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Riemann's Theta Function Riemann’s theta function Ching-Li Chai∗ version 12/02/2014 Riemann’s theta function q(z;W) was born in the famous memoir [13] on abelian functions. Its a cousins, theta functions with characteristics q b (z;W), are essentially translates of q(z;W). These theta functions can be viewed in several ways: (a) They were first introduced and studied as holomorphic function in the z and/or the W variable. (b) Geometrically these theta functions can be identified with sections of ample line bundles on abelian varieties and can be thought of as projective coordinates of abelian varieties. There values at the zero, known as thetanullwerte, give projective coordinates of moduli of abelian varieties. (c) They are matrix coefficients, for the Schrodinger¨ representation of the Heisenberg group and the oscillator (or Segal-Shale-Weil) representation of the metaplectic group. The geometric point of view will be summarized in x1 and the group theoretic viewpoint in x2. x1. Theta functions as sections of line bundles For any positive integer g, denote by Hg the Siegel upper-half space of genus g, consisting of all g × g symmetric complex matrices with positivep definite imaginary part. We will use the following version of exponential function: e(z) := exp(2p −1 · z) for any z 2 C. (1.1) DEFINITION. (i) The Riemann theta function q(z;W) of genus g is the holomorphic func- g tion in two variables (z;W) 2 C × Hg, defined by the theta series 1 t t q(z;W) := ∑ e( 2 n · W · n=2) · e( n · z): g m2Z g a (ii) Let a;b 2 R . The theta function q b (z;W) with characteristics a;b is the is the holomorphic g function on C × Hg defined by hai 1 t t q (z;W) = ∑ e(2 (n + a) · W · (n + a)) · e( (n + a) · (z + b)) b g m2Z 1 t t = e( 2 a ·W· a + a · (z + b)) · q(z + W · a + b;W) a+m t a g t a Note that q b+n (z;W) = e( a · n) · q b (z;W) for all m;n 2 Z , so that e( a · b) · q b (z;W) depends g only on a;bmodZ . ∗Partially supported by NSF grants DMS 1200271 1 a The theta function q b (z;W) satisfies the following functional equation hai hai q (z + W·m + n;W) = e(ta · n − tb · m) · e(− 1 tm·W·m − tm · z) · q (z;W) 8m;n 2 g: b 2 b Z g For any a;b 2 R , the family of holomorphic functions a;b t t 1 t t g uW·m+n(z) := e( a · n − b · m) · e(− 2 m·W·m − m · z) m;n 2 Z × a;b g g g on V with values in C forms a 1-cocycle m for the lattice W · Z + Z in C , in the sense that a;b a;b a;b g g m (z) · m (z + x1) = m (z) 8x1;x2 2 W · + : x1 x2 x1+x2 Z Z g g g This 1-cocyle defines a line bundle on the compact complex torus C =(W · Z + Z ) so that a q b (z;W) can be interpreted as a section of this line bundle. (1.2) THEOREM. (APPELL-HUMBERT) Let L be a lattice in a finite dimensional complex vector space V. (1) Every holomorphic line bundle on V=L is isomorphic to the quotient of the trivial line bundle on V via a 1-cocycle x ! ux (z) for L, where ux (z) is an entire function on V with values in × for every and u z u z u z for all . C x 2 L x1+x2 ( ) = x1 ( ) · x2 ( + x1) x1;x2 2 L (2) For any 1-cocycle ux (z) as in (1), there exists a quadruple (H;S;`;y), where (2a) H : V ×V ! C is a Hermitian form, conjugate linear in the second argument, such that × Im(H) is Z valued on L, and y : L ! C1 is a complex function with absolute values 1 −1 −1 Im(H)(x1;x2) such that y(x1 + x2) · y(x1) · y(x2) = (−1) for all x1;x2 2 L, (2b) S : V ×V ! C is a symmetric C-bilinear form, (2c) ` : V ! C is a C-linear function, and × (2d) y : L ! C1 is a complex function with absolute values 1 such that −1 −1 Im(H)(x1;x2) y(x1 + x2) · y(x1) · y(x2) = (−1) 8x1;x2 2 