelliptic curves we write the equations in terms of thetanulls. Fast genus curve arithmetic in the Jacobian of the curve is used in cryptography and is based on inverting the moduli map for genus curves and on some other relations on theta functions. We determine similar formulas and relations for genus 3 hyperelliptic curves and offer an algorithm for how this can be done for higher genus curves. It is still to be determined whether our formulas for g = 3 can be used in cryptographic applications as in g =. Keywords. Theta functions Riemann surfaces theta-nulls automorphisms.. Introduction to Theta Functions of Curves Let X be an irreducible smooth projective curve of genus g defined over the complex field C. We denote the moduli space of genus g by M g and the hyperelliptic locus in M g by H g. It is well known that dim M g = 3g 3 and H g is a (g ) dimensional subvariety of M g. Choose a symplectic homology basis for X say {A... A g B... B g } such that the intersection products A i A j = B i B j = and A i B j = δ ij. We choose a basis {w i } for the space of holomorphic -forms such that A i w j = δ ij

2 [ where δ ij is the Kronecker delta. The matrix Ω = w Bi j is the period matrix of X. The columns of the matrix [I Ω form a lattice L in C g and the Jacobian of X is Jac (X ) = C g /L. Let H g = {τ : τ is symmetric g g matrix with positive definite imaginary part} be the Siegel upper-half space. Then Ω H g. The group of all g g matrices M GL g (Z) satisfying M t JM = J with J = ( ) Ig I g ( ) R S is called the symplectic group and denoted by Sp g (Z). Let M = T U Sp g (Z) and τ H g where R S T and U are g g matrices. Sp g (Z) acts transitively on H g as M(τ) = (Rτ + S)(T τ + U). Here the multiplications are matrix multiplications. There is an injection M g H g /Sp g (Z) =: A g where each curve C (up to isomorphism) goes to its Jacobian in A g. If l is a positive integer the principal congruence group of degree g and of level l is defined as a subgroup of Sp g (Z) by the condition M I g mod l. We shall denote this group by Sp g (Z)(l). For any z C g and τ H g the Riemann s theta function is defined as θ(z τ) = u Z g e πi(ut τu+u t z) where u and z are g-dimensional column vectors and the products involved in the formula are matrix products. The fact that the imaginary part of τ is positive makes the series absolutely convergent over every compact subset of C g H g. The theta function is holomorphic on C g H g and has quasi periodic properties θ(z + u τ) = θ(z τ) and θ(z + uτ τ) = e πi(ut τu+z t u) θ(z τ) where u Z g ; see [ for details. The locus Θ := {z C g /L : θ(z Ω) = } is called the theta ( ) divisor of X. Any point e Jac (X ) can be uniquely written g as e = (b a) where a b R Ω g are the characteristics of e. We shall use the [ a notation [e for the characteristic of e where [e =. For any a b Q b g the

3 theta function with rational characteristics is defined as a translate of Riemann s theta function multiplied by an exponential factor [ a θ (z τ) = e b πi(at τa+a t (z+b)) θ(z + τa + b τ). () By writing out Eq. () we have [ a θ (z τ) = b e πi((u+a)t τ(u+a)+(u+a) t (z+b)). u Z g [ The Riemann s theta function is θ. The theta function with rational charac- teristics has the following properties: [ a + n θ (z τ) = e πiatm θ b + m [ a θ (z + m τ) = e πiatm θ b θ [ a b (z τ) [ a (z τ) b [ a b (z + τm τ) = e πi( bt m m t τm m t z) θ [ a b (z τ) where n m Z n. All of these properties are immediately verified by writing them out. A scalar obtained by evaluating a theta function with characteristic at z = is called a theta constant or thetanulls. When the entries [ of column vectors a and a b are from the set { } then the characteristics are called the half-integer b characteristics. The corresponding theta functions with rational characteristics are called theta characteristics. Points of order n on Jac (X ) are called [ the n - a periods. Any point p of Jac (X ) can be written as p = τ a + b. If is a b n - period then a b ( n Z/Z)g. The n-period p can be associated with an element of H (X Z/nZ) as follows: Let a = (a a g ) t and b = (b b g ) t. Then p = τa + b = ( ai B i ω a i = ( (ai B i ω + b i = ( ) t ω ω g C C B i ω g ) t + ( b A i ω ) (a i A ω b g B i ω g + b i A g ω g ) A i ω g ) ) t where C = a i B i +b i A i. We identify the point p with the cycle C H (X Z/nZ) where C = ā i B i + b i A i ā i = na i and b i = nb i for all i. ()

4 .. Half-Integer Characteristics and the Göpel Group In this section we study groups of half-integer characteristics. Any half-integer characteristic m Zg /Z g is given by m = m = ( m m m g m m m g [ m where m i m i Z. For m = m Zg /Z g we define e (m) = ( ) (m ) t m. We say that m is an even (resp. odd) characteristic if e (m) = (resp. e (m) = ). For any curve of genus g there are g ( g + ) (resp. g ( g ) ) even theta functions (resp. odd theta functions). Let a be another half-integer characteristic. We define ( ) t t t g ) m a = t t t g where t i (m i + a i ) mod and t i (m i + a i ) mod. For the rest of the thesis we only consider characteristics q in which each of the elements q i q i is either or. We use the following abbreviations: m = g m i m i m a = i= m a b = a b + b m + m a g (m ia i m i a i) i= ( ) m = e πi g j= mja j. a The set of all half-integer characteristics forms a group Γ which has g elements. We say that two half integer characteristics m and a are syzygetic (resp. azygetic) if m a mod (resp. m a mod ) and three half-integer characteristics m a and b are syzygetic if m a b mod. A Göpel group G is a group of r half-integer characteristics where r g such that every two characteristics are syzygetic. The elements of the group G are formed by the sums of r fundamental characteristics; see [ pg. 89 for details. Obviously a Göpel group of order r is isomorphic to C r. The proof of the following lemma can be found on [ pg. 9. Lemma. The number of different Göpel groups which have r characteristics is ( g )( g ) ( g r+ ) ( r )( r. ) ( ) If G is a Göpel group with r elements it has g r cosets. The cosets are called Göpel systems and are denoted by ag a Γ. Any three characteristics of a Göpel system are syzygetic. We can find a set of characteristics called a basis of the Göpel system which derives all its r characteristics by taking only combinations of any odd number of characteristics of the basis.

5 Lemma. Let g be a fixed integer r be as defined above and σ = g r. Then there are σ ( σ + ) Göpel systems which only consist of even characteristics and there are σ ( σ ) Göpel systems which consist of odd characteristics. The other σ ( r ) Göpel systems consist of as many odd characteristics as even characteristics. Proof. The proof can be found on [ pg. 9. Corollary. When r = g we have only one (resp. ) Göpel system which consists of even (resp. odd) characteristics. Let us consider s = σ Göpel systems which have distinct characters. Let us denote them by We have the following lemma. a G a G a s G. Lemma 3. It is possible to choose σ + characteristics from a a a s say ā ā ā σ+ such that every three of them are azygetic and all have the same character. The above σ + fundamental characteristics are even (resp. odd) if σ mod (resp. 3 mod ). The proof of the following lemma can be found on [ pg. 5. Lemma. For any half-integer characteristics a and h we have the following: θ [a(z τ)θ [ah(z τ) = ( ) h g e πi ae θ [e(z τ)θ [eh(z τ). (3) ae e We can use this relation to get identities among half-integer theta constants. Here e can be any half-integer characteristic. We know that we have g ( g + ) even characteristics. As the genus increases we have multiple choices for e. In the following we explain how we reduce the number of possibilities for e and how to get identities among theta constants. First we replace e by eh and z = z = in Eq. (3). Eq. (3) can then be written as follows: θ [aθ [ah = ( ) h g e πi aeh θ [eθ [eh. () aeh e We have e πi aeh ( ) h ( aeh = e πi ae h ae) e πi aeh. Next we put z = z = in Eq. (3) and add it to Eq. () and get the following identity: θ [aθ [ah = g e e πi ae ( + e πi aeh )θ [eθ [eh. (5) If ae h mod the corresponding terms in the summation vanish. Otherwise + e πi aeh =. In this case if either e is odd or eh is odd the corresponding terms in the summation vanish again. Therefore we need ae h mod and

