The Schottky problem and second order theta functions Bert van Geemen December Introduction The Schottky problem arose in the work of Riemann To a Riemann surface of genus g one can asso ciate a p erio d matrix which is an element of a space H of dimension g g Since the g Riemann surfaces themselves dep end on only g parameters if g the question arises as to how one can characterize the set of p erio d matrices of Riemann surfaces This is the Schottky problem There have b een many approaches and a few of them have b een succesfull All of them exploit a complex variety a ppav and a subvariety the theta divisor which one can asso ciate to a p oint in H When the p oint is the p erio d matrix of a Riemann surface this variety is g known as the Jacobian of the Riemann surface A careful study of the geometry and the functions on these varieties reveals that Jacobians and their theta divisors have various curious prop erties Now one attempts to show that such a prop erty characterizes Jacobians We refer to M Lectures III and IV for a nice exp osition of four such metho ds to vdG B D for overviews of later results and V for a newer approach In these notes we discuss a particular approach to the Schottky problem which has its origin the work of Schottky and Jung and unpublished work of Riemann It uses the fact that to a genus g curve one can asso ciate certain ab elian varieties of dimension g the Prym varieties In our presentation we emphasize an intrinsic line bundle on a ppav principally p olarized ab elian variety and the action of a Heisenberg group on this bundle A systematic study of the geometry asso ciated to these leads in a natural way to Prym varieties Moreover one nds several other remarkable prop erties of Jacobians which suggest geometrical solutions of the Schottky problem Acknowledgements I thank the organizers of the conference Variedas ab elianas y func tiones theta for the opp ertunity to present these lectures and for providing the pleasent working conditions in Morelia The Schottky problem Introduction We recall the basic results on p erio d matrices of Riemann surfaces References are ACGH C GH and CGV Then we briey discuss mo dular forms a reference is Ig December Period matrices Let C b e a Riemann surface of genus g we consider only compact Riemann surfaces in 2g these lectures On the homology group H C Z Z there is an alternating nondegenerate 1 intersection form A symplectic basis of H C Z is a basis f g satisfying 1 1 g 1 g i j i j i j ij with Kroneckers delta so if i j ij ii ij 0 The complex vector space of holomorphic one forms on C is denoted as usual by H C C 0 it has dimension g Given a path in C and an H C one can compute the integral C R R 0 0 as a map H C C thus it is an element of H C the dual We will view C C 0 vector space of H C If is a closed path the integral only dep ends on the homology C class of which we denote by the same symbol H C Z Thus we get a map 1 Z 0 H C Z H C 1 C This map is injective in fact much more is true Theorem Let f g b e a symplectic basis of H C Z Then there is a unique i j 1 0 basis f g of H C such that 1 g C Z j ij i R 0 0 H C form the dual basis of the basis f g of H C Thus the elements C j C i 1 A symplectic basis of H C Z thus determines a basis of H C For this we only 1 C use the We now use the to dene a complex g g matrix i j Denition The p erio d matrix of C with resp ect to the symplectic basis of i j H C Z is the matrix 1 Z M C with ij g ij j i 0 and H C as in Theorem j C 0 Remark The p erio d matrix determines the image of H C Z in H C In fact 1 C R 0 of H C we have using the basis C i Z Z Z Z i1 i2 ig g 1 2 i 0 since b oth sides give the same result when applied to the basis elements H C j C December Torellis theorem Torellis theorem asserts that one can recover the Riemann surface from its p erio d matrix There are many pro ofs of this theorem all of them use the Jacobian its theta divisor which are asso ciated to a p erio d matrix see also The Siegel upp erhalf space The Schottky problem The Schottky problem basically asks for equations which determine the p erio d matrices of Riemann surfaces among all g g matrices There are two well known prop erties of p erio d matrices so p erio d matrices are symmetric and ij j i I m the imaginary part of which is a symmetric real g g matrix denes a p ositive g t g denite quadratic form on R xI m x for all x R We write I m This leads to the following denition and theorem Denition The Siegel upp erhalf space H is g t H f M C I m g g g Theorem Let b e the p erio