A Reduction of the Batyrev-Manin Conjecture for 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.

Download Full-text

Kummer surfaces associated to (1, 2)-polarized abelian surfaces

Nagoya Mathematical Journal ◽

10.1017/s002776300001028x ◽

2011 ◽

Vol 202 ◽

pp. 127-143

Author(s):

Afsaneh Mehran

Keyword(s):

Double Cover ◽

Pezzo Surface ◽

Abelian Surface ◽

Kummer Surface ◽

Abelian Surfaces ◽

Del Pezzo Surface ◽

Singular Fibers ◽

Kummer Surfaces ◽

Del Pezzo

AbstractThe aim of this paper is to describe the geometry of the generic Kummer surface associated to a (1, 2)-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of typeI2.

Download Full-text

ON SEXTIC TRINOMIALS WITH GALOIS GROUP C6, S3 OR C3 × S3

Journal of Algebra and Its Applications ◽

10.1142/s0219498812501289 ◽

2012 ◽

Vol 12 (01) ◽

pp. 1250128 ◽

Cited By ~ 5

Author(s):

STEPHEN C. BROWN ◽

BLAIR K. SPEARMAN ◽

QIDUAN YANG

Keyword(s):

Galois Group ◽

Number Field ◽

Rational Points ◽

Genus 2

We characterize irreducible trinomials x6 + Ax + B with coefficients in a number field K which have Galois group C6, S3 or C3 × S3. This characterization relates these trinomials to the K-rational points on a genus 2 curve. We determine these trinomials explicitly in the case K = ℚ.

Download Full-text

Degeneration of Kummer surfaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004120000067 ◽

2020 ◽

pp. 1-33

Author(s):

OTTO OVERKAMP

Keyword(s):

Galois Representation ◽

Base Change ◽

Abelian Surface ◽

Kummer Surface ◽

Semistable Reduction ◽

Valued Field ◽

Monodromy Conjecture ◽

Finite Base ◽

Kummer Surfaces ◽

Special Fibre

Abstract We prove that a Kummer surface defined over a complete strictly Henselian discretely valued field K of residue characteristic different from 2 admits a strict Kulikov model after finite base change. The Kulikov models we construct will be schemes, so our results imply that the semistable reduction conjecture is true for Kummer surfaces in this setup, even in the category of schemes. Our construction of Kulikov models is closely related to an earlier construction of Künnemann, which produces semistable models of Abelian varieties. It is well known that the special fibre of a strict Kulikov model belongs to one of three types, and we shall prove that the type of the special fibre of a strict Kulikov model of a Kummer surface and the toric rank of a corresponding Abelian surface are determined by each other. We also study the relationship between this invariant and the Galois representation on the second ℓ-adic cohomology of the Kummer surface. Finally, we apply our results, together with earlier work of Halle–Nicaise, to give a proof of the monodromy conjecture for Kummer surfaces in equal characteristic zero.

Download Full-text

Kummer surfaces associated to (1, 2)-polarized abelian surfaces

Nagoya Mathematical Journal ◽

10.1215/00277630-1260477 ◽

2011 ◽

Vol 202 ◽

pp. 127-143 ◽

Cited By ~ 1

Author(s):

Afsaneh Mehran

Keyword(s):

Double Cover ◽

Pezzo Surface ◽

Abelian Surface ◽

Kummer Surface ◽

Abelian Surfaces ◽

Del Pezzo Surface ◽

Singular Fibers ◽

Kummer Surfaces ◽

Del Pezzo

Download Full-text

Every algebraic Kummer surface is the K3-cover of an Enriques surface

Nagoya Mathematical Journal ◽

10.1017/s0027763000003019 ◽

1990 ◽

Vol 118 ◽

pp. 99-110 ◽

Cited By ~ 11

Author(s):

Jong Hae Keum

Keyword(s):

Crucial Role ◽

K3 Surfaces ◽

Abelian Surface ◽

Enriques Surface ◽

Kummer Surface ◽

3 Dimensional ◽

Complex Torus ◽

Kummer Surfaces ◽

Dimension 2 ◽

Simply Connected

A Kummer surface is the minimal desingularization of the surface T/i, where T is a complex torus of dimension 2 and i the involution automorphism on T. T is an abelian surface if and only if its associated Kummer surface is algebraic. Kummer surfaces are among classical examples of K3-surfaces (which are simply-connected smooth surfaces with a nowhere-vanishing holomorphic 2-form), and play a crucial role in the theory of K3-surfaces. In a sense, all Kummer surfaces (resp. algebraic Kummer surfaces) form a 4 (resp. 3)-dimensional subset in the 20 (resp. 19)-dimensional family of K3-surfaces (resp. algebraic K3 surfaces).

