Riemann s theta formula

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1 Riemann s theta ormula Ching-Li Chai version 12/03/2014 There is a myriad o identities satisied by the Riemann theta unction θ(z;ω) and its close relatives θ [ a b] (z;ω). The most amous among these theta relations is a quartic relation known to Riemann, associated to a 4 4 orthogonal matrix with all entries ±1; see 1.3. It debuted as ormula (12) on p. 20 o [10], and was named Riemann s theta ormula by Prym. In the preace o [10] Prym said that he learned o this ormula rom Riemann in Pisa, where he was with Riemann or several weeks in early 1865, and that he wrote down a proo ollowing Riemann s suggestions. For any ixed abelian variety, these theta identities give a set o quadratic equations which deines this abelian variety. The coeicients o these quadratic equations are theta constants, or thetanullwerte, which vary with the abelian variety. At the same time, the Riemann theta identities give a set o quartic equations satisied by the theta constants, which gives a systems o deining equations o the moduli space o abelian varieties (endowed with suitable theta level structures). 1. Riemann s theta ormula We will irst ormulate a generalized Riemann theta identity, or theta unctions attached to a quadratic orm on a lattice. (1.1) DEFINITION. (THETA FUNCTIONS ATTACHED TO QUADRATIC FORMS) Let Q be a Q- valued positive deinite symmetric bilinear orm on an h-dimensional Q-vector space Γ Q, where h is a positive integer. Let Γ Γ Q be a Z-lattice in Γ Q, i.e. a ree abelian subgroup o Γ Q o rank h. Denote by Γ Q the Q-linear dual o Γ Q, and let Γ := {λ Γ Q λ(γ) Z.}. We identiy elements o Q g Q Γ Q with g-tuples o elements o Γ Q and similarly or Q g Q Γ Q. (i) The pairing Q g C g (n,z) t n z C on Q g C g and the natural pairing Γ Q Γ Q Q induces a pairing, : (Q g Q Γ Q ) (C g Q Γ Q C. (ii) Let Q : (Q g Γ Q ) (Q g Γ Q ) M g (Q) be the matrix-valued symmetric bilinear pairing Q : (u,v) = ((u 1,...,u g ),(v 1,...,v g )) Q(u,v) = ( Q(u i,v j ) ) 1 i, j g u,v Q g Q Γ. (iii) For every A Q g Q Γ Q, every B Q g Q Γ Q and every element Ω H g o the Siegel upperhal space o genus g, deine the theta unction θ Q,Γ [ A B] on the gh-dimensional C-vector space Partially supported by NSF grants DMS

2 C Q Γ Q attached to (Q,Γ) by θ Q,Γ [ A B](Z;Ω) := N Z g Z Γ e ( 1 2 Tr(Ω Q(N + A, N + A) ) e( N + A, Z + B ), where e(z) := exp(2π 1z) or all z C. Note that we have θ Q Q,Γ Γ [ (A,A ) (B,B ) ] ((Z,Z );Ω) = θ Q,Γ [ A B](Z;Ω) θ Q,Γ [ A B ] (Z ;Ω) or the orthogonal direct sum (Q Q,Γ Γ ) o (Q,Γ) and (Q,Γ ). In particular i (Q,Γ) is the orthogonal direct sum o h one-dimensional quadratic orms, then θ Q,Γ is a product o h usual theta unctions with characteristics. Let (Q,Γ) be a Q-valued positive deinite quadratic orm. Let T : L Q Γ Q be a Q-linear isomorphism o vector spaces over Q, and let L be a Z-lattice in L Q. Let T : Γ Q L Q be the Q-linear dual o T. Let Q : L Q L Q Q be the positive deinite quadratic orm on L Q induced by Q through the isomorphism T. Let 1 T : Q g Q L Q Γ Q be the linear map induced by T ; similarly or 1 T : C g Q Γ Q Cg Q L Q. Let K = (1 T )(Z g Z L)/ ( (Z g Z Γ) (1 T )(Z g Z L) ) [ (1 T )(Z g Z L) + (Z g Z L) ] /Z g Z L = (1 T ) 1 (Z g Z L )/ ( (Z g Z Γ ) (1 T ) 1 (Z g