scholarly journals Picard Curves with Small Conductor

2017 ◽  
pp. 97-122
Author(s):  
Michel Börner ◽  
Irene I. Bouw ◽  
Stefan Wewers
Keyword(s):  
Picard Curves

Some remarks on the Picard curves over a finite field

2007 ◽  
Vol 280 (7) ◽  
pp. 802-811 ◽  
Author(s):  
Yoh Takizawa
Keyword(s):  
Finite Field ◽  
Picard Curves

scholarly journals A note on some Picard curves over finite fields

2015 ◽  
Vol 34 ◽  
pp. 107-122
Author(s):  
Ahmad Kazemifard ◽  
Saeed Tafazolian
Keyword(s):  
Finite Fields ◽  
Picard Curves ◽  
Curves Over Finite Fields

scholarly journals Computing $L$-polynomials of Picard curves from Cartier-Manin matrices

2021 ◽  
pp. 1
Author(s):  
Sualeh Asif ◽  
Francesc Fité ◽  
Dylan Pentland ◽  
A. V. Sutherland
Keyword(s):  
Picard Curves

On the Jacobian Varieties of Picard Curves: Explicit Addition Law and Algebraic Structure

1999 ◽  
Vol 208 (1) ◽  
pp. 149-166 ◽  
Author(s):  
Jorge Estrada Sarlabous ◽  
Ernesto Reinaldo Barreiro ◽  
Jorge Alejandro Pieiro Barceló
Keyword(s):  
Algebraic Structure ◽  
Jacobian Varieties ◽  
Picard Curves

scholarly journals An inverse Jacobian algorithm for Picard curves

2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Joan-C. Lario ◽  
Anna Somoza ◽  
Christelle Vincent
Keyword(s):  
Class Number ◽  
Complex Multiplication ◽  
Hyperelliptic Curves ◽  
Period Matrix ◽  
Isomorphism Classes ◽  
Jacobian Problem ◽  
Picard Curves ◽  
The Given ◽  
Small Period ◽  
Jacobian Algorithm

AbstractWe study the inverse Jacobian problem for the case of Picard curves over $${\mathbb {C}}$$ C . More precisely, we elaborate on an algorithm that, given a small period matrix $$\varOmega \in {\mathbb {C}}^{3\times 3}$$ Ω ∈ C 3 × 3 corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most $$4$$ 4 . In particular, we obtain the complete list of maximal CM Picard curves defined over $${\mathbb {Q}}$$ Q . In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].

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