Download Full-text

Constructing abelian surfaces for cryptography via Rosenhain invariants

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157014000370 ◽

2014 ◽

Vol 17 (A) ◽

pp. 157-180 ◽

Cited By ~ 4

Author(s):

Craig Costello ◽

Alyson Deines-Schartz ◽

Kristin Lauter ◽

Tonghai Yang

Keyword(s):

Complex Multiplication ◽

Rational Functions ◽

The Other ◽

Algebraic Numbers ◽

Kummer Surface ◽

Abelian Surfaces ◽

Minimal Polynomials ◽

Genus 2 ◽

Kummer Surfaces ◽

Security Levels

AbstractThis paper presents an algorithm to construct cryptographically strong genus $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ curves and their Kummer surfaces via Rosenhain invariants and related Kummer parameters. The most common version of the complex multiplication (CM) algorithm for constructing cryptographic curves in genus 2 relies on the well-studied Igusa invariants and Mestre’s algorithm for reconstructing the curve. On the other hand, the Rosenhain invariants typically have much smaller height, so computing them requires less precision, and in addition, the Rosenhain model for the curve can be written down directly given the Rosenhain invariants. Similarly, the parameters for a Kummer surface can be expressed directly in terms of rational functions of theta constants. CM-values of these functions are algebraic numbers, and when computed to high enough precision, LLL can recognize their minimal polynomials. Motivated by fast cryptography on Kummer surfaces, we investigate a variant of the CM method for computing cryptographically strong Rosenhain models of curves (as well as their associated Kummer surfaces) and use it to generate several example curves at different security levels that are suitable for use in cryptography.

Download Full-text

Note on hyperelliptic surfaces and certain kummer surfaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100019034 ◽

1936 ◽

Vol 32 (3) ◽

pp. 342-354 ◽

Cited By ~ 1

Author(s):

H. F. Baker

Keyword(s):

Hyperelliptic Curve ◽

Meromorphic Functions ◽

Hyperelliptic Curves ◽

Inversion Problem ◽

Kummer Surface ◽

Functions Of Two Variables ◽

Hyperelliptic Surfaces ◽

Genus 2 ◽

Kummer Surfaces ◽

Hyperelliptic Surface

In 1907 Enriques and Severi published an extensive and fascinating account of hyperelliptic surfaces. In general a hyperelliptic surface is that expressed by the necessary relation connecting three meromorphic functions of two variables which have four columns of periods. Such functions arise naturally by associating the two variables, in accordance with Jacobi's inversion problem for hyperelliptic integrals of genus 2, with a pair of points of a hyperelliptic curve. When the primitive periods of the functions are those arising for the curve, and the set of three functions chosen is representative, in the sense that only one pair of (incongruent) values of the variables arises for given values of the functions, the surface is called by Enriques and Severi a Jacobian surface; but, if several sets of (incongruent) values of the variables arise for given values of the functions, say r sets, the surface is said to be of rank r. For example, when the three functions are all even, to each set of values of these there belong not only the values u, v of the variables, but also the values −u, − v, and r is thus even, being 2 at least, as in the case of the Kummer surface. In the paper referred to, many cases in which r > 1, corresponding to particular hyperelliptic curves possessing involutions of order r, are worked out. In general the method followed consists in arguing, from the character of the associated group of order r, to the character and equation of the hyperelliptic surface Φ of rank r; and from this the Jacobian surface F is inferred upon which there exists an involution of sets of r points, the surface Φ being the representation of this involution. The argumentation is always beautiful, but often not very brief. The hyperelliptic surfaces for which the primitive periods of the functions are not those of a hyperelliptic curve are also shown in the paper to arise from involutions on the Jacobian surface; with these I am not here concerned.

Download Full-text