Jacobian Variety of Generalized Fermat Curves

Carvacho, Mariela; Hidalgo, Rubén A.; Quispe, Saúl

doi:10.1093/qmath/haw009

Cited by 14 publications

(19 citation statements)

References 18 publications

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“…In [3], as a consequence of Kani-Rosen results [17], it was noted that JF 3,3 is isogenous to the product of four curves of genus one and three jacobians of genus two curves. The four genus one curves are given by C 1 : y 3 = x(x − 1)(x − λ 1 ), C 2 : y 3 = x(x − 1), C 3 : y 3 = x(x − λ 1 ), C 4 : y 3 = (x − 1)(x − λ 1 ), and the three genus two curves by E 1 : y 3 = x(x − 1)(x − λ 1 ) 2 , E 2 : y 3 = x(x − 1) 2 (x − λ 1 ), E 3 : y 3 = x 2 (x − 1)(x − λ 1 ).…”

Section: 1mentioning

confidence: 97%

“…Let C k,n : H 1,0 (F k,n ) → H 1,0 (F k,n ) the corresponding Cartier operator. In Theorem 3.8 we have constructed a standard basis for H 1,0 (F k,n ), given by the elements of the form θ r;α 3 ,...,α n+1 = z r dz y α 3 3 · · · y α n+1 n+1 , (r; α 3 , . .…”

Section: Evaluation Of the Cartier Operator On The Standard Basismentioning

confidence: 99%

“…K − {0, 1}, λ 1 λ 2 has genus g k,3 = 1 + k 3 (3k − 5)/2. In this case, the standard basis is θ r;α 3 ,α 4 ,α 5 = z r dz y α3 3 y α 4 4 y α 5 5 (r;α 3 ,α 4 ,α 5 )∈I k,3 and the value of C k,4 θ r;α 3 ,α 4 ,α 5 is given in the following table                                                                                          Classical Humber curves.Let us consider the case of the classical Humbert curves (these being of genus g = 5) in characteristic p = 3 (11) F 2,4 =…”

mentioning

confidence: 99%

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Automorphisms of generalized Fermat curves

Hidalgo

Kontogeorgis

Leyton-Álvarez

et al. 2017

Journal of Pure and Applied Algebra

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Section: 1mentioning

confidence: 97%

Section: Evaluation Of the Cartier Operator On The Standard Basismentioning

confidence: 99%

mentioning

confidence: 99%

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Automorphisms of generalized Fermat curves

Hidalgo

Kontogeorgis

Leyton-Álvarez

et al. 2017

Journal of Pure and Applied Algebra

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“…An elliptic decomposition of the jacobian variety JS µ . If µ = (µ 4 , µ 5 , µ 6 , µ 7 ) ∈ Ω, then in [2] we obtained that the jacobian variety JŜ µ ofŜ µ is isogenous to a product…”

Section: 5mentioning

confidence: 99%

About the Fricke–Macbeath curve

Hidalgo

2017

European Journal of Mathematics

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A closed Riemann surface of genus g ≥ 2 is called a Hurwitz curve if its group of conformal automorphisms has order 84(g − 1). In 1895, A. Wiman noticed that there is no Hurwitz curve of genus g = 2, 4, 5, 6 and, up to isomorphisms, there is a unique Hurwitz curve of genus g = 3; this being Klein's plane quartic curve. Later, in 1965, A. Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus g = 7; this known as the Fricke-Macbeath curve. Equations were also provided; that being the fiber product of suitable three elliptic curves. In the same year, W. Edge constructed such a genus seven Hurwitz curve by elementary projective geometry. Such a construction was provided by first constructing a 4-dimensional family of genus seven closed Riemann surfaces S µ admitting a group Gµ ≅ Z 3 2 of conformal automorphisms so that S µ Gµ has genus zero. In this paper we discuss the above curves in terms of fiber products of classical Fermat curves and we provide a geometrical explanation of the three elliptic curves in Macbeath's description. We also observe that the jacobian variety of each S µ is isogenous to the product of seven elliptic curves (explicitly given) and, for the particular Fricke-Macbeath curve, we obtain the well known fact that its jacobian variety is isogenous to E 7 for a suitable elliptic curve E.

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“…Using a suitable form of the quadrics, Hidalgo presented in [11] and [12] a family of Humbert-Edge's curves of type 5 whose fields of moduli are contained in R but none of their fields of definition are contained in R. Frías-Medina and Zamora presented in [9] a characterization of Humbert-Edge's curves using certain abelian groups of order 2 n and presented specializations admitting larger automorphism subgroups. Carvacho, Hidalgo and Quispe determined in [4] the decomposition of the Jacobian of generalized Fermat curves, and as a consequence for Humbert-Edge's curves. Auffarth, Lucchini Arteche and Rojas described in [1] the The first author is supported by Grants PAPIIT UNAM IN100419 "Aspectos Geométricos del moduli de curvas Mg" and CONACyT, México A1-S-9029 "Moduli de curvas y curvatura en Ag".…”

mentioning

confidence: 99%

Geometric aspects on Humbert-Edge's curves of type 5, Kummer surfaces and hyperelliptic curves of genus 2

Castorena,

Frías-Medina

2021

Preprint

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In this work we study the Humbert-Edge's curves of type 5, defined as a complete intersection of four diagonal quadrics in P 5 . We characterize them using Kummer surfaces and using the geometry of these surfaces we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge's curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we let see how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus g = n−1 2 and the moduli space of Humbert-Edge's curves of type n ≥ 5 where n is an odd number.

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Jacobian Variety of Generalized Fermat Curves

Cited by 14 publications

References 18 publications

Automorphisms of generalized Fermat curves

Automorphisms of generalized Fermat curves

About the Fricke–Macbeath curve

Geometric aspects on Humbert-Edge's curves of type 5, Kummer surfaces and hyperelliptic curves of genus 2

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