1 | | We plan to add code to compute the Hasse-Weil Zeta function of a cyclic cover of P1 over finite fields using p-adic methods. |

| 1 | We add a new method to compute the zeta function of a cyclic cover of {{{P^1}}}, this is the result of a forthcoming paper generalizing the work of Kedlaya, Harvey, Minzlaff and Gonçalves. |

| 2 | In particular, we add two classes for cyclic covers, one over a generic ring and a specialized one over finite fields. |

| 3 | This requires wrapping David Harvey's code for computing products of matrices already in Sage but not accessible to Sage, see #25366 |

| 4 | |

| 5 | Here is a quick example: |

| 6 | |

| 7 | {{{ |

| 8 | sage: p = 4999; |

| 9 | sage: x = PolynomialRing(GF(p),"x").gen(); |

| 10 | sage: C = CyclicCover(5, x^5 + 1) |

| 11 | sage: C |

| 12 | Cyclic Cover of P^1 over Finite Field of size 4999 defined by y^5 = x^5 + 1 |

| 13 | sage: C.frobenius_polynomial() |

| 14 | x^12 + 29994*x^10 + 374850015*x^8 + 2498500299980*x^6 + 9367502249700015*x^4 + 18731257498500149994*x^2 + 15606259372500374970001 |

| 15 | sage: C.genus() |

| 16 | 6 |

| 17 | }}} |