CHAPTER 1
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:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
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. Finally, if D is a canonical divisor, then
u(D) = -2K (mod J(W)).
IV. Aribitrary Unramified Coverings. For this section assume [b.bar]: W1 -> W0 is unramified but otherwise arbitrary. Continue the previous notation and let θj be the theta-divisor in J(Wj) for j = 0, l.
LEMMA 4: There exists a half-period, e1, in J(W1) so that
(1) [bar.a] θ0 [subset] θ1 + e1.
(e1 depends only on the canonical homology bases chosen.)
PROOF: Let D0 be an integral canonical divisor on W0 of degree 2p0-2. Then D0 -(2p0 - 2)z0 is a divisor of degree zero whose image under u0 is u0(D0) since u0(z0) = 0. [a.bar]D0 is canonical in W1 so we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Dividing by two gives
(2) [a.bar]K0 [equivalent to] K1 + (p0 - 1) u1 ([a.bar]z0) + e1
where 2e1 [congruent to] 0.
If g0 [member of] θ0, then there is a divisor D0 of degree p0 - 1 on W0 so that
u0(D0) + K0 [equivalent to] g0.
Thus
[a.bar]u0(D0 - (p0 - 1)z0) + [a.bar]K0 [equivalent to] [a.bar]g0
or
u1([a.bar]D0) - (p0 - 1)u1([a.bar]z0) + [a.bar]K0 [equivalent to] g0.
By formula (2) it follows that
[a.bar]g0 [equivalent to] u1([a.bar]D0) + K1 + e1.
Since the degree of [a.bar]D0 is n(p0 - 1) = p1 - 1 we have
u1([a.bar]D0) + K1 [member of] θ1.
Thus
[a.bar]g0 [member of] θ1 + e1.
q.e.d.
V. Abelian Covers. In this section assume [b.bar] : W1 -> W0 is a possibly ramified abelian cover; i.e., W0 = W1 / G where G is an abelian group of automorphisms of W1 whose elements may have fixed points. Let R be the set of characters of G; i.e., the set of homomorphisms of G into the multiplicative group of complex numbers of modulus one. Since G is a finite group, R is isomorphic to G although there is no canonical isomorphism. The group A is, however, canonically isomorphic to R. For suppose f is meromorphic on W1 and fn is a function lifted from W0. Then the divisor of f is invariant under G. Thus if T [member of] G, then f o T = χf(T)f where χf is easily seen to be a character. If f and g yield the same character, then f/g gives the identity character and so lies in M0. Thus the map f -> χf is an isomorphism of A into R. That this map is onto is seen by examining, for each χ [member of] R, the cyclic extension of M0 given by the fixed field for kernel χ. This completes a proof of Lemma 1 when M1 = MA. In the above situation where we have a function f whose divisor is invariant under G, we shall say that f is associated with the character χf.
Thus the field extension M1 over M0 can have as a vector space basis functions corresponding to each character of R. If f is an arbitrary function, then let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the fχ's which are not zero are linearly independent since they are associated with different characters.
VI. Unramified Abelian Covers. In this section assume [b.bar]: W1 -> W0 is an unramified abelian cover. In this context we give a proof of Lemma 2; that is, we now show that the kernel of [a.bar]: J(W0) -> J(W1) is isomorphic to A.
For the proof define a map from A into ker [a.bar] as follows. If f [member of] M*1 and fn [member of] M*0, then the divisor of f, (f), is invariant and so (f) = [bar.a]D0 where D0 is a divisor on W0 of degree zero. Let ωf = u0(D0). Then [a.bar]ωf = [a.bar]u0(D0) [equivalent to] u1 ([a.bar]D0) [equivalent to] 0, and so ωf [member of] ker [a.bar]. The map f -> ωf is easily seen to be well-defined on A and one-to-one, To show the map is onto suppose [a.bar]ω = 0. Then ω = u0(D0) where D0 is a divisor of degree zero on W0. Since [a.bar]D0 [equivalent to] 0, suppose (f)= [a.bar]D0. Since [a.bar]D0 is invariant under G, f o T = χf(T)f for some character χf. Consequently, fn [member of] M0. This completes the proof of Lemma 2 in the unramified abelian case.
Let us index ker [a.bar] by R rather than A; that is, ker [a.bar] = {ωχ | χ [member of] R} where ωχ is defined as follows. If u0(D0) = ωχ and (f) = [a.bar]D0, then f is associated with χ. The method of indexing, χ -> ωχ is an isomorphism of groups.
The reader is reminded that throughout this discussion canonical homology bases on W0 and W1 are fixed and all results are related to these bases. Thus the bases of analytic differentials are determined as are the theta functions and the half-period e1 of Lemma 4. Base points are also fixed and so maps u0 and u1 are defined, but these are in a sense auxiliary devices and the base points do not enter into the statements of the final results.
At this point let us introduce a convenient abuse of notation. If θ(u; B) is a theta function and τ = π i h + Bg, where g and h are real p-vectors, then there is an exponential function E(u) so that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
In this context write θ[τ](u) for the usual [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This notation will be extremely convenient and will lead to no confusion provided the canonical homology bases remain fixed.
A main result of this report is the following. By Lemma 3, there is an exponential, E(u), so that E(u)θ[e1]([a.bar]u; B1) is an nth order theta function with respect to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then there is a constant l ≠ 0 so that
(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
That this is a plausible result can be seen from Lemma 4. If ωχ [member of] ker [a.bar], then
[a.bar](θ0 - ωχ) [subset] θ1 + e1.
Thus if g0 [member of] θ0 - ωχ, then [a.bar]g0 [member of] θ1 + e1; that is, if θ[ωχ](g0; B0) vanishes, then so does θ[e1] ([a.bar]g0; B1). Thus the two sides of formula (3) are nth order theta functions and whenever the right-hand side vanishes, so does the left-hand side. Both sides have rational characters, half-integer characters in fact. That this is true for the left-hand side follows from the generalization of the usual transformation theory mentioned after the statement of Lemma 3. For the right-hand side it is true because the sum of the elements in an abelian group is always an element of order two. Formula (3) would be proven if we knew that e was not zero, but this does not seem immediate. The final theorem of this report among other things establishes formula (3) with an l ≠ 0.
