General derivative Thomae formula for singular half-periods
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General derivative Thomae formula for singular half-periods J. Bernatska1 Received: 1 February 2020 / Revised: 22 June 2020 / Accepted: 8 July 2020 © Springer Nature B.V. 2020
Abstract The paper develops second Thomae theorem in hyperelliptic case. The main formula, called general Thomae formula, provides expressions for values at zero of the lowest non-vanishing derivatives of theta functions with singular characteristics of arbitrary multiplicity in terms of branch points and period matrix. We call these values derivative theta constants. First and second Thomae formulas follow as particular cases. Some further results are derived. Matrices of second derivative theta constants (Hessian matrices of zero-values of theta functions with characteristics of multiplicity two) have rank three in any genus. Similar result about the structure of order three tensor of third derivative theta constants is obtained, and a conjecture regarding higher multiplicities is made. As a byproduct, a generalization of Bolza formulas are deduced. Keywords Second and third derivative theta constants · Characteristic via partition · Bolza formula Mathematics Subject Classification 14K25 · 32A15 · 32G15 · 14H15
1 Introduction Thomae formulas are of great interest in many areas of mathematics and physics such as quantum field theory, string theory, theory of integrable systems, number theory, p-adic analysis, etc. This paper provides a development of the classical work of Thomae [1]. First and second Thomae formulas give a representation of theta constants with non-singular even and odd characteristics in terms of branch points and period matrix of a hyperelliptic Riemann surface. In the present paper, a similar representation of derivative theta constants with singular characteristics is found for
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J. Bernatska [email protected]; [email protected] National University Kyiv Mohyla Academy, Kyiv, Ukraine
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J. Bernatska
all possible multiplicities. This problem was not considered in mathematical literature since the time of Thomae. Instead, generalizations of Thomae formulas to the case of Z N -curves, also called cyclic covers of CP1 or simply cyclic curves, were discovered. This was initiated by paper [2], where a generalization of first Thomae formula provided an expression for determinant of Dirac’s operator in terms of branch points of Z N -curve. This gave rise to a flow of publications. In [3] a rigorous proof of the mathematical result from [2] was given. Then, first Thomae formula was generalized to a special class of singular Z N -curve in [4], to general cyclic covers of CP1 in [5], to Abelian covers of CP1 in [6], and developed in other papers and a book of H. Farkas and Sh. Zemel Generalization of Thomae’s Formula for Z N Curves (2010) containing many examples. A detailed generalization of first Thomae formula for a trigonal cyclic curve with a specific choice of symplectic cohomology basis is given in [7]. The only generalization of second Thomae formula was obtained in [8] for trigonal cyclic curves. Thi
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