Abstract: Main aim of the present talk is to give an introduction to the theory of hyperelliptic surfaces and their three dimensional and one dimensional analogues. The report contains a constructive description of hyperelliptic manifolds in low dimension case and their basic properties. We present various versions of theorems by Farkas, Kra and Accola well known in the complex analysis. Also we give a partial answer to de Franchis problem posed in 1913 year. More precisely, we provide a structural description of all holomorphic mappings between genus three and genus two Riemann surfaces. As a result, we get an exact upper bound for the size of the set of such mappings.The results were published in the papers [1,2,3,4,5]. [1] Mednykh, A. D., Mednykh, I.A. The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces. (English. Russian original): Sib. Math. J. 57, No. 6, 1055-1065 (2016); translation from Sib. Mat. Zh. 57, No. 6, 1346-1360 (2016). [2] Mednykh, I. A. Discrete analogs of Farkas and Accola’s theorems on hyperelliptic coverings of a Riemann surface of genus 2. (English. Russian original): Math. Notes 96, No. 1, 84-94 (2014); translation from Mat. Zametki 96, No. 1, 70-82 (2014). [3] Mednykh, I. A. On the sharp upper bound for the number of holomorphic mappings of Riemann surfaces of low genus. (English. Russian original) Sib. Math. J. 53, No. 2, 259-273 (2012); translation from Sib. Mat. Zh. 53, No. 2, 325-344 (2012). [4] Mednykh, I. A. Classification up to equivalence of the holomorphic mappings of Riemann surfaces of low genus. (English. Russian original): Sib. Math. J. 51, No. 6, 1091-1103 (2010); translation from Sib. Mat. Zh. 51, No. 6, 1379-1395 (2010). [5] A.D. Mednykh, M. Reni, Twofold unbranched coverings of genus two 3-manifolds are hyperelliptic, Isr. J. Math., 123 (2001), 149–155.

Cite: Mednykh A.D. , Mednykh I.A.
«Holomorphic mapping of hyperelliptic Riemann surfaces and their generalization»
Конструктивные методы теории римановых поверхностей и приложения, 13-17 ноября 2023, г. Сочи 13-17 Nov 2023