VANISHING PROPERTIES OF THETA FUNCTIONS FOR ABELIAN COVERS OF RIEMANN SURF ACES

(unramified case)

Robert D. M. Accola

1. Introduction. The vanishing properties of hyperelliptic theta functions have been known since the last century. Recently, Farkas discovered special vanishing properties for theta functions associated with surfaces which admit fixed-point free automorphisms of period two. The author has discovered other vanishing properties for special surfaces admitting abelian automorphism groups of low order. The purpose of this report is to give a partial exposition of a theory that will subsume most of the above cases in a general theory. Due to limitations of time and space, a full exposition must be postponed.

Let W1 be a closed Riemann surface of genus p1, p1 ≥ 2, admitting a finite abelian group of automorphisms, G. The space of orbits of G, W/G (= W0), is naturally a Riemann surface so that the quotient map, [b.bar], is analytic. In this report we shall develop the theory in the case where no element of G other than the identity has a fixed point; that is, the map [b.bar] : W1 -> W0 is without ramification. We shall, however, state some theorems in a more general context especially when proofs are omitted.

II. Remarks on General Coverings. Let [b.bar] : W1 -> W0 be an arbitrary n-sheeted ramified covering of closed Riemann surfaces of genera p1 and p0 respectively. Let M1 be the field of meromorphic functions on W1 and let M0 be the lifts, via [b.bar]. of the field of meromorphic functions on W0. Then M0 is a sub field of M1 of index n. We now define an important abelian group, A, as follows:

Definition: A, {f [member of] M*1| fn [member of] M*0}/M*0.

Now let MA be the maximal abelian extension of M0 in M1.

Lemma 1: A is isomorphic to the (dual of the) Galois group of MA over M0.

PROOF: (omitted). A proof in the case where M1 = MA will follow in Section V.

Now, fix a point in W1, z1, and let z0 = [b.bar](z1). Fix canonical homology bases in W1 and W0 and choose bases for the analytic differentials dual to these homology bases. Thus maps u1 and u0 from W1 and W0 into their Jacobians, J(W1) and J(W0), are defined:

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The maps u1 and u0 are extended to divisors in the usual way. A map [a.bar] is now defined from divisors on W0 to those of W1 as follows: for x0 [member of] W0, [a.bar]x0 is the inverse image of x0 under [b.bar] with branch points counted according to multiplicity. Thus [a.bar]x0 always has degree n. [bar.a] is extended by linearity to arbitrary divisors on W0. Now we define a map from J(W0) -> J(W1), again denoted by [a.bar] as follows: if D0 is a divisor on W0 of degree zero, then [a.bar]u0(D0) = u1 ([a.bar]D0). [a.bar] is easily seen to be a homomorphism. Let MUA be a maximal unramified abelian extension of M0 in M1; thus M0 [subset] MUA [subset] MA [subset] M1.

LEMMA 2: The kernel of [a.bar]: J(W0) -> J(W1) is isomorphic to the Galois group of MUA over M0.

PROOF: (omitted). A proof in the case when MUA = M1 will follow in Section VI.

With the homology bases and the dual bases of analytic differentials chosen, let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the corresponding period matrices where E is the appropriate identity matrix. Finally let θ[χ0] and θ[χ1](u; B1) be the corresponding first order theta functions with arbitrary characteristic.

LEMMA 3: For any characteristic χ1 there is an exponential function E(u) so that E(u) θ [χ1] ([a.bar]u; B1), as a multiplicative function on J(W0), is an nth order theta function.

PROOF: (omitted). The proof is an immediate adaptation of the simplest parts of transformation theory.

III. Resume of the Riemann Vanishing Theorem. The proofs of the results in this report depend on Riemann's solution to the Jacobi inversion problem. We summarize here those portions of the theory that will be needed later.

Let W be a closed Riemann surface of genus p, p ≥ l, let a canonical homology basis be chosen, let a dual basis of analytic differentials be chosen, let a base point be chosen, and let u be the map of W into J(W).

Riemann's theorem asserts the existence of a point K in J(W) so that if we choose any e [member of] J(W), then there is an integral divisor D on W of degree p so that

u(D) + K = e(mod J(W)).

If θ(e) ≠ 0, then D is unique. If θ(e) = 0, then the above equation can be solved with an integral divisor of degree p - 1. Moreover, in this latter case, the order of vanishing of θ(u) at e equals i(D), the index of speciality of D. (By the Riemann-Roch theorem, i(D) equals the number of linearly independent meromorphic functions which are multiples of -D since the degree of D is p - 1.) Moreover, θ(u(D) + K) = 0 whenever D is an integral divisor of degree at most p - 1....