6 e eh mod in order to get nonzero terms in the summation. If e satisfies e e h mod for some h then e h is also a candidate for the left hand side of the summation. Only one of such two values e and e h is taken. As a result we have the following identity among theta constants θ [aθ [ah = ( ) h g e πi ae θ [eθ [eh (6) ae e where a h are any characteristics and e is a characteristics such that ae h mod e eh mod and e eh. By starting from the Eq. (3) with z = z and following a similar argument to the one above we can derive the identity θ [a + e πi ah θ [ah = g e πi ae {θ [e + e πi ah θ [eh} (7) where a h are any characteristics and e is a characteristic such that h + e h mod e eh mod and e eh. Remark. ae h mod and eh e mod implies a h + h mod. We use Eq. (6) and Eq. (7) to get identities among thetanulls in Chapter and in Chapter 3... Hyperelliptic Curves and Their Theta Functions e A hyperelliptic curve X defined over C is a cover of order two of the projective line P. Let z be the generator (the hyperelliptic involution) of the Galois group Gal(X /P ). It is known that z is a normal subgroup of the Aut(X ) and z is in the center of Aut(X ). A hyperelliptic curve is ramified in (g + ) places w w g+. This sets up a bijection between isomorphism classes of hyperelliptic genus g curves and unordered distinct (g+)-tuples w w g+ P modulo automorphisms of P. An unordered (g + )-tuple {w i } g+ i= can be described by a binary form (i.e. a homogenous equation f(x Z) of degree g + ). To describe H g we need rational functions of the coefficients of a binary form f(x Z) invariant under linear substitutions in X and Z. Such functions are called absolute invariants for g = ; see [7 for their definitions. The absolute invariants are GL (C) invariants under the natural action of GL (C) on the space of binary forms of degree g +. Two genus g hyperelliptic curves are isomorphic if and only if they have the same absolute invariants. The locus of genus g hyperelliptic curves with an extra involution is an irreducible g-dimensional subvariety of H g which is denoted by L g. Finding an explicit description of L g means finding explicit equations in terms of absolute invariants. Such equations are computed only for g = ; see [7 for details. Writing the equations of L in terms of theta constants is the main focus of Chapter. Computing similar equations for g 3 requires first finding the corresponding absolute invariants. This is still an open problem in classical invariant theory even for g = 3.

7 Let X P be the degree hyperelliptic projection. We can assume that is a branch point. Let B := {α α α g+ } be the set of other branch points. Let S = { g + } be the index set of B and η : S Zg /Z g be a map defined as follows: [ η(i ) = [ η(i) = where[ the nonzero element of the first row appears in i th column. We define η( ) to be. For any T B we define the half-integer characteristic as η T = η(k). a k T Let T c denote the complement of T in B. Note that η B Z g. If we view η T as an element of Zg /Z g then η T = η T c. Let denote the symmetric difference of sets that is T R = (T R) (T R). It can be shown that the set of subsets of B is a group under. We have the following group isomorphism: For γ = {T B #T g + mod }/T T c = Zg /Z g. [ γ γ Zg /Z g we have θ[γ( z τ) = e (γ)θ[γ(z τ). (8) It is known that for hyperelliptic curves g ( g + ) ( ) g+ g of the even theta constants are zero. The following theorem provides a condition for the characteristics in which theta characteristics become zero. The proof of the theorem can be found in [. Theorem. Let X be a hyperelliptic curve with a set B of branch points. Let S be the index set as above and U be the set of all odd values of S. Then for all T S with even cardinality we have θ[η T = if and only if #(T U) g + where θ[η T is the theta constant corresponding to the characteristics η T. When the characteristic γ is odd e (γ) =. Then from Eq. (8) all odd theta constants are zero. There is a formula which satisfies half-integer theta characteristics for hyperelliptic curves called Frobenius theta formula.

8 Lemma 5 (Frobenius). For all z i C g i such that z + z + z 3 + z = and for all b i Q g i such that b + b + b 3 + b = we have j S { } ɛ U (j) θ[b i + η(j)(z i ) = i= where for any A B ɛ A (k) = { if k A otherwise. Proof. See [ pg. 7. A relationship between theta constants and the branch points of the hyperelliptic curve is given by Thomae s formula. Lemma 6 (Thomae). For all sets of branch points B = {α α α g+ } there is a constant A such that for all T B #T is even θ[η T (; τ) = ( ) #T U A i<j ij T U (α i α j ) i<j ij / T U (α i α j ) where η T is a non singular even half-integer characteristic corresponding to the subset T of branch points. See [ pg. 8 for the description of A and [ pg. for the proof. Using Thomae s formula and Frobenius theta identities we express the branch points of the hyperelliptic curves in terms of even theta constants..3. Cyclic Curves and Their Theta Functions A cyclic cover X P is defined to be a Galois cover with cyclic Galois group C. We call it a normal cyclic cover of P if C is normal in G = Aut(X ) where Aut(X ) is the automorphism group of the curve X. Then Ḡ = G/C embeds as a finite subgroup of P GL( C) and it is called the reduced automorphism group of G. An affine equation of a cyclic curve can be given by the following: y m = f(x) = s (x α i ) di m = C < d i < m. (9) i= Note that when d i > for some i the curve is singular. Hyperelliptic curves are cyclic curves with m =. After Thomae many mathematicians for example Fuchs Bolza Fay Mumford et al. gave derivations of Thomae s formula in the hyperelliptic case. In 988 Bershdaski and Radul found a generalization of Thomae s formula for Z N curves of the form

9 y N = f(x) = Nm i= (x a i ). () In 988 Shiga showed the representation of the Picard modular function by theta constants. He considered the algebraic curve in the (x y) plane which is given by C(ɛ) : y 3 = x(x a )(x a )(x a ) () where ɛ = [a a a is a parameter on the domain Λ = {ɛ : a a a (a a )(a a )(a a ) }. He gave a concrete description of the Picard work [. His result can be considered an extension of the classical Jacobi representation λ = θ where θ θ3 i (z τ) indicates Jacobi s theta function and θ i is the convention for θ i ( τ) for the elliptic modular function λ(τ) to the special case of genus 3. In 99 Gonzalez Diez studied the moduli spaces parameterizing algebraic curves which are Galois covering of P with prime order and with given ramification numbers. These curves have equation of the form y p = f(x) = r (x a i ) mi ; p prime and p m i. () i= He expresses a i in terms of functions of the period matrix of the curve. Farkas (996) gave a procedure for calculating the complex numbers a i which appear in the algebraic equation y p = k (x a i ) with p k (3) i= in terms of the theta functions associated with the Riemann surface of the algebraic curve defined by the Eq. (3). He used the generalized cross ratio of four points according to Gunning. Furthermore he considered the more general problem of a branched two-sheeted cover of a given compact Riemann surface and obtained the relations between the theta functions on the cover and the theta function to the original surface. Nakayashiki in 997 gave an elementary proof of Thomae s formula for Z N curves which was discovered by Bershadsky and Radul. Enolski and Grava in 6 derived the analogous generalized Thomae s formula for the Z N singular curve of the form y N = f(x) = m m (x λ i ) N (x λ i+ ). () i= i= We summarize all the results in the following theorem.

