Classification of real moduli spaces over genus 2 curves

1995 ◽  
Vol 57 (2) ◽  
pp. 207-215 ◽  
Author(s):  
Shuguang Wang
Keyword(s):  
Genus 2 ◽  
2011 ◽  
Vol 63 (5) ◽  
pp. 992-1024 ◽  
Author(s):  
Nils Bruin ◽  
Kevin Doerksen

Abstract In this paper we study genus 2 curves whose Jacobians admit a polarized (4, 4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion, and we derive the relation their absolute invariants satisfy.As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered.Our main tool is a Galois theoretic characterization of genus 2 curves admitting multiple Richelot isogenies.


2016 ◽  
Vol 19 (A) ◽  
pp. 29-42 ◽  
Author(s):  
Abhinav Kumar ◽  
Ronen E. Mukamel

We compute equations for real multiplication on the divisor classes of genus-2 curves via algebraic correspondences. We do so by implementing van Wamelen’s method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for the Hilbert modular surface of discriminant $5$ and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.


2016 ◽  
Vol 30 (2) ◽  
pp. 572-600 ◽  
Author(s):  
Huseyin Hisil ◽  
Craig Costello
Keyword(s):  
Genus 2 ◽  

2017 ◽  
Vol 11 (1) ◽  
pp. 39-76 ◽  
Author(s):  
Jeffrey Achter ◽  
Everett Howe

2011 ◽  
Vol 131 (5) ◽  
pp. 936-958 ◽  
Author(s):  
Kristin Lauter ◽  
Tonghai Yang
Keyword(s):  
Genus 2 ◽  

2015 ◽  
Vol 18 (1) ◽  
pp. 170-197 ◽  
Author(s):  
Reinier Bröker ◽  
Everett W. Howe ◽  
Kristin E. Lauter ◽  
Peter Stevenhagen

AbstractWe study the problem of efficiently constructing a curve $C$ of genus $2$ over a finite field $\mathbb{F}$ for which either the curve $C$ itself or its Jacobian has a prescribed number $N$ of $\mathbb{F}$-rational points.In the case of the Jacobian, we show that any ‘CM-construction’ to produce the required genus-$2$ curves necessarily takes time exponential in the size of its input.On the other hand, we provide an algorithm for producing a genus-$2$ curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-$2$ curve having exactly $10^{2014}+9703$ (prime) points, and two genus-$2$ curves each having exactly $10^{2013}$ points.In an appendix we provide a complete parametrization, over an arbitrary base field $k$ of characteristic neither two nor three, of the family of genus-$2$ curves over $k$ that have $k$-rational degree-$3$ maps to elliptic curves, including formulas for the genus-$2$ curves, the associated elliptic curves, and the degree-$3$ maps.Supplementary materials are available with this article.


2009 ◽  
Vol 15 (5) ◽  
pp. 569-579 ◽  
Author(s):  
J.M. Miret ◽  
R. Moreno ◽  
J. Pujolàs ◽  
A. Rio

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