Abstract
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MATHEMATICS
R. N. ABDULAEV
THE HOMOGENEOUS RIEMANN PROBLEM ON CLOSED RIEMANN SURFACES
(Presented by Academician I. N. Vekua, August 3, 1964)
Let, on a closed Riemann surface \(R\) of genus \(\rho\), there be given a contour \(\Gamma\), consisting of a finite number of mutually nonintersecting smooth closed oriented curves, and a function \(G \in H(\Gamma)\), \(G \ne 0\). In [1] it was established that, for \(\chi = \operatorname{ind}_{\Gamma} G \ge 0\), the solvability of the homogeneous Riemann problem for analytic functions, defined by the boundary condition on the contour \(\Gamma\)
\[ \Phi^{+}=G\Phi^{-}, \tag{1} \]
is equivalent to the solvability of the congruence
\[ \sum_{j=1}^{\chi} u_\nu(D_j)\equiv \lambda_\nu \pmod{\text{periods}} \quad (\nu=1,2,\ldots,\rho), \tag{2} \]
where \(\lambda_1,\lambda_2,\ldots,\lambda_\rho\) are numbers depending on the coefficient \(G\) and on the surface \(R\) (for all notation see [1, 2]), and, in terms of the Riemann \(\vartheta\)-function, necessary and sufficient conditions for the solvability of this congruence were given. In the present work we obtain explicit expressions for the number \(K\) of linearly independent solutions of problem (1) for \(0<\chi\le \rho-1\) (Theorem 1) and for \(\rho\le \chi\le 2\rho-2\) (Theorem 2).
Define, on the \(\rho\)-dimensional complex space, the function \(\chi(z_1,z_2,\ldots,z_\rho)=\chi(z_\nu)\) as follows: \(\chi(z_\nu^0)=-1\) if \(\vartheta(z_\nu^0)=\vartheta(z_1^0,z_2^0,\ldots,z_\rho^0)\ne 0\), and \(\chi(z_\nu^0)=n\) if the function \(\vartheta(z_\nu)\), together with all its partial derivatives up to order \(n\) inclusive, vanishes at the point \((z_1^0,z_2^0,\ldots,z_\rho^0)\), but at least one derivative of the function \(\vartheta(z_\nu)\) of order \(n+1\) at this point does not vanish. From the properties of the Riemann \(\vartheta\)-function it follows that, if \((e_\nu)\equiv(g_\nu)\pmod{\text{periods}}\), then \(\chi(e_\nu)=\chi(g_\nu)\) [3].
The subsequent arguments are based on the following assertion [3]: if \(\chi(e_\nu+k_\nu)=n\ge 0\), then
\[ e_\nu=\sum_{j=1}^{\rho-1} u_\nu(D_j)\pmod{\text{periods}} \quad (\nu=1,2,\ldots,\rho), \tag{3} \]
and, in modern terms, \(\dim D_1D_2\ldots D_{\rho-1}=n+1\), and conversely, \((k_1,k_2,\ldots,k_\rho\) are fixed numbers depending only on the surface; [2], Ch. IX, equality (16)).
Lemma 1. Any integral divisor of order \(\chi\), \(0<\chi\le \rho-1\), \(D_1D_2\ldots D_\chi\), can be extended to a divisor of order \(\rho-1\), \(D_1D_2\ldots D_{\rho-1}\), in such a way that
\[ \dim D_1D_2\ldots D_\chi=\dim D_1D_2\ldots D_\chi D_{\chi+1}\ldots D_{\rho-1}. \]
Theorem 1. If \(0<\chi\le \rho-1\), then
\[ K=\min_{P_j\in R}\chi\left(\sum_{j=1}^{\rho-\chi-1}u_\nu(P_j)+\lambda_\nu+k_\nu\right)+1. \tag{4} \]
Proof. Consider first the case \(K=0\). By virtue of Theorem 2 of [1], there exists a divisor \(P'_1P'_2\ldots P'_{\rho-\chi-1}\) such that
\[ \vartheta\left(\sum_{j=1}^{\rho-\chi-1}u_\nu(P'_j)+\lambda_\nu+k_\nu\right)\ne 0, \tag{5} \]
and therefore
\[ \min_{P_j\in R}\chi\left(\sum_{j=1}^{\rho-\chi-1}u_\nu(P_j)+\lambda_\nu+k_\nu\right)=-1. \tag{6} \]
Conversely, from (6), by definition, (5) follows; whence, on the basis of Theorem 2 of [1], we have \(K=0\).