10 Theorem. Consider the algebraic curve X : y n = f(x) defined over the complex field C. Case : If f say f(x) = k i= (x λ i) then i) If n k say k = mn for some m N then for an ordered partition Λ = (Λ Λ n ) of { nm} we have θ[e Λ () n = C Λ (deta) n i<j (λ i λ j ) n l L q l(k i)q l (k j)+ (n )(n ) 6 where k i = j for i Λ j and e Λ Λ + Λ + + (n )Λ n D ς is the associated divisor class of the partition Λ L = { N } N + N q l (i) = N N + fraction part of ( l+i+ N ) N for l L ς is Riemann s constant and C Λ depends on the partition Λ having the property that for two different partitions Λ and Λ we have CΛ N = CN Λ. Moreover if n is a prime p the branch points λ i of the curve y n = x(x )(x λ ) (x λ k 3 ) can be given by E n i λ i = (λ(p k Q Q Q )) n where λ(p k Q Q Q ) = θ(e+φ Q (P k ))θ(e+φ Q (Q )) θ(e+φ Q (P k ))θ(e+φ Q (Q )) while Q Q and Q denote the points in the curve corresponding to the points and in P respectively P i s are points in the curve corresponding to the points λ i E i is a constant depending on the point P i and φ P is an injective map from X to C g /G. ii) If n k then if n = 3 and k = then the parameters λ λ λ 3 can be given as follows: [ λ = θ 3 6 [ 6 λ = θ [ λ 3 = θ Case : If f = let f(x) = m k= (x λ k+) m k= (x λ k) n. Then θ[e m (; Ω) N = N i= det AN i (πi) mn(n ) i<k m i<k m (λ i λ k ) N(N ) (λ i+ λ k+ ) N(N ). i I ( j J (λ i λ j ) i I j J (λ i λ j ) i I k I (λ i λ k ) j J k J (λ i λ j ) )(N ) where e m = ν((n ) i I P i + (N ) j J P j D ) is a nonsingular N characteristic J J = { m + } and I I = { 3 m + } with J + I = m + and I = I I J = J J m + and = (N ) m k= P k P is the Riemann divisor of the curve X.

11 Proof. For proof of the part i) of case see [. When n is prime the proof can be found in [. The main point of [9 is to prove part ii) of case. The proof of case can be found in [ Relations Among Theta Functions for Algebraic Curves with Automorphisms In this section we develop an algorithm to determine relations among theta functions of a cyclic curve X with automorphism group Aut(X ). The proof of the following lemma can be found in [6. Lemma 7. Let f be a meromorphic function on X and let m m (f) = b i i= be the divisor defined by f. Take paths from P (initial point) to b i and P to c i so that m bi i= P ω = m ci i= P ω. For an effective divisor P + + P g we have i= c i f(p ) f(p g ) = E θ( i k= θ( i Pi P Pi P ω b k P ω τ) ω c k P ω τ) (5) where E is a constant independent of P... P g the integrals from P to P i take the same paths both in the numerator and in the denominator denotes the Riemann s constant and ( P i Pi P ω = ω... ) P t i P ω g. P This lemma gives us a tool that can be used to find branch points in terms of theta constants. By considering the meromorphic function f = x on X and suitable effective divisors we can write branch points as ratios of thetanulls. We present some explicit calculations using the Lemma 7 in Chapter 3 and. The hard part of this method is the difficulty of writing complex integrals in terms of characteristics. Algorithm. Input: A cyclic curve X with automorphism group G σ G such that σ = n g(x σ ) = and σ G. Output: Relations among the theta functions of X Step : Let Γ = G/ σ and pick τ Γ such that τ has the largest order m. Step : Write the equation of the curve in the form y n = f(x m ) or y n = xf(x m ). Step 3: Determine the roots λ... λ r of f(x τ ) in terms of the theta functions. Step : Determine relations on theta functions using Gröbner basis techniques.

12 For step 3 we can use Lemma 7. If the curve in step 3 falls into one of the categories given in Theorem we can use the corresponding equation to invert the period map without worrying about the complex integrals.. Genus curves Let k be an algebraically closed field of characteristic zero and X be a genus curve defined over k. Consider a binary sextic i.e. a homogeneous polynomial f(x Z) in k[x Z of degree 6: f(x Z) = a 6 X 6 + a 5 X 5 Z + + a Z 6. The polynomial functions of the coefficients of a binary sextic f(x Z) invariant under linear substitutions in X Z of determinant one. These invariants were worked out by Clebsch and Bolza in the case of zero characteristic and generalized by Igusa for any characteristic different from. Igusa J-invariants {J i } of f(x Z) are homogeneous polynomials of degree i in k[a... a 6 for i = 3 5; see [7 for their definitions. Here J is the discriminant of f(x Z). It vanishes if and only if the binary sextic has a multiple linear factor. These J i are invariant under the natural action of SL (k) on sextics. Dividing such an invariant by another invariant with the same degree gives an invariant (eg. absolute invariant) under GL (k) action. The absolute invariants of X are defined in terms of Igusa invariants as follows: i := J J i := 78 J J 3J 6 J 3 i 3 := 86 J J 5. Two genus fields (resp. curves) in the standard form Y = f(x ) are isomorphic if and only if the corresponding sextics are GL (k) conjugate... Half Integer Theta Characteristics For genus two curve we have six odd theta characteristics and ten even theta characteristics. The following are the sixteen theta characteristics where the first ten are even[ and the last six are odd. For simplicity we denote them by θ i (z) a instead of θ i (z τ) where i =... for the even functions and i =... 6 b for the odd functions. [ [ θ (z) = θ (z τ) θ (z) = θ [ θ 3 (z) = θ 3 (z τ) θ (z) = θ [ [ θ 5 (z) = θ [ 5 (z τ) θ 6 (z) = θ 6 [ [ θ 7 (z) = θ 7 (z τ) θ 8 (z) = θ 8 (z τ) (z τ) (z τ) (z τ)

13 [ θ 9 (z) = θ 9 θ (z) = θ [ [ θ 3 (z) = θ 3 (z τ) θ (z) = θ [ [ (z τ) θ (z) = θ (z τ) θ (z) = θ [ [ θ 5 (z) = θ [ 5 (z τ) θ 6 (z) = θ 6 (z τ) (z τ) (z τ) (z τ) Remark. All the possible half-integer characteristics except the zero characteristic can be obtained as the sum of not more than characteristics chosen from the following 5 characteristics: {[ [ [ [ [ }. The sum of all 5 characteristics in the set determines the zero characteristic. Take σ = g r =. Then a Göpel group G contains four elements. The number of such Göpel groups is 5. Let G = { m m m m } be a Göpel group of even characteristics (we have six such groups). Let b b b b be the characteristics such that the G b G b G b b G are all the cosets of the group G. Then each of the systems other than G contains two odd characteristics and two even characteristics. Consider equations given by Eq. (6) and Eq. (7). If h denotes any one of the 3 characteristics m m m m then we have 6 possible characteristics for e which satisfy e h h. They are n b h nh bh where n is a characteristic in the Göpel group other than h and b is an even characteristic chosen from one of the systems b G b G b b G. The following three cases illustrate the possible values for characteristic h and for characteristic e. Without loss of generality we can take only three values for e which give rise to different terms on the right hand side of Eq. (6) and Eq. (7). Case : h = m. Take e { m b } and take a = b. Then from Eq. (6) and Eq. (7) we have ( m b ) ( ) θ [θ [m + e πi bm m θ [m θ [m m θ [b θ [b m = b m θ [ + θ [m + e πi bm [θ [m + θ [m m [θ [b + θ [b m =. Case : h = m. Take e { m b } and take a = b. Then from Eq. (6) and Eq. (7) we have ( m b ) ( ) θ [θ [m + e pii bm m θ [m θ [m m θ [b θ [b m = b m θ [ + θ [m + e pii bm [θ [m + θ [m m [θ [b + θ [b m =.

