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Riemann’s theta formula - University of Pennsylvania

Riemann"s theta formula

Ching-Li Chai

version 12/03/2014 There is a myriad of identities satisfied by the Riemann theta functionq(z;W)and its close relativesqa b(z;W). The most famous among these theta relations is a quartic relation known to Riemann, associated to a 44 orthogonal matrix with all entries1; see 1.3. It debuted as formula (12) on p.20 of [10], and was namedRiemann"s theta formulaby Prym. In the preface of [10] Prym said that he learned of this formula from Riemann in Pisa, where he was with Riemann for several weeks in early 1865, and that he wrote down a proof following Riemann"s suggestions. For any fixed abelian variety, these theta identities give a set ofquadraticequations which defines this abelian variety. The coefficients of these quadratic equations are theta constants, or "thetanullwerte", which vary with the abelian variety. At the same time, the Riemann theta identi-

ties give a set ofquarticequations satisfied by the theta constants, which gives a systems of defining

equations of the moduli space of abelian varieties (endowed with suitable theta level structures). x1. Riemann"s theta formula We will first formulate a generalized Riemann theta identity, for theta functions attached to a quadratic form on a lattice. (1.1) DEFINITION. (THETA FUNCTIONS ATTACHED TO QUADRATIC FORMS)LetQbe aQ- valuedpositive definitesymmetric bilinear form on anh-dimensionalQ-vector spaceGQ, where his a positive integer. LetGGQbe aZ-lattice inGQ, i.e. a free abelian subgroup ofGQof rankh. Denote byG_QtheQ-linear dual ofGQ, and letG_:=fl2G_Qjl(G)Z:g. We identify elements ofQg QGQwithg-tuples of elements ofGQand similarly forQg QG_Q. (i) The pairingQgCg3(n;z)7!tnz2ConQgCgand the natural pairingGQG_Q!Q induces a pairingh;i:(Qg

QGQ)(Cg

QG_Q!C.

(ii) Let

˜Q:(Qg

GQ)(Qg

GQ)!Mg(Q)be the matrix-valued symmetric bilinear pairing Q:(u;v) = ((u1;:::;ug);(v1;:::;vg))!˜Q(u;v) =Q(ui;vj)

1i;jg8u;v2Qg

QG: (iii) For everyA2Qg

QGQ, everyB2Qg

QG_Qand every elementW2Hgof the Siegel upper-

half space of genusg, define the theta functionqQ;G[A

B]on thegh-dimensionalC-vector space

Partially supported by NSF grants DMS1200271

1 C

QG_Qattached to(Q;G)by

q Q;G[A

B](Z;W):=å

N2Zg ZGe12

Tr(W˜Q(N+A;N+A)e(hN+A;Z+Bi);

wheree(z):=exp(2pp1z)for allz2C.

Note that we haveqQQ0;GG0h(A;A0)

(B;B0)i((Z;Z0);W) =qQ;G[A

B](Z;W)qQ0;G0hA0

B

0i(Z0;W)for the orthogonal

direct sum(QQ0;GG0)of(Q;G)and(Q0;G0). In particular if(Q;G)is the orthogonal direct sum ofhone-dimensional quadratic forms, thenqQ;Gis a product ofh"usual" theta functions with characteristics. Let(Q;G)be aQ-valued positive definite quadratic form. LetT:LQ!GQbe aQ-linear isomorphism of vector spaces overQ, and letLbe aZ-lattice inLQ. LetT_:G_Q!LQbe the Q-linear dual ofT. LetQ0:LQLQ!Qbe the positive definite quadratic form onLQinduced byQthrough the isomorphismT. Let 1 T:Qg

QLQ!GQbe the linear map induced byT;

similarly for 1 T_:Cg

QG_Q!Cg

QL_Q. Let

K= (1 T)(Zg

ZL)=(Zg

ZG)\(1

T)(Zg

ZL)!(1

T)(Zg

ZL)+(Zg

ZL)=Zg

ZL D= (1

T_)1(Zg

ZL_)=(Zg

ZG_)\(1

T_)1(Zg

ZL_)!(1

T_)1(Zg

ZL_)+(Zg

ZL_)=Zg

ZL_ (1.2) THEOREM. (GENERALIZEDRIEMANN THETA IDENTITY)For every A2Qg GQand every B2Qg

G_Q, the equality

(R Q;T ch)qQ0;L (1 T)1A (1 T_)B ((1

T_)Z;W) =#(D)gå

A

02K;B02De(hA;B0i)qQ;GhA+A0

B+B0i(Z;W)

holds for allW2Hgand all Z2Cg G_Q.

Note that each term on the right hand side of (R

Q;T ch) is independent of the choice ofB0in its congru- ence class moduloZg ZL_. (1.3)Theorem 1.2 is very easy to prove once stated in that form. In two examples below(Q0;L)is the diagonal quadratic formx21++x2honZh,Gis alsoZn, andTis (given by) a matrix such that

TtTis a multiple of the identity matrixIh.

(a) Whenh=4,QandQ0are both the diagonal quadratic formx21+x22+x23+x24onZ4,A=B=0 andTis given by the orthogonal matrix0 B

B@1 1 1 1

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