scholarly journals Counting points on genus-3 hyperelliptic curves with explicit real multiplication

2019 ◽  
Vol 2 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Simon Abelard ◽  
Pierrick Gaudry ◽  
Pierre-Jean Spaenlehauer
Keyword(s):  
Hyperelliptic Curves ◽  
Real Multiplication ◽  
Counting Points

scholarly journals Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

2018 ◽  
Vol 19 (3) ◽  
pp. 591-621 ◽  
Author(s):  
Simon Abelard ◽  
Pierrick Gaudry ◽  
Pierre-Jean Spaenlehauer
Keyword(s):  
Hyperelliptic Curves ◽  
Complexity Bounds ◽  
Counting Points

scholarly journals Isogenies for Point Counting on Genus Two Hyperelliptic Curves with Maximal Real Multiplication

2017 ◽  
pp. 63-94 ◽  
Author(s):  
Sean Ballentine ◽  
Aurore Guillevic ◽  
Elisa Lorenzo García ◽  
Chloe Martindale ◽  
Maike Massierer ◽  
...  
Keyword(s):  
Hyperelliptic Curves ◽  
Point Counting ◽  
Real Multiplication ◽  
Genus Two

Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials

1991 ◽  
Vol 43 (5) ◽  
pp. 1055-1064 ◽  
Author(s):  
Walter Tautz ◽  
Jaap Top ◽  
Alain Verberkmoes
Keyword(s):  
Cyclotomic Field ◽  
Prime Numbers ◽  
Explicit Construction ◽  
Hyperelliptic Curves ◽  
Double Cover ◽  
The Other ◽  
Roots Of Unity ◽  
Permutation Polynomials ◽  
Endomorphism Algebra ◽  
Real Multiplication

AbstractThe aim of this paper is to present a very explicit construction of one parameter families of hyperelliptic curves C of genus (p−1 )/ 2, for any odd prime number p, with the property that the endomorphism algebra of the jacobian of C contains the real subfield Q(2 cos(2π/p)) of the cyclotomic field Q(e2π i/p).Two proofs of the fact that the constructed curves have this property will be given. One is by providing a double cover with the pth roots of unity in its automorphism group. The other is by explicitly writing down equations of a correspondence in C x C which defines multiplication by 2cos(2π/ p) on the jacobian of C. As a byproduct we obtain polynomials which define bijective maps Fℓ → Fℓ for all prime numbers in certain congruence classes.

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