14 Case 3: h = m m. Take e { m b b } and take a = b b. Then from Eq. (6) and Eq. (7) we have ( ) ( ) m m θ [θ [m m + e pii bbm m m θ [m θ [m b b b b m θ [b b θ [b b m m = θ [ + θ [m m + e pii bbm [θ [m + θ [m [θ [b b + θ [b b m m =. The identities above express the even theta constants in terms of four theta constants; therefore we may call them fundamental theta constants.. Identities of Theta Constants θ[ θ[m θ[m θ[m m. We have only six Göpel groups such that all characteristics are even. The following are such Göpel groups and corresponding identities of theta constants. { [ [ [ [ } i) G = = m = m = m m = is a Göpel group. [ If b = [ b = then the corresponding Göpel systems are given by the following: {[ [ [ [ } G = {[ b G = [ [ [ } {[ [ [ [ } b G = b 3 G = {[ [ [ [ }. Notice that from all four cosets only G has all even characteristics as noticed in Corollary. Using Eq. (6) and Eq. (7) we have the following six identities for the above Göpel group: θ5θ 6 = θθ θθ 3 θ5 + θ6 = θ θ3 + θ θ7θ 9 = θθ 3 θθ θ7 + θ9 = θ + θ3 θ θ 8θ = θθ θ3θ θ8 + θ = θ + θ θ3 θ.

15 These identities express even theta constants in terms of four theta constants. We call them fundamental theta constants θ θ θ 3 θ. Following the same procedure we can find similar identities for each possible Göpel group. { [ [ [ [ } ii) G = = m = m = m m = is a Göpel group. [ [ If b = b = then the corresponding Göpel systems are given by the following: {[ [ [ G = {[ [ [ b G = {[ b G = {[ b 3 G = [ [ [ [ [ [ [ [ } } } }. We have the following six identities for the above Göpel group: { [ [ iii) G = = m = m = [ [ If b = b = by the following: θ3θ = θθ θ8θ θ3 + θ = + θ θ8 θ θ6θ 9 = θθ + θθ 8 θ6 + θ9 = θ θ8 + θ θ 5θ 7 = θθ 8 θθ θ5 + θ7 = θ θ + θ8 θ. [ m m = [ } is a Göpel group. then the corresponding Göpel systems are given {[ [ [ G = {[ [ [ b G = {[ [ b G = {[ [ b 3 G = [ [ } } [ [ } [ [ }.

16 We have the following six identities for the above Göpel group: θθ = θθ 3 θ7θ 9 θ + θ = + θ3 θ7 θ9 θ8θ 5 = θθ 7 θ3θ 9 θ8 + θ5 = θ3 + θ7 θ9 θ 6θ = θθ 9 + θ3θ 7 θ6 + θ = θ θ3 θ7 + θ9. { [ [ iv) G = = m = m = [ [ If b = b = by the following: [ [ m m = } is a Göpel group. then the corresponding Göpel systems are given {[ [ [ G = [ } {[ [ [ b G = [ } {[ [ b G = {[ [ b 3 G = [ [ [ [ } }. We have the following six identities for the above Göpel group: θθ 3 = θθ θ5θ 6 θ + θ3 = + θ θ5 θ6 θ8θ 7 = θθ 5 θθ 6 θ8 + θ7 = θ + θ5 θ6 θ 9θ = θθ 6 + θθ 5 θ9 + θ = θ θ θ5 + θ6. { [ [ v) G = = m = [ [ } m = m m = is a Göpel group. [ If b = [ b = then the corresponding Göpel systems are given by the following: {[ [ G = [ [ } {[ b G = [ [ [ }

17 b G = {[ {[ b 3 G = [ [ [ [ [ [ } }. We have the following six identities for the above Göpel group: θθ 6 = θθ 5 θ7θ 8 θ + θ6 = + θ5 θ7 θ8 θ3θ 9 = θθ 7 θ5θ 8 θ3 + θ9 = θ5 + θ7 θ8 θ θ = θθ 8 θ5θ 7 θ + θ = θ θ5 θ7 + θ8. { [ [ vi) G = = m = [ [ } m = m m = is a Göpel group. [ If b = [ b = then the corresponding Göpel systems are given by the following: {[ [ G = {[ b G = [ [ {[ [ b G = {[ [ b 3 G = [ [ [ [ [ [ [ } } } }. We have the following six identities for the above Göpel group: θθ 3 = θθ θ5θ 6 θ + θ3 = + θ θ5 θ6 θ8θ 7 = θθ 5 θθ 6 θ8 + θ7 = θ + θ5 θ6 θ 9θ = θθ 6 + θθ 5 θ9 + θ = θ θ θ5 + θ6. From now on we consider θ θ θ 3 and θ as the fundamental theta constants.

18 .3. Inverting the Moduli Map Let λ i i =... n be branch points of the genus g smooth curve X. Then the moduli map is a map from the configuration space Λ of ordered n distinct points on P to the Siegel upper half space H g. In this section we determine the branch points of genus curves as functions of theta characteristics. The following lemma describes these relations using Thomae s formula. The identities are known as Picard s formulas. We will formulate a somewhat different proof for Picard s lemma. Lemma 8 (Picard). Let a genus curve be given by Y = X(X )(X λ)(x µ)(x ν). (6) Then λ µ ν can be written as follows: λ = θ θ3 θ µ = θ 3θ8 θ θ ν = θ θ8 θ θ. (7) θ Proof. There are several ways to relate λ µ ν to theta constants depending on the ordering of the branch points of the curve. Let B = {ν µ λ } be the branch points of the curve in this order and U = {ν λ } be the set of odd branch points. Using Lemma 6 we have the following set of equations of theta constants and branch points: θ = A νλ(µ )(ν λ) θ = A µ(µ )(ν λ) θ3 = A µλ(µ λ)(ν λ) θ = A ν(ν λ)(µ λ) θ5 = A λµ(ν )(ν µ) θ6 = A (ν µ)(ν λ)(µ λ) θ7 = A µ(ν )(λ )(ν λ) θ8 = A µν(ν µ)(λ ) θ9 = A ν(µ )(λ )(µ λ) θ = A λ(λ )(ν µ) (8) where A is a constant. By choosing appropriate equations from the set Eq. (8) we have the following: ( θ λ = θ3 ) ( θ µ = 3 θ8 ) ( θ ν = θ8 ). θ θ θ θ θ θ Each value for (λ µ ν) gives isomorphic genus curves. Hence we can choose This completes the proof. λ = θ θ3 θ µ = θ 3θ8 θ θ ν = θ θ8 θ θ. θ

19 .. Automorphism Groups of Curves Let X be a genus curve defined over an algebraically closed field k of characteristic zero. We denote its function field by K := k(x ) and Aut(X ) = Aut(K/k) is the automorphism group of X. In any characteristic different from the automorphism group Aut(X ) is isomorphic to one of the groups given by the following lemma. Lemma 9. The automorphism group G of a genus curve X in characteristic is isomorphic to C C V D 8 D C 3 D 8 GL (3) or + S 5. The case G = + S 5 occurs only in characteristic 5. If G = Z 3 D 8 (resp. GL (3)) then X has equation Y = X 6 (resp. Y = X(X )). If G = C then X has equation Y = X 6 X. For the proof of the above lemma and the description of each group see [7. For the rest of this chapter we assume that char(k) =. One of the main goals of Section. is to describe each locus of genus curves with fixed automorphism group in terms of the fundamental theta constants. We have the following lemma. Lemma. Every genus two curve can be written in the form: where α = θ 8 θ y = x (x ) ( x θ θ3 ) ( θ x θ θ3 + θ θ θ θ α x + θ θ ) 3 θ θ α θ can be given in terms of θ θ θ 3 and θ α + θ + θ θ 3 θ θ 3 θ θ θ Furthermore if α = ± then V Aut(X ). α + =. Proof. Let us write the genus curve in the following form: Y = X(X )(X λ)(x µ)(x ν) where λ µ ν are given by Eq. (7). Let α := θ 8. Then θ Using the following two identities µ = θ 3 α ν = θ θ α. θ θ 8 + θ = θ + θ θ 3 θ θ 8θ = θ θ θ 3θ (9) we have α + θ + θ θ 3 θ θ 3 θ θ θ α + =. ()