d matrix of a Riemann surface Then H g The subset H of M C is actually complex manifold with the complex structure g g induced from that on M C In fact H is an op en subset of the vector space of symmetric g g 0 0 g g matrices if I m then also I m for any symmetric with suciently 1 small co ecients The dimension of H is g g g 2 We investigate H in more detail to see what kind of equations for p erio d matrices one g should exp ect The symplectic group To dene the p erio d matrix of a Riemann surface we had to choose a symplectic basis Any two symplectic bases are related by an element of I t S pg Z fA M Z AE A E g with E g 2g 0 0 0 I One can show that acts on H as follows g g a b 1 H H A A a bc d A g g g c d where a d are g g blo cks of A The p erio d matrices of a Riemann surface X are a orbit g in H Thus rather than study the p erio d matrices of Riemann surfaces in H one could study g g their images under the quotient map H A nH g g g g The action of on H is prop erly discontinuous but not xed p oints free and A is complex g g g C using the j invariant for elliptic curves variety with singularieties if g If g A g Riemann surfaces of genus December Mo duli spaces Let M b e the mo duli space of Riemann surfaces of genus g It is a g variety whose p oints corresp ond to isomorphism classes of Riemann surfaces Then we have a well dened holomorphic map j M A X g g g where is a p erio d matrix of X Torellis theorem implies that j is injective The Schottky problem can now b e reformulated as the problem of nding equations for the image of j 0 Denition Let J H b e the set of p erio d matrices of Riemann surfaces Its image g g in A is g 0 j M I mag eJ A nH g g g g g 0 The subvarieties J and j M are not closed see We dene the Jacobi lo cus J to b e g g g 0 the closure of J in H g g 0 J J H g g g Decomp osable matrices A H will b e called decomp osable if lies in the g g orbit of matrices in diagonal blo ck form the diagonal blo cks b eing matrices in upp er half planes 0 of lower dimension The set J J in H consists of decomp osable matrices the diagonal g g g blo cks b eing p erio d matrices of Riemann surfaces of lower genus This follows from a result of Hoyt Mo dular forms From Teichmuller theory one knows that the subset J is actually g 0 an irreducible subvariety of H of dimension g for g if g one has H J J g 1 1 1 Since J is a complex subvariety of H it is natural to ask for holomorphic functions f on g g i H such that f for all i implies J The fact that J is invariant under suggests g i g g g that we could try to nd such f which are invariant It is known that a variant of this idea i g will work A Siegel mo dular form of weight k is a holomorphic function on H which transforms in the g following way under g a b k f H C f A det c d f A g g c d in case g one has to add a certain growth condition on f for i The mo dular forms of weight k form a complex vector space which has nite dimension For suitable large k a basis f f of this vector space gives an everywhere dened map 0 N N H P f f k g 0 N k Since f A det c d f for each i we have A and thus the map i i k k k factors over A nH In this way we get a map g g g N A nH P k g g g December A fundamental result is that for suitable k the map embeds A so A A The k g g k g image of and thus A itself is a quasi pro jective variety that is a Zariski op en subset of a k g pro jective variety For this pro jective variety one can take the Satake compactication of A g which has the following nice settheoretic description as a disjoint union A A A A g g g 1 0 A and for suitable k here A is dened to b e a p oint Actually the maps extend to maps g 0 k one has A A A k k g g g N The closure of j M is a pro jective subvariety of P Thus it is dened by a nite set k g of homogeneous p olynomials Determining these p olynomials as well as the mo dular forms of weight k gives a solution to the Schottky problem The b est type of equations for the Jacobi lo cus J are thus homogeneous p olynomials in mo dular forms g Note that if F CX X is a homogeneous p olynomial of degree d and f f is 0 N 0 N a basis of M k then F f f is a mo dular form of weight k d Thus the g 0 N equations for J we lo ok for will b e mo dular forms g Algebraic geometry We just explained that the nicest equations for the p erio d matrices are mo dular forms These mo dular forms are obtained from homogeneous p olynomials N which dene the closure in P of the image of the comp osition j N M A nH P g g g g The map is given by mo dular forms on H To nd the image of p oint C M one g g has