Generalized theta functions and the Deuring polynomial
From the introduction: The Deuring polynomial, \(H(\lambda)\in\mathbb F_p[\lambda]\), is the coefficient of \(x^m\) in \((x-1)^m(x-\lambda)^m\), where \(m = (p-1)/2\), \(p\) being a prime \(\neq 2\), \[ H(\lambda) = (-1)^m \sum_{i=0}^m {\binom mi}^2 \lambda^i. \] Let us consider the elliptic curve in Legendre form \[ E_\lambda \equiv y^2 = x(x-1)(x-\lambda) \] over the field \(\overline{\mathbb F}\), an algebraic closure of \(\mathbb F_p\). The roots of \(H(\lambda)\) give the values of \(\lambda\in\mathbb F\) such that \(E_\lambda\) is supersingular. The Deuring polynomial can be obtained from the hypergeometric function, \(F(a, b; c, z)\); \[ H(\lambda) = (-1)^mF(1/2, 1/2; 1, \lambda) \bmod (p, \lambda^p). \] Moreover, by a theorem of Manin the number of rational points of \(E_\lambda\), with \(\lambda\in\mathbb F_p\), is congruent to \(1-H(\lambda) \bmod p\). In [J. Number Theory 110, No. 2, 279--316 (2005; Zbl 1064.11042), Lemma 5.5 and Remark 15], the author studied a relation between the \(p\)-generalized theta divisor and the \(p\)-division points of an elliptic curve. The \(p\)-division points of an elliptic curve can be obtained by the intersection of the generalized theta divisor with the image of this curve by a certain immersion within the moduli space of vector bundles of rank \(p\). In this work, he explores the above result to find a relation between the \(p\)-theta functions and the Deuring polynomial via a generalization for vector bundles of the Jacobi inversion theorem. Moreover, he finds an explicit function with divisor \(z_1 + \ldots+ z_{p-1} - (p-1)\infty\). Here \(z_1,\ldots, z_{p-1}\) are the non trivial \(p\)-torsion points of \(E_{\lambda^p}\) , defined over \(\mathbb F\).
scientific article; zbMATH DE number 5797833
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