Quadrangular $${{\mathbb {Z}}}_{p}^{l}$$-actions on Riemann surfaces

Let $$p \geqslant 3$$ be a prime integer and, for $$l 1$$ , let $$G \cong {{\mathbb {Z}}}_{p}^{l}$$ group of conformal automorphisms some closed Riemann surface S genus $$g 2$$ . By the Riemann–Hurwitz formula, either \leqslant g+1$$ or $$p=2\,g+1$$ If $$l=1$$ and then S/G is sphere with exactly three cone points if moreover 11$$ G unique p-Sylow subgroup $$\textrm{Aut}\hspace{0.55542pt}(S)$$ $$p=g+1$$ four 7$$ again subgroup. The above facts permitted many authors to obtain algebraic models corresponding groups $$\textrm{Aut}(S)$$ in these situations. Now, us assume 5$$ (1) $$p^{l} g-1$$ (2) has zero, $$p^{l-1}(p-3) 2(g-1)$$ $$2 l r-1$$ where $$r number S/G. we are case (2). $$r=3$$ $$l=2$$ happens classical Fermat curve degree p, whose well known. next case, $$r=4$$ studied this paper. We provide an representation S, description its automorphisms, discussion field moduli isogenous decomposition Jacobian variety.