Abstract
The theory of multiplicative functions and Prym differentials for scalar characters on a compact Riemann surface has found applications in function theory, analytic number theory, and mathematical physics.
We construct the matrix multiplicative functions and Prym m-differentials on a finite Riemann surface for a given matrix character with values in GL(n,ℂ) starting from a meromorphic function on the unit disk with finitely many poles. We show that these multiplicative functions and Prym m-differentials depend locally holomorphically on the matrix character.
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References
Dick R., “Krichever-Novikov-like bases on punctured Riemann surface,” Deutsches Elektronen-Synchrotron (DESY) 89-059. May, 1989.
Farkas H. M. and Kra I., Riemann Surfaces, Springer-Verlag, New York (1992) (Graduate Texts Math.; 71).
Appell P., “Généralisation des fonctions doublement périodiques de seconde espéce,” J. Math. Pures Appl., 9, 5–24 (1883).
Chueshev V. V., Multiplicative functions and Prym differentials on a variable compact Riemann surface. Part 2 [in Russian], Kemerovo State University, Kemerovo (2003).
Haupt O., “Zur theorie der Prymschen Funktionen 1 und N Ordnung,” Math. Ann., 77, No. 1, 24–64 (1916).
Iwasawa K., Algebraic Functions, Amer. Math. Soc., New York and Providence (1993) (Trans. Math. Monogr.; 188).
Bespomestnych A. A. and Chueshev V. V., “Multiplicative functions with matrix characters on compact Riemann surfaces,” J. Siberian Fed. Univ. Math. Phys., 2, No. 1, 31–39 (2009).
Krushkal’ S. L., Quasiconformal Mappings and Riemann Surfaces, Wiley, New York etc. (1979).
Kazantseva A. A. and Chueshev V. V., “The spaces of meromorphic Prym differentials on a finite Riemann surface,” Siberian Math. J., 53, No. 1, 72–86 (2012).
Krepitsina T. S. and Chueshev V. V., “Multiplicative functions and Prym differentials on variable tori,” Vestnik NGU. Ser. Mat. Mekh. Inform., 12, No. 1, 74–90 (2012).
Chuesheva O. A., “The spaces of meromorphic Prym differentials on finite tori,” J. Siberian Fed. Univ. Math. Phys., 7, No. 1, 162–172 (2014).
Kolmogorov A. N. and Fomin S. V., Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1972).
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Original Russian Text Copyright © 2015 Chuesheva O.A.
The author was supported by the Government of the Russian Federation (Contract 14.Y26.31.0006).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 56, No. 3, pp. 693–703, May–June, 2015; DOI: 10.17377/smzh.2015.56.318.
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Chuesheva, O.A. Prym differentials with matrix characters on a finite Riemann surface. Sib Math J 56, 549–556 (2015). https://doi.org/10.1134/S0037446615030180
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DOI: https://doi.org/10.1134/S0037446615030180