If \(K\ge 1\), then comparison (2) is solvable and \(K=\dim D_1D_2\ldots D_\chi\). Using Lemma 1, we obtain
\[ K=\dim D_1D_2\ldots D_\chi =\min_{P_j\in R}\dim D_1D_2\ldots D_\chi P_1P_2\ldots P_{\rho-\chi-1}= \]
\[ =\min_{P_j\in R}\chi\left(\sum_{j=1}^{\chi}u_\nu(D_j)+ \sum_{j=1}^{\rho-\chi-1}u_\nu(P_j)+k_\nu\right)+1= \]
\[ =\min_{P_j\in R}\chi\left(\sum_{j=1}^{\rho-\chi-1}u_\nu(P_j)+\lambda_\nu+k_\nu\right)+1. \]
The theorem is proved.
Lemma 2. If \(\dim P_1P_2\ldots P_\rho>1\), then, in order to satisfy the comparison
\[ \sum_{j=1}^{\rho}u_\nu(Q_j)\equiv \sum_{j=1}^{\rho}u_\nu(P_j)\pmod{\text{periods}} \qquad (\nu=1,2,\ldots,\rho), \]
at least one of the points \(Q_1,Q_2,\ldots,Q_\rho\) may be prescribed arbitrarily, and conversely.
Let now \(\chi\ge \rho\). In this case comparison (2) is always solvable, and, moreover, \(\chi-\rho\) points \(D_{\rho+1},\ldots,D_\chi\) may be prescribed arbitrarily.
Theorem 2. If \(\rho\le \chi\le 2\rho-2\), then
\[ K=\min_{P_j\in R}\chi\left(\lambda_\nu-\sum_{j=1}^{\chi-\rho+1}u_\nu(P_j)+k_\nu\right)+\chi-\rho+2. \tag{7} \]
Proof. From the Riemann–Roch theorem it follows that \(K\ge \chi-\rho+1\). Consider first the case \(K=\chi-\rho+1\). It is known ((2), lemma on p. 154) that, in the equivalence class of divisors satisfying comparison (2), one can find such a divisor \(D'_1D'_2\ldots D'_\rho D'_{\rho+1}\ldots D'_\chi\) that \(\dim D'_1D'_2\ldots D'_\rho=1\). Then the function
\[ \vartheta\left(u(P)-\left(\sum_{j=1}^{\rho}u_\nu(D'_j)+k_\nu\right)\right) \]
is not identically equal to zero, and its zeros are the points \(D'_1,D'_2,\ldots,D'_\rho\) [2]. Taking a point \(D'\) different from \(D'_1,\ldots,D'_\rho\), we shall have
\[ \vartheta\left(\lambda_\nu-\sum_{j=\rho+1}^{\chi}u_\nu(D'_j)-u_\nu(D')+k_\nu\right) =-\vartheta\left(u_\nu(D')-\right. \]
\[ \left.-\left(\lambda_\nu-\sum_{j=\rho+1}^{\chi}u_\nu(D'_j)+k_\nu\right)\right) = C\,\vartheta\left(u_\nu(D')-\left(\sum_{j=1}^{\rho}u_\nu(D'_j)+k_\nu\right)\right)\ne 0 \]
(\(C\) is a nonzero constant), whence it follows that
\[ \min_{P_j \in R} \chi \left( \lambda_\nu - \sum_{j=1}^{\chi-\rho+1} u_\nu(P_j) + k_\nu \right) = -1. \tag{8} \]
Conversely, suppose that (8) holds, or, what is the same thing, there exists a divisor \(Q'_1 Q'_2 \cdots Q'_{\chi-\rho+1}\) such that
\[ \vartheta \left( \lambda_\nu - \sum_{j=1}^{\chi-\rho+1} u_\nu(Q'_j) + k_\nu \right) \ne 0. \tag{9} \]
Inequality (9) means that the congruence
\[ \sum_{j=1}^{\rho-1} u_\nu(D_j) \equiv \lambda_\nu - \sum_{j=1}^{\chi-\rho+1} u_\nu(Q'_j) \pmod{\text{periods}} \qquad (\nu = 1, 2, \ldots, \rho) \tag{10} \]
is unsolvable; hence, on the basis of Lemma 2, we conclude that if the divisor \(D_1D_2 \cdots D_\rho\) is a solution of the congruence