20 If α = ± then µν = λ. It is well known that this implies that the genus curve has an elliptic involution. Hence V Aut(X ). Remark 3. i) From the above we have that θ 8 = θ implies that V Aut(X ). Lemma determines a necessary and equivalent statement when V Aut(X ). ii) The last part of Lemma. shows that if θ 8 = θ then all coefficients of the genus curve are given as rational functions of the four fundamental theta functions. Such fundamental theta functions determine the field of moduli of the given curve. Hence the curve is defined over its field of moduli. Corollary. Let X be a genus curve which has an elliptic involution. Then X is defined over its field of moduli. This was the main result of [5..5. Describing the Locus of Genus Two Curves with Fixed Automorphism Group by Theta Constants The locus L of genus curves X which have an elliptic involution is a closed subvariety of M. Let W = {α α β β γ γ } be the set of roots of the binary sextic and A and B be subsets of W such that W = A B and A B =. We define the cross ratio of the two pairs z z ; z 3 z by (z z ; z 3 z ) = z ; z 3 z z ; z 3 z = z z 3 z z : z z 3 z z. Take A = {α α β β } and B = {γ γ β β }. Jacobi [8 gives a description of L in terms of the cross ratios of the elements of W : α β α β : α β α β = γ β γ β : γ β γ β. We recall that the following identities hold for cross ratios: and (α α ; β β ) = (α α ; β β ) = (β β ; α α ) = (β β ; α α ) (α α ; β ) = ( β ; α α ) = (β ; α α ). Next we use this result to determine relations among theta functions for a genus curve in the locus L. Let X be any genus curve given by the equation Y = X(X )(X a )(X a )(X a 3 ). We take A B. Then there are five cases for α A B where α is an element of the set { a a a 3 }. For each of these cases there are three possible relationships for cross ratios as described below: i) A B = { }: The possible cross ratios are

21 (a ; ) = (a 3 a ; ) (a ; ) = (a a 3 ; ) (a ; ) = (a a 3 ; ). ii) A B = { }: The possible cross ratios are (a ; ) = (a a 3 ; ) (a ; ) = (a 3 a ; ) (a ; ) = (a a 3 ; ). iii) A B = {a }: The possible cross ratios are ( ; a ) = (a 3 a ; a ) (a ; a ) = ( a 3 ; a ) ( ; a ) = (a a 3 ; a ). iv) A B = {a }: The possible cross ratios are ( ; a ) = (a a 3 ; a ) ( ; a ) = (a 3 a ; a ) (a ; a ) = ( a 3 ; a ). v) A B = {a 3 }: The possible cross ratios are (a ; a 3 ) = ( a ; a 3 ) ( ; a 3 ) = (a a ; a 3 ) ( ; a 3 ) = (a a ; a 3 ). We summarize these relationships in Table.. Lemma. Let X be a genus curve. Then Aut(X ) = V if and only if the theta functions of X satisfy (θ θ )(θ 3 θ )(θ 8 θ )( θ θ 3θ 8θ θ θ θ θ + θ θ 3θ + θ 3θ θ ) (θ 3θ 8θ θ θ θ θ + θ θ 3θ θ θ 3θ θ )( θ 8θ 3θ + θ 8θ θ θ +θ θ 3θ 8θ θ 3θ θ )( θ θ 8θ θ θ θ + θ 8θ θ θ + θ θ 3θ 8θ ) ( θ θ 8θ 3θ + θ θ θ + θ θ 3θ θ 3θ θ θ )( θ θ 8θ θ + θ θ θ θ θ 3θ θ + θ θ θ )( θ 8θ θ + θ θ 8θ θ θ θ θ + θ 3θ 8θ θ ) () (θ θ 8θ θ θ θ θ θ θ 3θ 8θ + θ 8θ θ )(θ θ 3θ 8 θ θ 8θ θ θ θ 3θ θ + θ 3θ 8θ )(θ θ 8θ 3 θ θ 8θ θ + θ θ 3θ θ 3θ 8θ θ ) (θ θ 8θ θ θ 3θ θ + θ θ 3θ 8 θ 3θ 8θ θ ) =. However we are unable to determine a similar result for cases D 8 or D by this argument. Instead we will use the invariants of genus curves and a more computational approach. In the process we will offer a different proof for the lemma above.

22 Table. Relation of theta functions and cross ratios Cross ratio f(a a a 3 ) = theta constants ( ; a ) = (a 3 a ; a ) a a + a a 3 a a θ θ 3 θ 8 θ θ θ θ θ + θ θ 3 θ + θ 3 θ θ (a ; a ) = ( a 3 ; a ) a a a + a 3 a a 3 a θ3 θ 8 θ θ θ θ θ + θ θ 3 θ θ θ 3 θ θ 3 ( ; a ) = (a a 3 ; a ) a a a a 3 a + a 3 θ8 θ 3 θ + θ 8 θ θ θ + θ θ 3 θ 8 θ θ 3 θ θ ( ; a ) = (a a 3 ; a ) a a a a 3 a + a 3 θ θ 8 θ θ θ θ + θ8 θ θ θ + θ θ 3 θ 8 θ 5 ( ; a ) = (a 3 a ; a ) a a a + a a 3 a θ θ 8 θ 3 θ + θ θ θ + θ θ 3 θ θ 3 θ θ θ 6 (a ; a ) = ( a 3 ; a ) a a a 3 a a + a 3 a θ θ 8 θ θ + θ θ θ θ θ 3 θ θ + θ θ θ 7 (a ; a 3 ) = ( a ; a 3 ) a a a 3 a a 3 a + a 3 θ8 θ θ + θ θ 8 θ θ θ θ θ + θ 3 θ 8 θ θ 8 ( ; a 3 ) = (a a ; a 3 ) a 3 a a a 3 a + a 3 θ8 θ 9 ( ; a 3 ) = (a a ; a 3 ) a 3 a + a a 3 a 3 a θ θ 8 θ θ θ θ θ θ θ 3 θ 8 θ + θ 8 θ θ (a ; ) = (a a 3 ; ) a + a 3 a + a a 3 θ θ 3 θ 8 θ θ 8 θ θ θ θ 3 θ θ + θ 3 θ 8 θ (a ; ) = (a 3 a ; ) a a a a + a 3 θ θ 8 θ 3 θ θ 8 θ θ + θ θ 3 θ θ 3 θ 8 θ θ (a ; ) = (a a 3 ; ) a a + a 3 a a 3 θ θ 8 θ θ θ 3 θ θ + θ θ 3 θ 8 θ 3 θ 8 θ θ 3 (a ; ) = (a 3 a ; ) a a a 3 θ 8 θ (a ; ) = (a a 3 ; ) a a 3 a θ 3 θ 5 (a ; ) = (a a 3 ; ) a 3 a a θ θ Lemma. i) The locus L of genus curves X which have a degree elliptic subcover is a closed subvariety of M. The equation of L is given by 878J J J J J J 956J J J 3 69J 3 J J J J J J 3 J 399J J J J J 3 J J 6 879J J J 3 J J J J J 6 78J 5 J 5 597J 3 8J 3 J J J J J 6 J 6 + 7J J J 6 J 3 795J J J J J J 6 78J 5 J J J J J 6

