Abstract:
Unlike the Abel map of a symmetric power of a Riemann surface onto its Jacobian, the Abel–Prym map can generically not be inversed by means of conventional technique related to the Jacobi inversion problem and its main ingredient, namely Riemann's vanishing theorem. This occurs because replacing the Riemann theta function by the Prym one in this theorem gives twice as many zeros as the dimension of the Prym variety. However, if the Riemann surface has a second involution commuting with the one defining the Prym variety and satisfying a certain additional condition, an analogue of the Jacobi inversion can be defined, and expressed in terms of the Prym theta function. We formulate these conditions and refer to pairs of involutions satisfying them as pairs of the first type. We formulate necessary conditions for a pair of involutions to be a pair of the first type, and give a series of examples of curves with such pairs of involutions, which are mainly the spectral curves of Hitchin systems, and also the spectral curve of the Kovalevskaya system.
Bibliography: 14 titles.
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