Z L ) ) [ (1 T ) 1 (Z g Z L ) + (Z g Z L ) ] /Z g Z L (1.2) THEOREM. (GENERALIZED RIEMANN THETA IDENTITY) For every A Q g Γ Q and every B Q g Γ Q, the equality (R Q,T ch ) [ ] θ Q,L (1 T ) 1 A (1 T ((1 T )Z;Ω) = #( ) g )B A K,B e( A, B ) θ Q,Γ[ A+A ] B+B (Z;Ω) holds or all Ω H g and all Z C g Γ Q. Note that each term on the right hand side o (R Q,T ch ) is independent o the choice o B in its congruence class modulo Z g Z L. (1.3) Theorem 1.2 is very easy to prove once stated in that orm. In two examples below (Q,L) is the diagonal quadratic orm x x2 h on Zh, Γ is also Z n, and T is (given by) a matrix such that T tt is a multiple o the identity matrix I h. (a) When h = 4, Q and Q are both the diagonal quadratic orm x1 2 +x2 2 +x2 3 +x2 4 on Z4, A = B = 0 and T is given by the orthogonal matrix , the equation (RQ,T ch ) is the classical Riemann s theta ormula [10, p. 20] (b) When h ) = 2, Γ = L = Z 2, B = 0, Q is x1 2 + x2 2, Q is 2x x2 2, T is given by the matrix 1 2 ( 1 1, and (R Q,T ch ) becomes 1 1 [ [ ] a b θ (z,2ω) θ (w,2ω) = 2 0] g 0 or all z,w C g and all a,b Q g. c 2 1 Z g /Z g θ [ c+(a+b)/2 2 0 ] (z + w,2ω) θ [ c+(a b)/2 0 ] (z w,2ω)

3 Much more about theta identities can be ound in [8, Ch. II 6], [9, 6], [2, Ch. IV 1], and classical sources such as [4, Ch. VII 1], [5] and [10]. 2. Theta relations (2.1) Notation. Let d 1,d 2,...,d g be positive integers such that 4 d 1 d 2 d g, ixed in this section. Let δ = (d 1,...,d g ). For any positive integer n, let For any positive integer m, let K(nδ) := g j=1 n 1 di 1 Z/Z. K m := g j=1 m 1 Z/Z, and let K m := Hom(K m,c ). For any non-negative integer n and any a K(2 n δ), we will use the ollowing notation q n (a) = q n,δ (a) := θ[ a 0 ](0,2 n Ω), Q n (a) = Q n,δ (a) := θ[ a 0 ](2 n z,2 n Ω) or theta constants and theta unctions, where Ω H g has been suppressed. Clearly the ollowing symmetry condition holds: (Θ ev ) q n (a) = q n ( a) a K(2 n δ). (2.2) The generalized Riemann theta identities implies a whole amily o relations between theta unctions and theta constants. Among them are the quadratic relations (Θ n,δ quad ) between theta unctions with theta constants as coeicients below, as well as the quartic relations (Θ n,δ quad ) between theta constants, or all n 0. The theta constants satisy strong non-degeneracy conditions, represented by (Θnv n+1,δ ) below. (2.2.1) For any a,b,c K(2nδ) satisying a b c (mod K(nδ)) and any l K2, we have 0 = [ l(η) q n+1 (c + r) ] [ l(η) Q n (a + b + η) Q n (a b + r) ] (Θ n,δ quad ) [ l(η) q n+1 (b + r) ] [ l(η) Q n (a + c + η) Q n (a c + r) ]. (2.2.2) For any a,b,c,d K(2nδ) satisying a b c d (mod K(nδ)) and any l K2, we have 0 = [ l(η) q n (a + b + η)q n (a b + η) ] [ l(η) q n (c + d + η)q n (c d + η) ] (Θ n,δ quar) [ l(η) q n (a + d + η)q n (a d + η) ] [ l(η) q n (b + c + η)q n (b d + η) ]. 3