23 933J J J 6 J 7 J + J 6 J 3 J J J J J 3 J 3 J 6 995J J J 6 + 3J J J 5 J J 6 5J 5 J J J J 5 8J 7 J + 8J J J J J 6 J =. () ii) The locus of genus curves X with Aut(X ) = D 8 is given by the equation of L and 76J J + 56J 3 + 7J J 8J 3 J 6 88J J J J 6 =. (3) iii) The locus of genus curves X with Aut(X ) = D is J J + J 3 J 6 5J J + 8J J J J 6 36J 6 = 86J J J J J 3J J J 3 338J J J 6 768J J + 8J 3 J + 96J 5 =. () Our goal is to express each of the above loci in terms of the theta characteristics. We obtain the following result. Theorem 3. Let X be a genus curve. Then the following hold: i) Aut(X ) = V if and only if the relations of theta functions given Eq. () holds. ii) Aut(X ) = D 8 if and only if the Eq. I in [8 is satisfied. iii) Aut(X ) = D if and only if the Eq. II and Eq. III in [8 are satisfied. Proof. Part i) of the theorem is Lemma. Here we give a somewhat different proof. Assume that X is a genus curve with equation Y = X(X )(X a )(X a )(X a 3 ) whose classical invariants satisfy Eq. (). Expressing the classical invariants of X in terms of a a a 3 substituting them into () and factoring the resulting equation yields (a a a 3 ) (a a 3 a ) (a 3 a a ) (a a a a 3 a + a 3 ) (a 3 a + a a 3 a 3 a ) ( a + a 3 a + a a 3 ) (a a a a + a 3 ) (a a a + a 3 a a 3 a ) (a a a 3 a a 3 a + a 3 ) (a 3 a a a 3 a + a 3 ) (a a + a a 3 a a ) (a a a a 3 a + a 3 ) (a a a + a a 3 a ) (a a + a 3 a a 3 ) (a a a 3 a a + a 3 a ) =. (5) It is no surprise that we get the 5 factors of Table.. The relations of theta constants follow from Table..

24 ii) Let X be a genus curve which has an elliptic involution. Then X is isomorphic to a curve with the equation Y = X(X )(X a )(X a )(X a a ). If Aut(X ) = D 8 then the SL (k)-invariants of such curve must satisfy Eq. (3). Then we get the equation in terms of a and a. By writing the relation a 3 = a a in terms of theta constants we get θ = θ 3. All the results above lead to part ii) of the theorem. iii) The proof of this part is similar to part ii). We express the conditions of the previous lemma in terms of the fundamental theta constants only. Lemma 3. Let X be a genus curve. Then we have the following: i) V Aut(X ) if and only if the fundamental theta constants of X satisfy (θ 3 θ )(θ θ 3)(θ θ )(θ θ )(θ 3 θ )(θ θ ) ( θ + θ 3 + θ θ )(θ θ 3 + θ θ )( θ θ 3 + θ + θ ) (θ + θ 3 + θ + θ )(θ θ + θ 3 θ + θ θ 3 θ θ θ 3θ ) ( θ3 θ θ θ θ 3 θ + θ θ θ 3θ ) (θ θ + θ θ + θ θ (6) θ θ θ 3θ ) ( θ θ + θ 3 θ + θ θ 3 θ θ θ 3θ ) =. ii) D 8 Aut(X ) if and only if the fundamental theta constants of X satisfy Eq.(3) in [6. iii) D 6 Aut(X ) if and only if the fundamental theta constants of X satisfy Eq.() in [6. Proof. Notice that Eq. () contains only θ θ θ 3 θ θ 8 and θ. Using Eq. (9) we can eliminate θ 8 and θ from Eq. (). The J invariant of any genus two curve is given by the following in terms of theta constants: J = θ θ3 θ 8θ8 θ (θ θ θ 3θ ) (θ θ θ θ 3) (θ θ 3 θ θ ). Since J the factors (θθ θ3θ ) (θθ θθ 3) and (θθ 3 θθ ) cancel in the equation of the V locus. The result follows from Theorem 3. The proof of part ii) and iii) is similar and we avoid details. Remark. For part ii) and iii) the equations are lengthy and we don t show them here. But by using the extra conditions θ = θ 3 or θ = θ 3 we could simplify the equation of the D 8 locus as follows: i)when θ = θ 3 we have

25 (θ θ )(θ θ θ 3 )(θ + θ + θ 3 )(θ + θ θ 3 )( θ θ θ + θ 3 )( θ θ 3 + θ θ )( θ θ θ θ θ θ θ 8 θ 6 3 θ θ 8 θ 3 6 θ θ 6 θ θ θ θ θ 8 θ θ 8 θ 8 θ θ 8 θ θ 3 6 θ 8 θ 3 θ 6 θ 6 θ θ 3 6 θ 6 θ θ θ 6 θ 6 θ θ 6 θ θ 8 3 θ θ + 8 θ 8 θ 6 θ θ 8 3 θ 6 θ θ 3 θ θ θ 3 θ + 8 θ θ θ θ θ θ (7) +7 θ 8 θ θ θ 6 θ 3 ) =. ii) When θ = θ 3 we have (θ θ )(θ 3 + θ θ )( θ + θ θ 3 )( θ + θ + θ 3 ) (θ 3 + θ θ + θ )( θ + θ 3 + θ θ )(6 θ θ θ 3 6 θ θ θ θ 8 θ θ θ 8 θ θ θ 6 θ θ θ θ 3 68 θ θ 8 3 θ θ 8 θ θ 8 6 θ 3 3 θ 8 θ 8 θ θ 8 θ θ 3 6 θ 8 θ 3 θ + 6 θ 6 θ θ 3 +6 θ 6 θ θ 6 3 θ 6 θ 6 θ 3 68 θ 6 θ θ 8 3 θ θ +6 θ θ 6 θ 3 θ θ 8 3 θ 6 θ θ 3 θ 8 88 θ θ 3 θ (8) 8 θ θ θ θ 8 θ 6 ) =. Define the following as A = ( θ θ ) B = ( θ 3 θ ) C = ( θ θ ) D = ( θ 8 θ ) E = ( θ θ ). Using the two identities given by Eq. (9) we have + A B C D E = A DEA + BCA + C B DECB + D E =. Then we formulate the following lemma. Lemma. Let X be a genus curve. Then V Aut(X ) if and only if the theta constants of X satisfy

