Tata Lectures on Theta II

We shall work over the complex field ¢.

Given a finite number of distinct elements a_i ∈ ¢, i ∈ S, let \( f\left( t \right) = \prod\limits_{i \in S} {\left( {t - a_i } \right)} \). We form the plane curve C₁ defined by the equation The polynomial s² -f(t) is irreducible, so (s² -f(t)) is a prime ideal, and C₁ is a 1-dimensional affine variety in ¢². In fact, C₁ is smooth. To prove this, we will calculate the dimension of the Zariski-tangent space at each point, i.e., the space of solutions \( (\dot s,\dot t) \) ∈ ¢² to the equation That is equivalent to the equation if s ≠ 0, the solutions are all linearly dependent since \( \dot s = \frac{{\dot t}} {{2s}}\left( {\sum\limits_{j \in S} {\prod\limits_{i \ne j} {\left( {t - a_i } \right)} } } \right) \) ; if s = 0, we get from the equation of the curve \( \prod\limits_{i \in S} {\left( {t - a_i } \right)} = 0 \), hence t = a_i for some i; thus \( 0 = \dot t \cdot \prod\limits_{j \ne i} {\left( {a_i - a_j } \right)} \), so \( \dot t = 0 \). Thus at all points, the Zariski tangent space is one-dimensional.

Let’s recall that a hyperelliptic curve C is determined by an equation s² = f(t), where f is a polynomial of degree 2g+1; C has one point at infinity, and (t)_∞ = 2 · ∞ We shall study the structure of Pic C = {group of divisors modulo linear equivalence}.

Let X be a variety. Then a vector field D on X is given equivalently by: In fact, given D(x), \( f \in \Gamma \left( {U,\mathcal{O}_X } \right) \), define Df by When X is an abelian variety, then translations on X define isomorphisms for all x ∈ X (O = identity), so we may speak of translation-invariant vector fields. It is easy to see that for all D(O) ∈ T_{x, o′} there is a unique translation-invariant vector field with this value at O. In general, the vector fields on X form a Lie algebra under commutators: 4.c EQ For translation-invariant vector fields, the commutativity of X implies that bracket is zero (see Abelian Varieties, D. Mumford, Oxford Univ. Press, p. 100.

In classical mechanics, one encounters the class of problems: M = real 2n-dimensional manifold, with a closed non-degenerate differential 2-form ω

The appearance of \( \vec \Delta \) in the main theorem of §5 looks quite mysterious. It appeared as a result of an involved evaluation of the integrals in Riemann’s derivation. As in the Appendix to §3, Ch. II, we would like to introduce the concept of theta characteristics in order to give a more intrinsic formulation of (5.3) and clarify the reason for the peculiar looking constant \( \vec \Delta \). It cannot be eliminated but it can be made to look more natural in this setting.

In this section we want to combine Riemann’s theta formula (II.6) with the Vanishing Property (6.7) of the last section. An amazing cancellation takes place and we can prove that for hyperelliptic Ω, ϑ(\( \vec z \), Ω) satisfies a much simpler identity discovered in essence by Frobenius. We shall make many applications of Frobenius’ formula. The first of these is to make more explicit the link between the analytic and algebraic theory of the Jacobian by evaluating the constants c_k of Theorem 5.3. The second will be to give explicitly via thetas the solutions of Neumann’s dynamical system discussed in §4. Other applications will be given in later sections. Because one of these is to the Theorem characterizing hyperelliptic Ω by the Vanishing Property (6.7), we want to derive Frobenius’ theta formula using only this Vanishing and no further aspects of the hyperelliptic situation. Therefore, we assume we are working in the following situation:

As a consequence of the formula expressing the polynomial U^D (t) in terms of theta functions, we can directly relate the cross-ratios of the branch points a_i to the “theta-constants” ϑ[η](0,Ω). This result goes back to Thomae: Beitrag zur bestimmung von ϑ(0,...,0) durch die Klassenmoduln algebraischer Funktionen, Crelle, 71 (1870).

The goal of this section is to prove that the fundamental Vanishing property of §6 characterizes hyperelliptic Jacobians. The method will be to show that any abelian variety X_Ω which has the Vanishing property must have a covering of degree 2^g+1 which occurs as an orbit of the g commuting flows of the Neumann dynamical system.

On any hyperelliptic jacobian Jac C, there is one meromorphic function which is most important, playing a central role in the function theory on Jac C. When g = 1, this function is Weierstrass’ p-function, so, at the risk of precipitating some confusion in notation, we want to call this function \( \mathfrak{p}\left( {\vec z} \right) \) too.

