Computing Supersingular Isogenies on Kummer Surfaces

2018 ◽  
pp. 428-456 ◽  
Author(s):  
Craig Costello
Keyword(s):  
Kummer Surfaces

scholarly journals Kummer surfaces in characteristic $2$

1974 ◽  
Vol 50 (9) ◽  
pp. 718-722 ◽  
Author(s):  
Tetsuji Shioda
Keyword(s):  
Kummer Surfaces ◽  
Characteristic 2

scholarly journals Kummer surfaces and the computation of the Picard group

2012 ◽  
Vol 15 ◽  
pp. 84-100 ◽  
Author(s):  
Andreas-Stephan Elsenhans ◽  
Jörg Jahnel
Keyword(s):  
Abelian Varieties ◽  
Upper Bounds ◽  
Picard Group ◽  
A Value ◽  
Kummer Surfaces

AbstractWe test R. van Luijk’s method for computing the Picard group of a K3 surface. The examples considered are the resolutions of Kummer quartics in ℙ3. Using the theory of abelian varieties, the Picard group may be computed directly in this case. Our experiments show that the upper bounds provided by van Luijk’s method are sharp when sufficiently many primes are used. In fact, there are a lot of primes that yield a value close to the exact one. However, for many but not all Kummer surfaces V of Picard rank 18, we have ${\rm rk}\,{\rm Pic}(V_{\overline {\mathbb F}_{\hspace *{-.8pt}p}}) \geq 20$ for a set of primes of density at least 1/2.

scholarly journals A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

2004 ◽  
Vol 47 (3) ◽  
pp. 398-406
Author(s):  
David McKinnon
Keyword(s):  
Number Field ◽  
Open Subset ◽  
Rational Points ◽  
Rational Curves ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Geometric Genus ◽  
Finite Set ◽  
Genus 2 ◽  
Kummer Surfaces

AbstractLet V be a K3 surface defined over a number field k. The Batyrev-Manin conjecture for V states that for every nonempty open subset U of V, there exists a finite set ZU of accumulating rational curves such that the density of rational points on U − ZU is strictly less than the density of rational points on ZU. Thus, the set of rational points of V conjecturally admits a stratification corresponding to the sets ZU for successively smaller sets U.In this paper, in the case that V is a Kummer surface, we prove that the Batyrev-Manin conjecture for V can be reduced to the Batyrev-Manin conjecture for V modulo the endomorphisms of V induced by multiplication by m on the associated abelian surface A. As an application, we use this to show that given some restrictions on A, the set of rational points of V which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Batyrev-Manin type.

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