26 (B A)(A C)(B C)( A)( B)( C)( C + A + A C DE AC A BC + ADEBC + AB + DEBC + ADEB A + ABC AB C A B + ADE B C BC + B C)( DEBC ABC + B C + AC + AB C ADEB + A + A C + ABC DEC ADEC A C A BC AC ADE) =. (9) 3. Genus 3 curves 3.. Introduction to Genus 3 Curves In this section we focus on genus 3 cyclic curves. The locus L 3 of genus 3 hyperelliptic curves with extra involutions is a 3-dimensional subvariety of H 3. If X L 3 then V Aut(X ). The normal form of the hyperelliptic genus 3 curve is given by y 3 = x 8 + a 3 X 6 + a x + a x + and the dihedral invariants of X 3 are u = a + a 3 u = (a + a 3)a u 3 = a a 3. The description of the locus of genus 3 hyperelliptic curves in terms of dihedral invariants or classical invariants is given in [7. We would like to describe the locus of genus 3 hyperelliptic curves with extra involutions and all its sub loci in terms of theta functions. The list of groups that occur as automorphism groups of genus 3 curves has been computed by many authors. We denote the following groups by G and G : G = x y x y 6 xyxy G = x y x y (xy) (x y). In Table we list all possible hyperelliptic genus 3 algebraic curves; see [ for details. In this case Aut(X ) has a central subgroup C of order such that the genus of X C is zero. In the second column of the table the groups which occur as full automorphism groups are given and the third column indicates the reduced automorphism group for each case. The dimension δ of the locus and the equation of the curve are given in the next two columns. The last column is the GAP identity of each group in the library of small groups in GAP. Note that C C and C are the only groups which don t have extra involutions. Thus curves with automorphism group C C or C do not belong to the locus L 3 of genus 3 hyperelliptic curves with extra involutions. In Table 3 we list the automorphism groups of genus 3 nonhyperelliptic curves. In the table the second column represents the normal cyclic subgroup C such that g(x C ) =. For the last 3 cases in the table the automorphism groups of the curves are not normal homocyclic covers of P. The only cyclic curves are curves with automorphism groups C S 3 C 3 C 6 C 9 and two other groups given by (6 3) and (8 33) in GAP identity. In this chapter we write the equations of the cyclic curves of genus 3 by using theta constants.

27 Table. Genus 3 hyperelliptic curves and their automorphisms Aut(X ) Aut (X ) δ equation y = f(x) Id. C {} 5 x(x )(x 5 + ax + bx 3 + cx + dx + e) ( ) C C C 3 x 8 + a 3 x 6 + a x + a x + ( ) 3 C C x(x )(x + ax + b) ( ) C C 7 x 7 ( ) 5 C 3 D (x + ax + )(x + bx + ) (8 5) 6 C D 8 D 8 x 8 + ax + (6 ) 7 C C D (x )(x + ax + ) (8 ) 8 D D 6 x(x 6 + ax 3 + ) ( ) 9 G D x(x 6 ) ( 5) G D 6 x 8 (3 9) C S S x 8 + x + (8 8) Table 3. Genus 3 non hyperelliptic curves and their automorphisms # Aut(X ) C Aut(X )/C equation Id. V V {} x + y + ax y + bx + cy + = () D 8 V C take b = c (83) 3 S V S 3 take a = b = c () C S 3 V S take a = b = c = or y = x(x ) (966) 5 6 C V y = x(x )(x t) (63) 6 8 C A y = x 3 (833) 7 C 3 C 3 {} y 3 = x(x )(x s)(x t) (3) 8 C 6 C 3 C take s = t (6) 9 C 9 C 3 C 3 y 3 = x(x 3 ) (9) L 3 () x 3 y + y 3 z + z 3 x = (68) S 3 a(x + y + z ) + b(x y + x z + y z )+ (6) c(x yz + y xz + z xy) = C x + x (y + az ) + by + cy 3 z + dy z () +eyz 3 + gz = either e = or g =

28 Figure describes the inclusions among all subloci for genus 3 curves. In order to study such inclusions the lattice of the list of automorphism groups of genus 3 curves needs to be determined. Let s consider the locus of the hyperelliptic curve whose automorphism group is V = { α β αβ}. Suppose α is the hyperelliptic involution. Since the hyperelliptic involution is unique the genus of the quotient curve X β is. Also we have α = C V and β = C V. Therefore the locus of the hyperelliptic curve with automorphism group V can be embedded into two different loci with automorphism group C. One comes from a curve that has hyperelliptic involution and the other comes from a curve which does not have hyperelliptic involution. Similarly we can describe the inclusions of each locus. The lattice of the automorphism groups for genus 3 curves is given Figure. 3.. Theta Functions for Hyperelliptic Curves For genus three hyperelliptic curves we have 8 odd theta characteristics and 36 even theta characteristics. The following shows the corresponding characteristics for each theta function. The first 36 are for the even functions and the last 8 are[ for the odd functions. For simplicity we denote them by θ i (z) instead of a θ i (z τ) where i = for the even functions and i = for the b odd functions. [ θ (z) = θ (z τ) θ (z) = θ [ (z τ) [ [ θ 3 (z) = θ 3 (z τ) θ (z) = θ (z τ) [ θ 5 (z) = θ [ 5 (z τ) θ 6 (z) = θ 6 (z τ) [ [ θ 7 (z) = θ 7 (z τ) θ 8 (z) = θ 8 (z τ) θ 9 (z) = θ 9 [ θ (z) = θ [ [ θ 5 (z) = θ 5 [ (z τ) θ (z) = θ (z τ) [ (z τ) θ (z) = θ (z τ) [ [ θ 3 (z) = θ 3 (z τ) θ (z) = θ (z τ) [ (z τ) θ 6 (z) = θ 6 θ 7 (z) = θ 7 [ [ (z τ) θ 8 (z) = θ 8 (z τ) (z τ)

29 Dimension of Loci 5 C hyperelliptic non hyperelliptic C 3 V V (C ) C C 3 S 3 D 8 C C D C D 8 C 6 6 S C G G C S C L 3() Figure. Inclusions among the loci for genus 3 curves with automorphisms. θ 9 (z) = θ 9 [ θ (z) = θ [ θ 3 (z) = θ 3 [ (z τ) θ (z) = θ [ [ (z τ) θ (z) = θ [ (z τ) θ (z) = θ (z τ) (z τ) (z τ)

30 [ θ 5 (z) = θ [ 5 (z τ) θ 6 (z) = θ 6 [ θ 7 (z) = θ [ 7 (z τ) θ 8 (z) = θ 8 [ θ 9 (z) = θ [ 9 (z τ) θ 3 (z) = θ 3 [ θ 3 (z) = θ [ 3 (z τ) θ 3 (z) = θ 3 [ θ 33 (z) = θ 33 [ θ (z) = θ (z τ) θ 3 (z) = θ 3 [ (z τ) (z τ) (z τ) (z τ) (z τ) [ θ 35 (z) = θ [ 35 (z τ) θ 36 (z) = θ 36 (z τ) [ θ 37 (z) = θ [ 37 (z τ) θ 38 (z) = θ 38 (z τ) [ [ θ 39 (z) = θ 39 (z τ) θ (z) = θ (z τ) [ (z τ) θ (z) = θ (z τ) [ θ 3 (z) = θ 3 θ 5 (z) = θ 5 [ θ 7 (z) = θ 7 [ θ 9 (z) = θ 9 [ θ 5 (z) = θ 5 [ [ (z τ) θ (z) = θ (z τ) θ 6 (z) = θ 6 [ (z τ) (z τ) [ (z τ) θ 8 (z) = θ 8 (z τ) (z τ) θ 5 (z) = θ 5 [ [ (z τ) θ 5 (z) = θ 5 (z τ) (z τ) [ [ θ 53 (z) = θ 53 (z τ) θ 5 (z) = θ 5 (z τ) [ θ 55 (z) = θ [ 55 (z τ) θ 56 (z) = θ 56 (z τ) [ θ 57 (z) = θ [ 57 (z τ) θ 58 (z) = θ 58 (z τ) [ θ 59 (z) = θ [ 59 (z τ) θ 6 (z) = θ 6 (z τ) [ θ 6 (z) = θ [ 6 (z τ) θ 6 (z) = θ 6 (z τ) [ θ 63 (z) = θ 63 (z τ) θ 6 (z) = θ 6 [ (z τ)

