The genus 1 case - review Theta Function Review g = 1 case We recall the main de¯nitions of theta functions in the 1-dim'l case: De¯nition 0 Let· ¿ 2¸C such that Im¿ > 0: For "; " real numbers and z 2 C then: " £ (z;¿) = "0 n ¡ ¢ ¡ ¢ ¡ ¢ ³ ´o P 1 " " ² t ²0 l²Z2 exp2¼i 2 l + 2 ¿ l + 2 + l + 2 z + 2 The series· is uniformly¸ and absolutely convergent on compact subsets " C £ H: are called Theta characteristics "0 The genus 1 case - review Theta Function Properties Review for g = 1 case the following properties of theta functions can be obtained by manipulation of the series : · ¸ · ¸ " + 2m " 1. £ (z;¿) = exp¼i f"eg £ (z;¿) and e; m 2 Z "0 + 2e "0 · ¸ · ¸ " " 2. £ (z;¿) = £ (¡z;¿) ¡"0 "0 · ¸ " 3. £ (z + n + m¿; ¿) = "0 n o · ¸ t t 0 " exp 2¼i n "¡m " ¡ mz ¡ m2¿ £ (z;¿) 2 "0 The genus 1 case - review Remarks on the properties of Theta functions g=1 1. Property number 3 describes the transformation properties of theta functions under an element of the lattice L¿ generated by f1;¿g. 2. The same property implies that that the zeros of theta functions are well de¯ned on the torus given· ¸ by C=L¿ : In fact there is only a " unique such 0 for each £ (z;¿): "0 · ¸ · ¸ "i γj 3. Let 0 ; i = 1:::k and 0 ; j = 1:::l and "i γj 2 3 Q "i k θ4 5(z;¿) ³P ´ ³P ´ i=1 "0 k 0 l 0 2 i 3 "i + " ¿ ¡ γj + γ ¿ 2 L¿ Then i=1 i j=1 j Q l 4 γj 5 j=1 θ 0 (z;¿) γj is a meromorphic function on the elliptic curve de¯ned by C=L¿ : The genus 1 case - review Analytic vs. Algebraic Review for g = 1 case The last property enables us to construct analytic functions on the elliptic curve de¯ned by C=L¿ : This explicit construction enables us to move between Analytic properties of C=L¿ to its Algebraic properties and vice versa. The following example is a classic: Consider an a±ne equation above C for the elliptic curve C; y 2 = x(x ¡ 1)(x ¡ ¸): To obtain ¿ from ¸ perform the following steps : 1. Choose a homology basis a1; a2 on C: dx 2. The di®erential y is the unique holomorphic non vanishing di®erential on C; up to a constant R R dx dx !2 3. If !1 = ;!2 = then ¿ = a1 y a2 y !1 The genus 1 case - review The expression for ¸ Now we like to invert the map we de¯ned in the previous slide and obtain ¸ from ¿: We do it using Theta functions as follows: 1. · ¸ 0 θ4 (z;¿) 0 · ¸ 1 θ4 (z;¿) 0 is a meromorphic function on the torus with zero of order 4 and at 1 1 1 the point 2 + 2 ¿ and pole of the same order at 2 ¿: 2. · ¸ 1 θ4 (0;¿) 0 ¸(¿) = · ¸ 0 θ4 (0;¿) 0 The genus 1 case - review Comments on the expression for ¸ 1. The quotients for ¸ in the previous slide is by no means unique. We can choose other quotients for example : · ¸ 0 θ4 (0;¿) 1 ¸ (¿) = · ¸ 1 0 θ4 (0;¿) 0 2. We can show using Theta functions identities that 1 ¡ ¸ = ¸1: or equivalently the famous Jacobi identity: · ¸ · ¸ · ¸ 1 0 0 θ4 (0;¿) + θ4 (0;¿) = θ4 (0;¿) 0 1 0 · ¸ 0 3. There is a formula that expresses the value of θ4 (0;¿) as a 0 polynomial in ¸: We discuss it further in the higher dimensional case The genus 1 case - review Genus 1 case - summary In the previous we saw for g = 1 curves how to: 1. Construct meromorphic functions through the transcendental invariant ¿: 2. Express Algebraic invariants of the curve through its analytical invariant using Theta functions with characteristics. What about algebraic curves with g > 1? The genus 1 case - review Higher dimensional case generalization To generalize for curves of higher genus we de¯ne the higher dimensional Theta functions. De¯nition The upper half Siegel space Hg ; Hg = g £ g matrices ¿ such that I ¿ is symmetric I Im¿ de¯nes positive quadratic form. Let us de¯ne the higher dimensional Theta functions. The de¯nition is similar to the 1 dimensional case but Hg ful¯lls the role of the usual H: The genus 1 case - review Higher dimensional Theta function de¯nition De¯nition· ¸ " For "; "0; real vectors ¿²H and z²Cg ; "0 g · ¸ ½ ³ ´ ³ ´ ³ ´ µ ¶¾ " X 1 " t " ² t ²0 £ (z;¿) = exp2¼i l + ¿ l + + l + z + "0 2 2 2 2 2 l²Z2g These series is uniformly and absolutely convergent on compact subsets g C £ Hg : The genus 1 case - review Higher dimensional Theta function properties By series manipulation we obtain the following properties of Theta function:· ¸ · ¸ " + 2m " I £ (z;¿) = exp¼i f"t eg £ (z;¿) "0 + 2e "0 · ¸ · ¸ " " I £ (z;¿) = £ (¡z;¿) ¡"0 "0 · ¸ " + 2m I £ (z + n + mt ¿; ¿) = "0 + 2e n o · ¸ t t 0 " + 2m exp 2¼i n "¡m " ¡ kmt z ¡ kmt ¿m £ (z;¿) 2 "0 + 2e The last property implies the following analogous to the 1- dimensional case: The genus 1 case - review Higher dimensional Theta function de¯nition · ¸ · ¸ " γj I Let i ; i = 1:::k and ; j = 1:::l and "0 γ0 ³ i ´ ³ j ´ Pk 0 Pl 0 i=1 "i + "i ¿ ¡ j=1 γj + γj ¿ 2 L¿ ( the lattice generated by g ¿ in C ) 2 3 Q "i k θ4 5(z;¿) i=1 "0 Then 2 i 3 Q l 4 γj 5 j=1 θ 0 (z;¿) γj g is a meromorphic function on the lattice de¯ned by C =L¿ : We use this property enables us to construct functions on algebraic curves of any genus. The genus 1 case - review Higher dimensional theta functions and Algebraic curves Let X be an algebraic curve if genus g: We can choose on X a homology basis a1:::ag ; b1:::bg such that ai aRj = bi bj = 0 and ai bj = 1. Let !1:::!g be holomorphic di®erentials and !j = ±ij ai De¯nition R The period matrix is: ¿ij = !j : bi Riemann showed that ¿ belongs to Hg : Recall the Jacobian mapping: Choose a base point P0 in X and de¯ne a map Á : X 7! Cg by: Z P Z P Á(P) = !1::: !g : P0 P0 Then X is embedded into Cg : Changing the base point causes the mapping to be changed by an element of L¿ . The genus 1 case - review Higher dimensional theta functions and Algebraic curves We can use quotients of Theta functions to construct meromorphic g functions on C =L¿ and hence on X : through restriction. Riemann characterized the zeros of the Theta function to show that these g quotients inside the C =L¿ are not vanishing identically on X : Theorem Let ¢ be a divisor such that 2¢ is the canonical divisor of the zeros of the holomorphic di®erential. Then a e is a 0 of a theta function if and only of there exists a divisor of degree g ¡ 1 V and such that e = ¢ + V g in C =L¿ The genus 1 case - review Remarks on the Higher dimensional case The dimension of Hg is g(g + 1)=2: The dimension of the space of generic algebraic curve X is 3g ¡ 3: An outstanding problem is: Problem Characterize the period matrices inside Hg : ( Schottky problem) Riemann discovered Riemann Surfaces through his investigations of Analytic continuation of Algebraic functions and branched covers of the Sphere. We de¯ne the branch points before formulating His theorem: De¯nition f : X 7! CP1 be a map of degree n: Then ¸ is a branch point of the mapping if jf ¡1 (¸) j < n: We formulate the following theorem that characterizes these branched covers: The genus 1 case - review Branched Covers of the Sphere and Riemann's existence theorem Theorem Let f : X 7! CP1 we can attach to X the following data that characterizes it up to a holomorphic mapping: 1. branch points ¸1:::¸r 2. permutations σ¡1...σr in Sn that are induced by the¢ mapping of the ¡1 open cover ¼1 X ¡ f (¸1:::¸r ) 7! CP ¡ (¸1:::¸r ) : The genus 1 case - review Examples k Qj Let X be a curve with the equation: y = i=1(x ¡ ¸i ): This curve is k to 1 cover of the Sphere branched above j points ¸1:::¸j : It can be shown (k(j¡1)¡(j+1) that for j points ¸1:::¸j : The genus is 2 : We can state a problem analogous to the Schottky problem: Problem Characterize the period matrices of cyclic curves k to 1. Find identities involving Theta functions that satis¯ed by such curves. If k = 2 then the case is the case of Hyperelliptic curves. Frobenius found identities satis¯ed on periods of such curves. Mumford showed that this characterizes the Hyperelliptic case. The genus 1 case - review Thomae Formula-General Background Our approach to tackle the problems above is through Thomae formulas for cyclic covers. These formulas express values of Theta functions evaluated at periods of algebraic curves X through polynomials of branch points. Then identities among theta functions will be identities among the polynomials. One then tries to ¯nd connections between the polynomials themselves. We explain Thomae formula in two cases: 1. Hyperelliptic curves 2. cyclic covers of degree 3 The genus 1 case - review Thomae formula - Hyperelliptic covers Let ¸1:::¸2n be branched points for Hyperelliptic curve of genus n ¡ 1: For every partition of f¸1::::¸ng ¤ = ¤0; ¤1 for equal parts form the polynomials p¤ = p¤0 p¤1 : where if ¤0 = (¸i1 ::::¸in ) then Y ¡ ¢ p¤0 = ¸il ¡ ¸ij il 6=ij Similarly forP ¤1: For a partition above form the following divisor: D(¤ ) = ¸ ¡ 1 ¡ 1 : Choosing a base point to be one of the 0 ¸i 2¤0 i 1 2 g ¸i we can identify the divisor with an element of C and hence θ[D(¤0)](0;¿): is a valid de¯nition.