4 (2.2.3) For any n 0, any a 1,a 2,a 3 K(2 n+1 δ) and any l 1,l 2,l 3 K4, there exists an element b K 8 K(2δ) and an element λ 2K4 such that (Θ n+1,δ nondeg ) 0 3 i=1 [ η K 4 (l i + λ)(η) q n+1 (a i + b + η) ]. (2.3) Equations deining abelian varieties. The geometric signiicance o these theta relations are two olds, or abelian varieties and also their moduli. Recall that d 1,...,d n are positive integers with d 1 d g. Let N = ( g i=1 d i) Let A Ω,δ := C g /(Ω Z g + D(δ) Z g be the abelian variety whose period lattice is generated by the columns o (Ω D), where D(δ) is the diagonal matrix with d 1,...,d g as its diagonal entries. It was proved by Leschetz that the theta unctions { {Q 0,δ (a) a K(δ) } deine a projective embedding j : A Ω,δ P N i 4 d 1 d g. Mumord showed in [6, I 4] that the quadratic equations (Θ 0,δ quar) in the projective coordinates o P N cuts out the abelian variety A Ω,δ as a subvariety inside P N i 4 d 1. In particular an abelian variety is determined by their theta constants q 0,δ (a) s i the level δ is divisible by 4. The group law on the abelian variety can also be recovered rom these theta constants. 2. The next question is: do the Riemann quartic equations (Θ n,δ quar) on the theta constants cut out the moduli o abelian varieties? The answer given in [6, II 6] is basically yes i 8 d 1 with a suitable non-degeneracy condition: Suppose that 8 d 1 d g, and {q(a) a K(δ)} is a amily o complex numbers indexed by K(δ). Assume that this given g i=1 d i)-tuple o numbers has the ollowing properties. q(a) = q( a) or all a K(δ). All quartic equations in (Θ 0,δ quar) hold. All quartic equations in (Θ 0,δ quar) hold. (The non-degeneracy condition) There exists a amily o complex numbers {q 1 (u) u K(δ)} indexed by K(2δ) which satisies q 1 (u) = q( u) or all u K(2δ), the quartic equations (Θ 1,δ quar), the condition (Θ 1,δ nondeg ), and q 0 (a) q 0 (b) = Then there exists an element Ω H g such that q 1 (u) q 1 (v) u,v K(2δ), u+v=a, u v=b q 0,δ (a) := θ[ a 0 ](0,Ω) a K(δ). a,b K(δ). 4

5 (2.4) Adelic Heisenberg groups and theta measures. The analysis in [6] o theta relations is based on a inite adelic version o the Heisenberg group, which is a central extension o A 2g by the multiplicative group scheme G m over a base ield k in which 2 is invertible. Here A = p Q p, the restricted product o p-adic numbers, over all primes p which are invertible in k. One gets such a group scheme Ĝ(L ) whenever one is handed a symmetric ample line bundle o degree one over a g- dimensional principally polarized abelian variety over k, plus a compatible amily o theta structure or torsion points o order invertible in k, which induces an isomorphism rom A 2g to the set o all torsion points as above. Such a Heisenberg group Ĝ(L ) is isomorphic to the standard inite adelic Heinsenberg group Heis(2g,A ) over the algebraic closure k alg o k. The deinition o Heis(2g,A ) is similar to that o the real Heisenberg group Heis(2g,R) in [1, 2.1], with the ollowing changes: (a) the ield R is replaced by the ring A, (b) the unit circle group C 1 is replaced by the multiplicative group scheme G m over k alg, and (c) the isomorphism e : R/Z C 1 is replaced by an isomorphism rom A /Ẑ = p Q p /Z p to the group µ (k alg ) o all roots o 1 in k alg. The group Ĝ(L ) acts on the direct limit lim n Γ(A,L n ) =: Γ(L ), where n runs through all positive integers which are invertible in k. This action o Ĝ(L ) on Γ(L ) is an A -version o the dual Schrödinger representation discussed in [1, 2.3] At this point the representation-theoretic ormalism or theta unctions discssed in [1, 2] carry over to th present situation. The insight gained rom the systematic use o the adelic Schrödinger representation produces not only the two theorems in (2.3), but also a new way to look at theta constants: There exists a measure µ on A g which satisies the properties (i) (iv) below. All theta relations are encoded in the simple equality in (i), and the non-degeneracy condition becomes (iv). (i) There exists another measure ν on A g such that ξ (µ µ) = ν ν as measures on A g Ag, where ξ : A g Ag Ag Ag is the map (x,y) (x + y,x y). (ii) The algebraic theta constants are integrals over suitable compact open subsets o A g. (iii) The algebraic theta unction attached to a non-zero global section s 0 o the one-dimensional vector space Γ(A,L ) as above, is the unction x θ alg (x) = (U (1, x) δ 0 )dµ on A 2g, where δ 0 is the characteristic unction or the compact open subset p Z p A 2g, and U (1, x) δ 0 is the result o the action on δ 0 by the the element U (1, x) Heis(2g,A ) under the dual Schrödinger representation. (iv) For every x A 2g, there exists an element η 1 2 p Z p such that θ alg (x + η) 0. When the base ield is C, the theta measure µ Ω attached to Ω H g is µ Ω (V ) = n V Q g e ( 1 2 tn Ω n ) or all compact open subset V A g, and the companion measure ν Ω is ν Ω (V ) = n V Q g e ( 1 2 tn Ω n ). 5

6 (2.5) The best introduction to the circle o ideas in this section [9]; the readers may also consult [2, Ch. IV] and the original papers [6]. Anyone who had more than a casual look at the papers [6] would agree that the results are both undamental and deep, opening up a completely new direction in the study o theta unctions. These paper are however, not easy to read, and the ideas in them have not been developed very ar. 1 It is indeed curious that there has been no killer application o the theory o algebraic theta unctions so ar. However it should be a sae bet that this anomaly won t last much longer. Reerences [1] C.-L. Chai, Riemann s theta unction, this volume. [2] J. Igusa, Theta Functions. Springer-Verlag, [3] G. Kemp, Complex Abelian Varieties and Theta Functions. Springer-Verlag, [4] A. Krazer, Lehrbuch Der Thetaunktionen. Corred reprint o the 1903 irst edition, Chelsea, [5] A. Krazer and F. Prym, Neue Grundlagen einer Theorie der Allgemeiner Thetaunctionen. B. G. Teubner, [6] D. Mumord, On the equations deining abelian varieties, I III. Inv. Math. 1 (1966), ; 3 (1967), , [7] D. Mumord, with Appendices by C. P. Ramanujam and Y. Manin. Abelain Varieties. 2nd ed., TIFR and Oxord Univ. Press, Corrected re-typeset reprint, TIFR and Amer. Math. Soc., [8] D. Mumord, with the assistance o C. Musili, M. Nori, E. Previato and M. Stillman, Tata Lectures on Theta I. Birkh auser, [9] D. Mumord, with M. Nori and P. Norman, Tata Lectures on Theta III. Birkh auser, [10] F. Prym, Untersuchungen über die Riemann sche Thetaormel und Riemann sche Charakteristikentheorie. Druck und Verlag von B. O. Teubner, Leipzig, [11] B. Riemann, Theorie der Abel schen Functionen. In Bernhard Riemann s Gesammelte Mathematische Werke Und Wissenschatlicher Nachlass, 2nd ed., 1892 (reprinted by Dover, 1953), pp ; J. reine angew. Math. 54 (1857), pp [12] B. Riemann, Über das Verschwinden der Theta-Functionen. In Bernhard Riemann s Gesammelte Mathematische Werke Und Wissenschatlicher Nachlass, 2nd ed., 1892 (reprinted by Dover, 1953), pp ; J. reine angew. Math. 65 (1866), pp The quotes Mumord s words in the preace o [9]. 6