The \mathrm {al} function of a cyclic trigonal curve of genus three

• Autores: Shigeki Matsutani, Emma Previato
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 66, Fasc. 3, 2015, págs. 311-349
• Idioma: inglés
• DOI: 10.1007/s13348-015-0138-y
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• Resumen

  • A cyclic trigonal curve of genus three is a ℤ3 Galois cover of ℙ1 , therefore can be written as a smooth plane curve with equation ?3=?(?)=(?−?1)(?−?2)(?−?3)(?−?4) . Following Weierstrass for the hyperelliptic case, we define an “ al ” function for this curve and al(?)? , ?=0,1,2 , for each one of three particular covers of the Jacobian of the curve, and ?=1,2,3,4 for a finite branchpoint (??,0) . This generalization of the Jacobi sn , cn , dn functions satisfies the relation:

    ∑?=14∏2?=0al(?)?(?)?′(??)=1 which generalizes sn2?+cn2?=1 . We also show that this can be viewed as a special case of the Frobenius theta identity.