L; such that u (z) = p1 · (H(z; ) + S(z; )) · p1 · (H( ; ) + S( ; )) · (`( )) · ( ) 8 2 : x e 2 −1 x x e 4 −1 x x x x e x y x x L The quadruple (H;S;`;y) is uniquely determined by the 1-cocycle ux (z). Conversely every quadruple (H;S;`;y) satisfying conditions (2a)–(2d) determines a 1-cocyle for (V;L). (3) Let L and L 0 be two line bundles attached to two quadruples (H;S;`;y) and (H0;S0;`0;y0) as in (2). (3a) L is isomorphic to L 0 if and only if H = H0 and y = y0. (3b) L is algebraically equivalent to L 0 if and only if H = H0. (3c) L is ample if and only if Im(H) is a polarization of (V;L), i.e. H is positive definite. 2 (1.3) It is explained in the article on Riemann forms that for any polarization m on a compact complex torus V=L for any choice of canonical Z-basis v1;:::;v2g of L with elementary divisors g d1j···jdg, there exists a unique element W 2 Hg such that the C-linear map from V to C which g sends vi to di times the i-th standard basis of C induces a biholomorphic isomorphism from V=L g g g to C =(W · Z + D · Z ), where D is the g × g diagonal matrix with d1;:::;dg as diagonal entries. Under this isomorphism, the polarization m becomes the alternating pairing t t g mD : (W · m1 + n1;W · m2 + n2) 7! m1 · D · n2 − n1 · D · m2 8m1;n2;m2;n2 2 Z g g g on the lattice LW;D := W · Z + D · Z , and the Hermitian form on C whose imaginary part is the R-bilinear extension of mD is given by the formula t −1 0 g HW : (z;w) 7! z · Im(W) · w 8z;w 2 C : g (1.3.1) An easy calculation shows that for any given a;b 2 R , the restriction to L(W;D) of the a;b a;b 1-cocycle u corresponds to the quadruple (HW;SW;0;y0 ·cD ), where SW is the symmetric C- bilinear form t −1 g SW : (z;w) 7! − ·Im(W) · w 8z;wC ; g or equivalently SW is the unique CC-bilinear form on C which coincides with −HW;D on the subset g g g g C ×(D · Z ) ⊂ C ×C , y0 is the quadratic unitary character on LW;D defined by tm·n g y0(W · m + D · n) = (−1) 8m;n 2 Z ; a;b and cD is the unitary character on L(W;D) defined by a;b t t g cD (W · m + D · n) = e( a · D · n − b · m) 8m;n 2 Z a;b a0;b0 0 −1 g 0 g a;b Note that cD = cD if and only if a − a 2 D ·Z and b − b 2 Z . Let LW;D be the line bundle g a;b 0 −1 g on C =LW;D given by the restriction to LW;D of the 1-cocyle u . Clearly for every a 2 a+D ·Z , h a0 i a;b q b (z;W) defines a section of the line bundle LW;D. 0 − g 0 − − g More generally, for every positive integer r, every a 2 a+D 1·Z and every b 2 r 1b+r 1Z , h a0 i −1 a;b ⊗r q b0 (z;r W) defines a section of the line bundle (LW;D) . Underlying this statement is the fact a0;b0 g g g that the pull-back of via the isogeny = ! = −1 of degree r is isomorphic to Lr−1W;D C LW;D C Lr W;D a;b ⊗r 0 0 (LW;D) , for every a ;b as above. g (1.3.2) PROPOSITION. (1) For every line bundle L on the compact complex torus C =LW;D with g a;b polarization mD, there exists a;b 2 R such that L is isomorphic to LW;D. Moreover for any 0 0 g a;b a0;b0 0 −1 g 0 g a ;b 2 R , LW;D is isomorphic to LW;D if and only if a 2 a + D ·Z and b 2 b + Z . g a;b g 0 (2) dimCG C =LW;D;LW;D = det(D) = ∏i=1 di. Moreover as a runs through a set of representa- −1 g g h a0 i a;b tives of (a + D ·Z )=Z , the theta functions q b (z;W) give rise to a C-basis of G(LW;D). 3 g a;b ⊗r g g (3) For every positive integer r, we have dimCG C =LW;D;(LW;D) = r · ∏i=1 di. Moreover as 0 − g g 0 a runs through a set of representatives of (a + D 1·Z )=Z and b runs through a set of represen- −1 −1 g g h a0 i −1 tatives of (r b + r Z )=Z , the global sections corresponding to q b0 (z;r W) form a C-basis a;b ⊗r of G (LW;D) .
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