31 Remark 5. Each half-integer characteristic other than the zero characteristic can be formed as a sum of not more than 3 of the following seven characteristics: {[ [ [ [ [ [ }. [ The sum of all characteristics of the above set gives the zero characteristic. The sums of three characteristics give the rest of the 35 even characteristics and the sums of two characteristics give odd characteristics. It can be shown that one of the even theta constants is zero. Let s pick S = { } and U = { 3 5 7}. Let T = U. Then [ By Theorem the theta constant corresponding to the characteristic η T = is zero. That is θ =. Next we give the relation between theta characteristics and branch points of the genus 3 hyperelliptic curve in the same way we did in the genus case. Once again Thomae s formula is used to get these relations. We get 35 equations with branch points and non-zero even theta constants. By picking suitable equations we were able to express branch points in terms of thetanulls similar to Picard s formula for genus curves. Let B = {a a a 3 a a 5 } be the finite branch points of the curves and U = {a a 3 a 5 } be the set of odd branch points. Theorem. Any genus 3 hyperelliptic curve is isomorphic to a curve given by the equation where Y = X(X )(X a )(X a )(X a 3 )(X a )(X a 5 ) a = θ 3θ θ 3 θ a = θ 3θ 3 θ 9 θ a 3 = θ θ3 θ a = θ θ7 θ 6 θ5 θ 3 a 5 = θ 3θ θ6. θ 9 Proof. Thomae s formula expresses the thetanulls in terms of branch points of hyperelliptic curves. To invert the period map we are going to use Lemma 6. For simplicity we order the branch points in the order of a a a 3 a a 5 and. Then the following set of equations represents the relations of theta constants and a... a 5. We use the notation (i j) for (a i a j ). θ = A ( 6) (3 6) (5 6) ( 3) ( 5) (3 5) ( ) ( 7) ( 7) θ = A (3 6) (5 6) (3 5) ( ) ( ) ( ) (3 7) (5 7) θ 3 = A (3 6) ( 6) (3 ) ( ) ( 5) ( 5) ( 7) ( 7) (5 7) θ = A ( 6) (3 6) (5 6) ( 3) ( 5) (3 5) ( ) ( 7) ( 7) θ 5 = A ( 6) (5 6) ( 5) ( ) ( 3) ( 3) ( 7) ( 7) (3 7) θ 6 = A ( 6) ( 6) (3 ) (3 5) ( 5) ( ) ( 7) ( 7)

32 θ 7 = A ( 6) (3 6) ( 6) ( 5) ( 3) ( ) (3 ) ( 7) (5 7) θ 8 = A ( 6) (3 6) ( 3) ( ) ( 5) ( 5) ( 7) ( 7) (5 7) θ 9 = A ( 6) (3 6) ( 3) ( ) ( 5) ( 5) ( 7) (3 7) θ = A (3 6) (5 6) (3 5) ( ) ( ) ( ) ( 7) ( 7) ( 7) θ = A (3 6) ( 6) (5 6) (3 ) (3 5) ( 5) ( ) ( 7) ( 7) θ 3 = A ( 6) ( 6) (5 6) ( 3) ( ) ( 5) ( 5) ( 7) (3 7) θ = A ( 6) (5 6) ( 5) ( 3) ( ) (3 ) ( 7) (3 7) ( 7) θ 5 = A ( 6) (5 6) ( 5) ( 3) ( ) (3 ) ( 7) (5 7) θ 6 = A ( 6) ( 3) ( ) ( 5) (3 ) (3 5) ( 5) ( 7) θ 7 = A ( 6) ( 6) ( 3) ( 5) (3 5) ( ) ( 7) ( 7) θ 8 = A ( 6) ( 6) ( 3) ( 5) (3 5) ( ) ( 7) (3 7) (5 7) θ 9 = A (3 6) ( 6) ( ) ( 5) ( 5) (3 ) (3 7) ( 7) θ = A ( 6) ( 3) ( ) ( 5) (3 ) (3 5) ( 5) ( 7) θ = A ( 6) ( 6) (5 6) ( ) ( 5) ( 5) ( 3) ( 7) (3 7) θ = A ( 6) (3 6) ( 6) ( 3) ( ) (3 ) ( 5) ( 7) (5 7) θ 3 = A ( 6) ( 6) (3 ) (3 5) ( 5) ( ) (3 7) ( 7) (5 7) θ = A ( 6) (5 6) ( ) ( 3) ( 3) ( 5) ( 7) (5 7) θ 5 = A (3 6) ( ) ( ) ( 5) ( ) ( 5) ( 5) (3 7) θ 6 = A ( 6) ( 6) ( 3) ( 5) (3 5) ( ) ( 7) ( 7) θ 7 = A ( 6) (5 6) ( 5) ( 3) ( ) (3 ) ( 7) (3 7) ( 7) θ 8 = A ( 6) (3 6) ( 3) ( ) ( 5) ( 5) ( 7) ( 7) (5 7) θ 9 = A ( 6) ( 6) ( 6) (3 5) ( ) ( ) ( ) (3 7) (5 7) θ 3 = A (5 6) ( ) ( 3) ( ) ( 3) ( ) (3 ) (5 7) θ 3 = A ( 6) ( 6) (3 6) ( ) ( 3) ( 3) ( 5) ( 7) (5 7) θ 3 = A ( 6) ( 6) ( 3) ( 5) (3 5) ( ) ( 7) (3 7) (5 7) θ 33 = A ( 6) (5 6) ( 3) ( ) (3 ) ( 5) ( 7) (5 7) θ 3 = A ( 6) (3 6) ( ) ( 5) ( 5) ( 3) ( 7) (3 7) θ 35 = A ( 6) ( ) ( 3) ( 5) ( 3) ( 5) (3 5) ( 7) θ 36 = A ( 6) ( 6) (5 6) ( ) ( 5) ( 5) (3 ) (3 7) ( 7)

33 Our expectation is to write down the branch points as quotients of thetanulls. By using the set of equations given above we have several choices for a... a 5 in terms of theta constants. Branch Points a a a 3 a a 5 ( θ 36 θ θ33 θ 9 ( θ θ 9 θ θ 7 ( θ θ θ33 θ 7 ( θ θ 9 θ θ 6 ( θ θ θ 3 θ 7 Possible Ratios ) ( ) θ ( θ 3 θ θ3 θ ) ( θ 36 θ 7 θ5 θ 9 ) ( θ θ 3 θ θ 6 ) ( θ θ 7 θ 5 θ 3 ) ( θ θ 36 θ 9 θ 6 Let s select the following choices for a a 5 : ) ( θ ) 9 θ θ6 θ ) 3 θ 3 θ9 θ ) ) ( θ 7 θ θ6 θ 5 ) ( θ ) θ 3 θ9 θ 33 ) ) ( θ 3 θ θ 6 θ 9 a = θ 3θ θ 3 θ a = θ 3θ 3 θ 9 θ This completes the proof. a 3 = θ θ3 θ a = θ θ7 θ 6 θ5 θ 3 a 5 = θ 3θ θ6. θ 9 Remark 6. i) Unlike the genus case here only θ θ 6 θ 7 θ θ 5 θ θ 3 are from the same Göpel group. ii) For genus case such relations are known as Picard s formulae. The calculations proposed by Gaudry on genus arithmetic on theta function in cryptography is mainly based on Picard s formulae Theta Identities for Hyperelliptic Curves Similar to the genus case we can find identities that hyperelliptic theta constants are satisfied. We would like to find a set of identities that contains all possible even theta constants. A Göpel group Eq. (6) and Eq. (7) all play a main role in this task. Now consider a Göpel group for genus 3 curves. Any Göpel group G contains 3 = 8 elements. The number of such Göpel groups is 35. We have Göpel groups such that all of the characteristics of the groups are even. The following is one of the Göpel groups which has only even characteristics: { G = c = c 6 = [ [ [ c = c [ 3 c = [ c 5 = [ [ c 7 = [ } c 8 =. By picking suitable characteristics [ b b and [ b 3 we can find the [ Göpel systems for group G. Let s pick b = b = and b 3 = then the corresponding Göpel systems are given by the following: