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An Alternative Approach to Algebro-Geometric Solutions of the AKNS Hierarchy

We develop an alternative systematic approach to the AKNS hierarchy based on elementary algebraic methods. In particular, we recursively construct Lax pairs for the entire AKNS hierarchy by introducing a fundamental polynomial formalism and establish the basic algebro-geometric setting including associated Burchnall-Chaundy curves, Baker-Akhiezer functions, trace formulas, Dubrovin-type equations for analogs of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.


Introduction
The principal aim of this paper is an alternative elementary algebraic approach to the entire AKNS hierarchy in the spirit of previous treatments of the Korteweg-de Vries (KdV), Boussinesq, and Toda hierarchies. More precisely, we advocate a fundamental polynomial formalism to recursively construct Lax pairs for the AKNS hierarchy, that is, pairs (D, E n+1 ) of matrix-valued differential expressions of order one (i.e., D) and n + 1 (i.e., E n+1 ) with D of the Dirac-type. In addition, we establish the basic algebro-geometric setup for special classes of solutions of the AKNS hierarchy including solitons, rational solutions, algebro-geometric quasi-periodic solutions, and limiting cases thereof. Our treatment includes a systematic approach to Burchnall-Chaundy curves, Baker-Akhiezer functions, trace formulas, Dubrovin-type equations describing the dynamics of Dirichlet and Neumann divisors, and theta function representations for algebro-geometric solutions.
Before we enter a description of the contents of each section, it seems appropriate to comment on existing treatments of this subject and to justify the addition of yet another detailed account on this topic. The theory of commuting matrix-valued differential expressions and, more generally, the algebro-geometric approach to matrix hierarchies of soliton equations has been developed in great generality by Dubrovin and Krichever. Corresponding authoritative accounts can be found, for instance, in [5], Chs. 3, 4, [14], [15], [16], [30], [31], [35], [36], [44], and the references therein. In contrast to these references, our own approach relies on two basic ingredients, an elementary polynomial approach to Lax pairs (or zero-curvature pairs) of the AKNS hierarchy and its explicit connection with a fundamental meromorphic quantity φ (cf. (3.10), (4.23)) which allows for a unified algebro-geometric treatment of the entire AKNS hierarchy.
In section 2 we describe a recursive approach to Lax pairs (and zero-curvature pairs) of the AKNS hierarchy following Al'ber's treatment of the KdV and nonlinear Schrödinger hierarchies [1], [2], [3] and establish its connection with the Burchnall-Chaundy theory 1 [7], [8], [9] and hence with hyperelliptic curves. Combining the recursive formalism of Section 2 with a polynomial approach to represent positive divisors of degree n of a hyperelliptic curve of genus n originally developed by Jacobi [33] and applied to the KdV case by Mumford [41], Section III.a.1 and McKean [38] (see also [17], [45]), a detailed analysis of the stationary AKNS hierarchy is provided in Section 3. This includes, in particular, the theta function representation of algebro-geometric solutions of the stationary AKNS hierarchy. The corresponding time-dependent formalism is then developed in detail in Section 4. Appendix A collects the relevant material for hyperelliptic Riemann surfaces and their theta functions. Appendix B contains a simple illustration of the Riemann-Roch theorem (cf. Theorem B.1).
We emphasize that our treatment comprises, in particular, the important special case of the nonlinear Schrödinger (NS) equation (cf. (3.87)), whose algebro-geometric solutions have been studied, for instance, in [5], Ch. 4, [15], [30], [31], [37], [39], [44]. Similarly, the case of the modified Korteweg-de Vries (mKdV) equation (cf. (3.89), whose algebrogeometric solutions have been studied, for instance, in [21], [22], are included as a special case of our formalism. Moreover, the present elementary approach is not at all restricted to the AKNS hierarchy but applies quite generally to 1+1-dimensional hierarchies of soliton equations. In fact, the KdV case has been treated in [27], the case of the Toda and Kac-van Moerbeke hierarchies in [6], and the case of the Boussinesq hierarchy in [13].
Finally, we mention that a combination of the AKNS formalism developed in this paper and the Picard-type techniques introduced in a recent explicit characterization of all elliptic solutions of the KdV hierarchy [26] (see also [25]) are expected to yield a similar characterization of all elliptic solutions of the AKNS hierarchy, a topic that continues to attract considerable interest (see, e.g., [5], Ch. 7, [10], [47]).
Suppose p, q ∈ C ∞ (R) (or meromorphic on C) and introduce the Dirac-type matrixvalued differential expression (2.1) In order to explicitly construct higher-order matrix-valued differential expressions E n+1 , n ∈ N 0 (= N ∪ {0}) commuting with D, which will be used to define the stationary AKNS hierarchy later, one can proceed as follows.
Next, we introduce polynomials F n , G n+1 , H n with respect to z ∈ C, and note that (2.6) respectively, (2.7) become (2.12) (2.10)-(2.12) yield that and hence where the integration constant R 2n+2 (z) is a monic polynomial in z of degree 2n + 2. Thus one may write Explicitly, one obtains for the first few polynomials in (2.9), , H 0 = ip, etc.
One can use (2.10)-(2.12) and (2.14) to derive differential equations for F n and H n separately by eliminating G n+1 . One obtains for F n , and upon dividing (2.17) by q 2 and differentiating the result with respect to x, Similarly one obtains for H n , (2.17) and (2.19) can be used to derive recursion relations for f ℓ and h ℓ in the homogeneous case where all c ℓ = 0, ℓ ∈ N (cf. Lemma 4.5). This has interesting applications to the high-energy expansion of the Green's matrix of D as briefly discussed in Remark 4.6.
Next, introducing a deformation parameter t n ∈ R in (p, q) (i.e., (p(x), q(x)) → (p(x, t n ), q(x, t n ))), the time-dependent AKNS hierarchy (cf., e.g., [42], Chs. 3, 5 and the references therein) is defined as the collection of evolution equations (varying n ∈ N 0 ), 27) or equivalently, by Explicitly, one obtains for the first few equations in (2.28) or (2.29), and obtained the recursion relation (2.2) and the remaining stationary results of this section as a consequence of (2.9) and (2.23). Similarly, starting with  In particular, the hyperelliptic curve K n in (2.26) is then obtained from the characteristic equation for V n+1 (z, x), Similarly, the time-dependent part (2.28)-(2.30), being based on the Lax equation (2.27), can equivalently be developed from the zero-curvature equation In fact, since the latter approach (2.37) is almost universally adopted in the contemporary literature on the AKNS hierarchy, we decided to recall its proper origin in connection with the Lax pair [E n+1 , D] and based our treatment on matrix-valued differential expressions instead.

The Stationary AKNS Formalism
In this section we continue our discussion of the AKNS hierarchy and concentrate on the stationary case. Following [27], where the analogous treatment of the stationary KdV hierarchy can be found, we outline the connections between the polynomial approach described in Section 2 and a fundamental meromorphic function φ(P, x) defined on the hyperelliptic curve K n . Moreover, we discuss in detail the associated stationary Baker-Akhiezer function Ψ(P, x, x 0 ), the common eigenfunction of D and E n+1 (we recall that [E n+1 , D] = 0), and associated positive divisors of degree n on K n (which should be considered as the analogs of Dirichlet and Neumann divisors in the KdV context).
We recall the hyperelliptic curve (2.26), where n ∈ N 0 will be fixed throughout this section and denote its compactification (adding the points ∞ ± ) by the same symbol. Thus K n becomes a (possibly singular) two-sheeted hyperelliptic Riemann surface of arithmetic genus n in a standard manner. We shall introduce a bit more notation in this context (see Appendix A for more details). Points P on K n are represented as pairs P = (z, y) satisfying (3.1) together with ∞ ± = (∞, ±∞), the points at infinity. The complex structure on K n is defined in the usual way by introducing local coordinates ζ P 0 : P → (z − z 0 ) near points P 0 ∈ K n which are neither branch nor singular points of K n , ζ ∞ ± : P → 1 z near ∞ ± , and similarly at branch and/or singular points of K n . The holomorphic sheet exchange map (involution) * is defined by * : K n → K n P = (z, y) → P * = (z, −y).

(3.2)
A detailed description of K n and its complex structure in the two most frequently discussed cases in applications where either E m ∈ R, 0 ≤ m ≤ 2n + 1 or Finally, positive divisors on K n of degree are denoted by Given these preliminaries, let Ψ(P, x, x 0 ) denote the common normalized eigenfunction of D and E n+1 , whose existence follows from the commutativity of D and E n+1 (cf., eg., [7], [8] in the case of scalar differential expressions), that is, due to for a given n ∈ N 0 , or equivalently, due to the requirement, Explicitly, this yields for some fixed x 0 ∈ R with the assumed normalization, Ψ(P, x, x 0 ) is called the Baker-Akhiezer (BA) function. Closely related to Ψ(P, x, x 0 ) is the following meromorphic function φ(P, x) on K n , defined by Since φ(P, x) will be the fundamental object for the stationary AKNS hierarchy, we next seek its connection with the recursion formalism of Section 2. Recalling (2.23), one infers and hence by (3.8), , P = (z, y) ∈ K n . (3.10) By (2.9) we may write, one infers from (3.10) that the divisor (φ(P, x)) of φ(P, x) is given by ν(x) = (ν 1 (x), . . . ,ν n (x)),μ(x) = (μ 1 (x), . . . ,μ n (x)).
Next we summarize a variety of properties of φ(P, x) and Ψ(P, x, x 0 ).
In order to motivate our introduction of the basic quantity φ(P, x) we started with the common eigenfunction ψ(P, x, x 0 ) of D and E n+1 . However, given (2.14) we could have defined φ(P, x) as in (3.10) and then verified that Ψ = ψ 1 ψ 2 defined by (3.8) and (3.22) satisfies (3.6) and (3.7).
Combining the polynomial approach of Section 2 with (3.11) and (3.12) readily yields trace formulas for the AKNS invariants. We indicate the first few of these below. etc. Here Proof. Finally, we shall provide an explicit representation of Ψ, φ, p, and q in terms of the Riemann theta function associated with K n . We freely employ the notation established in Appendix A. In order to avoid the trivial case n = 0 (considered in Example 3.8) we assume n ∈ N for the remainder of this argument.
Assuming K n to be nonsingular, that is, we choose, without loss of generality, the base point P 0 = (E 0 , 0) and denote by A P 0 (.), α P 0 (.) the Abel maps as defined in (A.30)-(A.32), and defineΞ P 0 , the vector of Riemann constants, by Next, consider the normal differential of the third kind ω ∞ + ,∞ − , which has simple poles at ∞ + and ∞ − , corresponding residues +1 and −1, vanishing a-periods, and is holomorphic otherwise on K n . Hence we have (cf. (A.36), (A.37)) where the numbers {λ j } 1≤j≤n are determined by the normalization (3.36). The Abelian differential of the second kind ω Next, we formulate the following auxiliary result.
is multi-valued meromorphic on K n and its divisor satisfies then ψ(., x) is unique up to a constant multiple (which may depend on x).
For completeness we also mention another theta function representation for the product p(x)q(x), originally due to [30].
Corollary 3.6. Assume the hypotheses of Theorem 3.5. Then Next, using By the uniqueness of ψ 1 (P, x, x 0 ) as discussed in Lemma 3.4 one concludes and hence where we combined (3.41) and (A.35) in the second equality. Since p(x)q(x) only depends on c 1,x (x) as opposed to c 1 (x) itself, it suffices to consider the following expansion near ∞ + .
In the particular case of the nonlinear Schrödinger (NS) hierarchy, where p(x, t n ) = ±q(x, t n ), n ∈ N 0 , (3.87) (3.80) further restricts A to be unimodular, that is, In the special case of the modified Korteweg-de Vries (mKdV) hierarchy, where Next, we briefly consider the trivial case n = 0 excluded in Theorem 3.5.
Example 3.8. Assume n = 0. Then Finally, we mention an interesting characterization of all algebro-geometric AKNS potentials due to De Concini and Johnson [11] in the special case where D generates a self-adjoint operator in L 2 (R) ⊗ C 2 . In this case the algebro-geometric potentials are characterized by the fact that the corresponding spectrum consists of finitely many intervals and the Lyapunov exponent vanishes a.e. on the spectrum. In this context we might also point out that a detailed study of Floquet theory for periodic and selfadjoint AKNS operators (not necessarily of algebro-geometric type) generated by D can be found in [28].

The Time-Dependent AKNS Formalism
In our final section we indicate how to generalize the polynomial approach of Sections 2 and 3 to the time-dependent AKNS hierarchy. Our starting point is a stationary n-gap solution ( for some fixed n ∈ N 0 and a given set of integration constants {c ℓ } 1≤ℓ≤n+1 . Our principal aim is to construct the r th AKNS flow for t 0,r ∈ R and some fixed r ∈ N 0 . In terms of Lax pairs this amounts to solving [E n+1 (t 0,r ), D(t 0,r )] = 0. (4.5) As a consequence one obtains that since the AKNS flows are isospectral deformations of D(t 0,r ).
We emphasize that the integration constants {c ℓ } inẼ r+1 and {c ℓ } in E n+1 , in general, are independent of each other (even if r = n). Hence we shall employ the notatioñ E r+1 ,Ṽ r+1 ,F r ,G r+1Hr ,f l ,g l ,h l ,c l , etc., in order to distinguish it from E n+1 , V n+1 , F n , G n+1 , H n , f l , g l , h l , c l , etc. In addition, we followed a more elaborate notation inspired by Hirota's τ -function approach and indicated the individual r th AKNS flow by a separate time variable t r ∈ R. (The latter notation suggests considering all AKNS flows simultaneously by introducing t = (t 0 , t 1 , t 2 , ........).) Instead of working directly with (4.4), (4.5) and (4.6), it is more convenient to take the zero-curvature equations (2.37) as our point of departure, that is, we start from where (cf. (2.9)) 0 (x) = 1, for fixed t 0,r ∈ R, n ∈ N 0 , r ∈ N 0 . Here f ℓ (x, t r ) and f ℓ (x) are defined as in (2.2) with (p(x), q(x)) replaced by (p(x, t r ), q(x, t r )) and (p (0) (x), q (0) (x)), respectively. Explicitly, (4.8) and (4.9) are equivalent to p tr = −iH r,x − 2zH r + 2ipG r+1 , (4.14) respectively. In particular, (2.14) holds in the present t r -dependent setting, that is, In analogy to (3.11) and (3.12) we write and define in analogy to (3.10), the following meromorphic function φ(P, x, t r ) on K n , the fundamental ingredient for constructing algebro-geometric solutions of the timedependent AKNS hierarchy, , (4.23) P = (z, y) ∈ K n .
(4.40) 21 Proof. (4.30) follows from (4.9) and (4.23). (4.31) can be proven as follows. Using (4.3) and (4.30) one infers by a straight forward (but rather lengthy) calculation that where C is independent of x (but may depend on P and t r ). By inspection of (4.23), the left-hand side of (4.42) is meromorphic on K n while the right-hand side of (4.42) is not meromorphic at ∞ + and ∞ − unless C = 0. Hence one infers C = 0 and thus (4.31). Next we consider the t r -dependence of F n (z, x, t r ), G n+1 (z, x, t r ), and H n (z, x, t r ).
(4.50) (iv). ψ 1 (P, x, x 0 , t r , t 0,r )ψ 2 (P * , x, x 0 , t r , t 0,r ) + ψ 1 (P * , x, x 0 , t r , t 0,r )ψ 2 (P, x, x 0 , t r , t 0,r ) = 2G n+1 (z, x, t r ) F n (z, x 0 , t 0,r ) . The dynamics of the zeros µ j (x, t r ) and ν j (x, t r ) of F n (z, x, t r ) and H n (z, x, t r ), in analogy to Lemma 3.2, is then described in Lemma 4.4. Assume (4.14)-(4.22) and let (x, t r ) ∈ R 2 . Then The initial condition The trace relations in Lemma 3.3 extend in a one-to-one manner to the present timedependent setting by substituting (p(x), q(x)) → (p(x, t r ), q(x, t r )), (4.58) (µ j (x), ν j (x)) → (µ j (x, t r ), ν j (x, t r )), 1 ≤ j ≤ n, It remains to provide the explicit theta function representation of Ψ, φ, p, and q. We rely on the notation established in Section 3 and Appendix A in the following, assuming K n to be nonsingular as in (3.33). As in Section 3 we assume n ∈ N for the remainder of this argument.
In addition to (3.34)-(3.42) we need to introduce the Abelian differentials of the second kind ω with {c ℓ } 1≤ℓ≤r ,c 0 = 1 the integration constants inF r . Moreover, writing Before we can prove the main result of this section we need the following auxiliary result which is of independent interest due to its implications for the Green's matrix of the differential expression D.
Lemma 4.5. Let P ∈ K n \{∞ + , ∞ − } and (x, t r ) ∈ R 2 . Then, for P = (ζ −1 , y) near ∞ ± , one obtains the asymptotic expansions wheref k (x, t r ) andĥ k (x, t r ) denote the homogeneous coefficients f k (x, t r ) and h k (x, t r ) in (4.11) and (4.13) (i.e., the ones satisfying (2.2) with all integration constants c ℓ = 0, ℓ ∈ N). Explicitly,f k (x, t r ) andĥ k (x, t r ) can be computed from the recursion relations, by (2.17). Moreover, where we chose the branch Insertion of (4.72) into (4.70) yields the recursion relation (4.67) which represents the homogeneous solutions for the f ℓ due to the lack of any possible integration constants in (4.67). (4.66) and (4.68) are proved in the same manner using (2.19).
The recursion technique in Lemma 4.5 represents the AKNS analog of the recursive KdV approach in Sections 2-4 of [46]. Lemma 4.5 has interesting consequences for the asymptotic high-energy expansion of the Green's matrix G(z, x, x ′ ) of D (i.e., the integral kernel of the resolvent (D − z) −1 ) as described in the following remark.
Remark 4.6. Let D be given by (2.1) (p, q not necessarily algebro-geometric coefficients) and assume that for some z ∈ C, and all x 0 ∈ R, ψ 1,± (z, .), ψ 2,± (z, .) ∈ L 2 ((x 0 , ±∞)) satisfy (2.22), that is, Then the Green's matrix G(z, x, x ′ ), x = x ′ of D is given by where the Wronskian is x-independent. Note that G(z, x, x ′ ) is continuous at x = x ′ in its off-diagonal elements but discontinuous on the diagonal. In the special algebro-geometric context we may replace where (cf. (4.69)) denote homogeneous quantities encountered in Lemma 4.5. Since G(P, x, x ′ ) is discontinuous at x = x ′ , we introduced the arithmetic mean of the corresponding one-sided limits following the usual treatment of first-order systems (see, e.g., [4], Sect. 9.4). In fact, the arithmetic mean in (4.79) leads to the characteristic function of D (in the terminology of [4], Sect. 9.5), the fundamental object for studying spectral properties of D. The asymptotic expansions (4.65) and (4.66) forF (P, x) andĤ(P, x) as P → ∞ ± then determine the off-diagonal asymptotic high-energy expansions of of the arithmetic mean of the diagonal Green's matrix in (4.79). Similarly, using (2.10) or (2.12), one obtains the asymptotic high-energy expansion of (4.79) as P → ∞ ± for its diagonal elements. Even though (4.65), (4.66), (4.79), and (4.81) were derived in the special algebro-geometric context, we emphasize, however, that the asymptotic expansion of (4.79) as P → ∞ ± only involves the homogeneous coefficientsf k (x),ĥ k (x) which are universal differential polynomials in (p(x), q(x)). Thus, identifying Ψ ± (z, x) and Ψ(P, x, x 0 ), Ψ(P * , x, x 0 ) as in (4.77) yields the universal high-energy expansion of the arithmetic mean of the diagonal Greens matrix G(z, x, x ′ ) of D as z → ∞ in the general (not necessarily algebro-geometric) case. The recursive and hence systematic approach to this high-energy expansion, based on (4.65)-(4.68), appears to be new.
The theta function representations for φ, Ψ, and (p, q) then finally read as follows.
Since Corollary 3.6 extends to the present time-dependent setting in a straightforward manner we record the corresponding result without proof, The open constant q(x 0 , t 0,r ) in (4.86)-(4.88) is inherent to the AKNS formalism as discussed in Lemma 3.7.
We note again that the results for Ψ and (p, q) in Theorem 4.7 are known and can be found, for instance, in [5], Ch.4, [14], [15], [30], [31], and [44]. Our main new contribution to this circle of ideas is the elementary alternative derivation of Theorem 4.7 based on the fundamental meromorphic function φ on K n and its connection with the polynomial recursion formalism of Section 2.

Appendix A. Hyperelliptic Curves and Theta Functions
We briefly summarize our notation and some of the basic facts on hyperelliptic curves and their theta functions as employed in Sections 3 and 4. For background information on this standard material we refer, for instance, to [18], Chs. I-III, IV, [19], [29], Ch. 2, [34], Ch. X, [40], Ch. 2.
Consider the points and introduce an appropriate set of n + 1 (nonintersecting) cuts C j joining E j and E k(j) , where E k(j) = E j for some j is permitted in order to include singular curves. Denote where the finite index set J ⊂ {0, 1, . . . , 2n + 1} has cardinality n + 1 and define the cut plane Π, Next, introduce the holomorphic function and the set of branch and/or singular points. M n becomes a Riemann surface upon introducing appropriate charts (U P 0 , ζ P 0 ) defined in a standard manner. Let Similarly, local coordinates for branch and/or singular points P 0 ∈ B s are defined as ζ P 0 (P ) = (z − z 0 ) r/2 for appropriate r = 1 or 2. For the reader's convenience we provide a detailed treatment of branch points in the nonsingular case (where E m = E m ′ for m = m ′ ) for the two most frequently occurring situations, the self-adjoint case where {E m } 0≤m≤2n+1 ⊂ R and the case where {E m } 0≤m≤2n+1 = {ǫ ℓ , ǫ ℓ } 0≤ℓ≤n consists of complex conjugate pairs at the end of this appendix.
In addition, it is useful to consider the subsets Π ± ⊂ M n (i.e., upper and lower sheets) .10) and the associated charts (A.11) (A.8), (A.9), and the corresponding charts for P 0 ∈ B s define a complex structure on M n . We shall denote the resulting Riemann surface by K n . In general, K n is a (possibly singular) curve of (arithmetic) genus n.
Next, consider the holomorphic sheet exchange map (involution) * : .12) and the two meromorphic projection maps π : π has poles of order 1 at ∞ ± and R 2n+2 (z) 1/2 has poles of order n + 1 at ∞ ± . Moreover, Thus K n is a two-sheeted ramified covering of the Riemann sphere C ∞ ( ∼ = C ∪ {∞}), K n is compact (sinceπ is open and C ∞ is compact), and K n is hyperelliptic (since it admits the meromorphic functionπ of degree two).
(A. 16) One infers that for n ∈ N, dπ/y is a holomorphic differential on K n with zeros of order n − 1 at ∞ ± and hence form a basis for the space of holomorphic differentials on K n .
Next we introduce a canonical homology basis {a j , b j } 1≤j≤n for K n where the cycles are chosen such that their intersection matrix reads Introducing the invertible matrix C in C n , the normalized differentials ω j , 1 ≤ j ≤ n, form a canonical basis for the space of holomorphic differentials on K n . The matrix τ in C n of b-periods, In the charts (U ∞ ± , ζ ∞ ± ≡ ζ) induced by 1/π near ∞ ± one infers Associated with the homology basis {a j , b j } 1≤j≤n we also recall the canonical dissection of K n along its cycles yielding the simply connected interiorK n of the fundamental polygon ∂K n given by The Riemann theta function associated with K n is defined by θ(z) = n∈Z n exp[2πi(n, z) + πi(n, τ n)], z = (z 1 , . . . , z n ) ∈ C n , (A. 26) where (u, v) = n j=1 u j v j denotes the scalar product in C n . It has the fundamental properties θ(z 1 , . . . , z j−1 , −z j , z j+1 , . . . , z n ) = θ(z), A divisor D on K n is a map D : K n → Z, where D(P ) = 0 for only finitely many P ∈ K n . The set of all divisors on K n will be denoted by Div(K n ). With L n we denote the period lattice L n := {z ∈ C n | z = m + τ n, m, n ∈ Z n } (A. 28) and the Jacobi variety J(K n ) is defined by The Abel maps A P 0 (.) respectively α P 0 (.) are defined by with P 0 ∈ K n a fixed base point. (In the main text we agree to fix P 0 = (E 0 , 0) for convenience.) In connection with (A.25) we shall also need the maps (cf. (3.34)) with path of integration lying inK n .
Let M(K n ) and M 1 (K n ) denote the set of meromorphic functions (0-forms) and meromorphic differentials (1-forms) on K n . The residue of a meromorphic differential ν ∈ M 1 (K n ) at a point Q 0 ∈ K n is defined by where γ Q 0 is a counterclockwise oriented smooth simple closed contour encircling Q 0 but no other pole of ν. Holomorphic differentials are also called Abelian differentials of the first kind (dfk). Abelian differentials of the second kind (dsk) ω (2) ∈ M 1 (K n ) are characterized by the property that all their residues vanish. They are normalized, for instance, by demanding that all their a-periods vanish, that is, If ω (2) P 1 ,n is a dsk on K n whose only pole is P 1 ∈K n with principal part ζ −n−2 dζ, n ∈ N 0 near P 1 and ω j = ( ∞ m=0 d j,m (P 1 )ζ m ) dζ near P 1 , then b j ω (2) P 1 ,n = 2πi n + 1 d j,n (P 1 ). (A.35) Any meromorphic differential ω (3) on K n not of the first or second kind is said to be of the third kind (dtk). A dtk ω (3) ∈ M 1 (K n ) is usually normalized by the vanishing of its a-periods, that is, P 1 ,P 2 associated with two points P 1 , P 2 ∈K n , P 1 = P 2 by definition has simple poles at P 1 and P 2 with residues +1 at P 1 and −1 at P 2 and vanishing a-periods. If ω where the path from Q to P lies inK n (i.e., does not touch any of the cycles a j , b j ).
We shall always assume (without loss of generality) that all poles of dsk's and dtk's on K n lie onK n (i.e., not on ∂K n ). (iv) (Jacobi's inversion theorem). Assume n ∈ N, then α P 0 : Div(K n ) → J(K n ) is surjective.
For notational convenience we agree to abbreviate and, for Q = (Q 1 , . . . , Q n ) ∈ σ n K n (σ n K n the n-th symmetric power of K n ), Moreover, σ n K n can be identified with the set of positive divisors 0 < D ∈ Div(K n ) of degree n.