As with the Neumann dynamical system, our purpose now is to introduce a dynamical system interesting in its own right, and then to show that it can, in some cases, be integrated explicitly by the theory of hyperelliptic Jacobians. More precisely, we can, following the ideas in the previous section, define an embedding of Jac C in an infinite dimensional space: On R₁, we consider a simple class of vector fields X: those which assign to f a tangent vector in \( X_f \in T_{R_1 ,f} \cong R_1 \) given by . Integrating this vector field means finding an analytic function f(x,y) s.t. By the Cauchy-Kowalevski Theorem, for all f(x,0) analytic in |x| < ε, there exists f(x,y) analytic in |x|,|y| < η solving this. What we want to do is to set up a sequence X₁,X₂,...of such vector fields called the Kortweg-de Vries hierarchy which a) commute [X_i,X_j] =0 — we must define this carefully — and b) are Hamiltonian in a certain formal sense, such that c) for all g, and for all hyperelliptic curves C of genus g:

Given an arbitrary compact Riemann surface X, of genus g, wouldn’t it be handy if we had a holomorphic function E: X × X → ¢ such that E(x,y) = 0 if and only if x = y? Although such a function doesn’t exist, it turns out that it “almost” does! To understand part of the problem and how to fix it, let’s look at the simplest case: Example. Let X = IP¹. The function x-y works on IP¹-{∞} but not on all of IP¹. So consider instead the “differential”: where \( \sqrt {dx} ,\sqrt {dy} \) are defined as follows.

In this section we will study what happens to Fay’s identities when the 4 points a,b,c,d come together in various stages. The result will be identities involving derivatives of theta functions. First, we need some notation. For the following formulas, let

The corollaries of Fay’s trisecant identity can be used to construct special solutions to many equations occurring in Mathematical Physics. In this section we will consider the following equations:

In this section we will define and describe the generalized Jacobian of the simplest singular curves: the curves obtained by identifying 2g points of IP¹ in pairs. We will then determine their theta functions and theta divisors. Finally, we will apply this theory to understand analytically and geometrically the limits of the solutions to the KdV equation that were discussed in the previous section, when the hyperelliptic curve becomes singular of the above form.

The history of algebraic equations is very long. The necessity and the trial of solving algebraic equations existed already in the ancient civilizations. The Babylonians solved equations of degree 2 around 2000 B.C. as well as the Indians and the Chinese. In the 16th century, the Italians discovered the resolutions of the equations of degree 3 and 4 by radicals known as Cardano’s formula and Ferrari’s formula. However in 1826, Abel [1] (independently about the same epoch Galois [7]) proved the impossibility of solving general equations of degree ≥ 5 by radicals. This is one of the most remarkable event in the history of algebraic equations. Was there nothing to do in this branch of mathematics after the work of Abel and Galois? Yes, in 1858 Hermite [8] and Kronecker [15] proved that we can solve the algebraic equation of degree 5 by using an elliptic modular function. Since \( \sqrt[n]{a} = \exp \left( {\left( {{1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}} \right)\log a} \right) \) which is also written as exp((1/n) ∫ ₁ ^a (1/x)dx), to allow only the extractions of radicals is to use only the exponential. Hence under this restriction, as we learn in the Galois theory, we can construct only compositions of cyclic extensions, namely solvable extentions. The idea of Hermite and Kronecker is as follows; if we use another transcendental function than the exponential, we can solve the algebraic equation of degree 5. In fact their result is analogous to the formula \( \sqrt[n]{a} = \exp (1/n)\int_1^a {(1/x)dx).} \) . In the quintic equation they replace the exponential by an elliptic modular function and the integral ∫(1/x)dx by elliptic integrals. Kronecker [15] thought the resolution of the equation of degree 5 by an elliptic modular function would be a special case of a more general theorem which might exist. Kronecker’s idea was realized in few cases by Klein [11], [13]. Jordan [10] showed that we can solve any algebraic equation of higher degree by modular functions. Jordan’s idea is clarified by Thomae’s formula, 8 Chap, m (cf. Lindemann [16]). In this appendix, we show how we can deduce from Thomae’s formula the resolution of algebraic equations by a Siegel modular function which is explicitely expressed by theta constants (Theorem 2). Therefore Kronecker’s idea is completely realized. Our resolution of higher algebraic equations is also similar to the formula \( \sqrt[n]{a} = \exp (1/n)\int_1^a {(1/x)dx).} \) In our resolution the exponential is replaced by tne Siegel modular function and the integral ∫(1/x)dx is replaced by hyperelliptic integrals. The existance of such resolution shows that the theta function is useful not only for non-linear differential equations but also for algebraic equations.