\[ \sum_{j=1}^{\rho} u_\nu(D_j) \equiv \lambda_\nu - \sum_{j=1}^{\chi-\rho} u_\nu(Q'_j) \pmod{\text{periods}} \qquad (\nu = 1, 2, \ldots, \rho), \tag{11} \]
then \(\dim D_1D_2 \cdots D_\rho = 1\); consequently,
\[ \dim D_1D_2 \cdots D_\rho Q'_1Q'_2 \cdots Q'_{\rho-\chi} \le \chi - \rho + 1. \]
On the other hand, by the Riemann–Roch theorem,
\[ \dim D_1D_2 \cdots D_\rho Q'_1Q'_2 \cdots Q'_{\rho-\chi} \ge \chi - \rho + 1. \]
Finally we have
\[ K = \dim D_1D_2 \cdots D_\rho Q'_1Q'_2 \cdots Q'_{\rho-\chi} = \chi - \rho + 1. \]
Let now \(K > \chi - \rho + 1\). From Lemma 2 it follows that congruence (10) is solvable with respect to the points \(D_1, D_2, \ldots, D_{\rho-1}\), whatever the points \(Q'_1, Q'_2, \ldots, Q'_{\chi-\rho+1}\) may be. Denote by \(\Delta\) the equivalence class of divisors satisfying congruence (2), and by \(\delta\) the set of all divisors of order \(\rho - 1\) which are sub-divisors of divisors of the class \(\Delta\). From the above-mentioned lemma (\((2)\), p. 154) it follows that
\[ \min_{D \in \delta} \dim D = \dim \Delta - \chi + \rho + 1 = K - \chi + \rho - 1. \]
As is easily seen, the divisors \(D_1D_2 \cdots D_{\rho-1}=D\) which are solutions of congruence (10) exhaust the whole set \(\delta\), when the points \(Q'_1, Q'_2, \ldots, Q'_{\chi-\rho+1}\) independently describe the entire surface \(R\). Thus,
\[ \begin{aligned} K &= \chi - \rho + 1 + \min_{D \in \delta} \dim D \\ &= \min_{D \in \delta} \chi \left( \sum_{j=1}^{\rho-1} u_\nu(D_j) + k_\nu \right) + \chi - \rho + 2 \\ &= \min_{P_j \in R} \chi \left( \lambda_\nu - \sum_{j=1}^{\chi-\rho+1} u_\nu(P_j) + k_\nu \right) + \chi - \rho + 2. \end{aligned} \]
The theorem is proved. Formula (7) can be obtained from the relation
\[ K = K' + \chi - \rho + 1, \]
where \(K'\) is the number of linearly independent solutions of the problem conjugate to the original one. The conjugate problem is equivalent to a certain congruence with \(2\rho - 2 - \chi\) unknown points, and, therefore, Theorem 1 may be used to calculate the number \(K'\). Finally, we note that for \(\chi > 2\rho - 2\),
\[ K = \chi - \rho + 1. \]
The theorems stated above are applicable to the study of the Hilbert problem determined by prescribing on the boundary of a finite surface (in particular, of a multiply connected domain) the condition
\[ \operatorname{Re}[(a - ib)F] = 0, \]
since this problem, in a known way \((4)\), is reduced to problem (1) on the double of the given Riemann surface.
Tbilisi Mathematical Institute
Academy of Sciences of the Georgian SSR
Received
17 VI 1964
CITED LITERATURE
- R. N. Abdullaev, DAN, 152, No. 6 (1963).
- N. G. Chebotarev, Theory of Algebraic Functions, Moscow—Leningrad, 1948, p. 3.
- B. Riemann, Abel’s Functions, Collected Works, 1948.
- Yu. L. Rodin, DAN, 129, No. 6 (1959).