Kummer Surface Research Articles

Isogenies on Kummer Surfaces

We first give a cleaner and more direct approach to the derivation of the Fast model of the Kummer surface. We show how to construct efficient ( N , N ) (N,N) -isogenies, for any odd N N , both on the general Kummer surface and on the Fast model.

Failing to Hash Into Supersingular Isogeny Graphs

Abstract An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of ‘hard supersingular curves’ that is equations for supersingular curves for which computing the endomorphism ring is as difficult as it is for random supersingular curves. A related open problem is to produce a hash function to the vertices of the supersingular $\ell $-isogeny graph, which does not reveal the endomorphism ring, or a path to a curve of known endomorphism ring. Such a hash function would open up interesting cryptographic applications. In this paper, we document a number of (thus far) failed attempts to solve this problem, in the hope that we may spur further research, and shed light on the challenges and obstacles to this endeavour. The mathematical approaches contained in this article include: (i) iterative root-finding for the supersingular polynomial; (ii) gcd’s of specialized modular polynomials; (iii) using division polynomials to create small systems of equations; (iv) taking random walks in the isogeny graph of abelian surfaces, and applying Kummer surfaces and (v) using quantum random walks.

Higher Chow cycles on a family of Kummer surfaces

Abstract We construct a collection of families of higher Chow cycles of type $(2,1)$ on a two-dimensional family of Kummer surfaces, and prove that for a very general member, they generate a subgroup of rank $\ge 18$ in the indecomposable part of the higher Chow group. Construction of the cycles uses a finite group action on the family, and the proof of their linear independence uses Picard–Fuchs differential operators.

Nodal quintic surfaces and lines on cubic fourfolds (with an appendix by John Christian Ottem)

We study nodal quintic surfaces with an even set of 16 nodes as analogues of singular Kummer surfaces. The interpretation of the natural double cover of an even 16 -nodal quintic as a certain Fano variety of lines could be viewed as a replacement for the additive structure of the cover of a singular Kummer surface by its associated abelian surface. Most of the results in this article can be seen as refinements of known facts and our arguments rely heavily on techniques developed by Beauville (1979), Murre (1972), and Voisin (1986), Results due to Shen (2012, 2014) are particularly close to some of the statements. In this sense, the text is mostly expository (but with complete proofs), although our arguments often differ substantially from the original sources.

Constructions of Kummer structures on generalized Kummer surfaces

Constructions of Kummer structures on generalized Kummer surfaces

Number of Kummer structures and Moduli spaces of generalized Kummer surfaces

Number of Kummer structures and Moduli spaces of generalized Kummer surfaces

Geometric aspects on Humbert-Edge curves of type 5, Kummer surfaces and hyperelliptic curves of genus 2

AbstractIn this work, we study the Humbert-Edge curves of type 5, defined as a complete intersection of four diagonal quadrics in ${\mathbb{P}}^5$ . We characterize them using Kummer surfaces, and using the geometry of these surfaces, we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we show how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-1}{2}$ and the moduli space of Humbert-Edge curves of type $n\geq 5$ where $n$ is an odd number.

Explicit uniform bounds for Brauer groups of singular K3 surfaces

Let k be a number field. We give an explicit bound, depending only on [k:Q] and the discriminant of the Neron-Severi lattice, on the size of the Brauer group of a K3 surface X/k that is geometrically isomorphic to the Kummer surface attached to a product of isogenous CM elliptic curves. As an application, we show that the Brauer-Manin set for such a variety is effectively computable. Conditional on GRH, we can also make the explicit bound depend only on [k:Q] and remove the condition that the elliptic curves be isogenous. In addition, we show how to obtain a bound, depending only on [k:Q], on the number of C-isomorphism classes of singular K3 surfaces defined over k, thus proving an effective version of the strong Shafarevich conjecture for singular K3 surfaces.

On Kummer‐like surfaces attached to singularity and modular forms

AbstractWe study a family of lattice polarized K3 surfaces which is an extension of the family of Kummer surfaces derived from principally polarized Abelian surfaces. Our family has two special properties. First, it is coming from a resolution of a simple K3 singularity. Second, it has a natural parameterization by Hermitian modular forms of four complex variables. In this paper, we show two results: (1) we determine the transcendental lattice and the Néron–Severi lattice of a generic member of our family. (2) We give a detailed description of the double covering structure associated with our K3 surfaces.

ENRIQUES INVOLUTIONS AND BRAUER CLASSES

AbstractWe prove that every element of order 2 in the Brauer group of a complex Kummer surface X descends to an Enriques quotient of X. In generic cases, this gives a bijection between the set ${\mathcal Enr}(X)$ of Enriques quotients of X up to isomorphism and the set of Brauer classes of X of order 2. For some K3 surfaces of Picard rank $20,$ we prove that the fibers of ${\mathcal Enr}(X)\to \mathrm {{Br}}(X)[2]$ above the nonzero points have the same cardinality.

A Diophantine problem about Kummer surfaces

A Diophantine problem about Kummer surfaces

Kummer quartic double solids

We study equivariant birational geometry of (rational) quartic double solids ramified over (singular) Kummer surfaces.

The generalized Plücker–Klein map

Abstract The intersection of two quadrics is called a biquadric. If we mark a non-singular quadric in the pencil of quadrics through a given biquadric, then the given biquadric is called a marked biquadric. In the classical papers of Plücker and Klein, a Kummer surface was canonically associated with every three-dimensional marked biquadric (that is, with a quadratic line complex provided that the Plücker–Klein quadric is marked). In Reid’s thesis, this correspondence was generalized to odd-dimensional marked biquadrics of arbitrary dimension . In this case, a Kummer variety of dimension corresponds to every biquadric of dimension . Reid only constructed the generalized Plücker–Klein correspondence. This map was not studied later. The present paper is devoted to a partial solution of the problem of creating the corresponding theory.

K3 surfaces with maximal finite automorphism groups containing M 20

It was shown by Mukai that the maximum order of a finite group acting faithfully and symplectically on a K3 surface is $960$ and that the group is isomorphic to the group $M_{20}$. Then Kondo showed that the maximum order of a finite group acting faithfully on a K3 surface is $3\,840$ and this group contains the Mathieu group $M_{20}$ with index four. Kondo also showed that there is a unique K3 surface on which this group acts faithfully, which is the Kummer surface $\Km(E_i\times E_i)$. In this paper we describe two more K3 surfaces admitting a big finite automorphism group of order $1\,920$, both groups contains $M_{20}$ as a subgroup of index 2. We show moreover that these two groups and the two K3 surfaces are unique. This result was shown independently by S. Brandhorst and K. Hashimoto in a forthcoming paper, with the aim of classifying all the finite groups acting faithfully on K3 surfaces with maximal symplectic part.

G-invariant Hilbert schemes on Abelian surfaces and enumerative geometry of the orbifold Kummer surface

For an Abelian surface A with a symplectic action by a finite group G, one can define the partition function for G-invariant Hilbert schemes $$\begin{aligned} Z_{A, G}(q) = \sum _{d=0}^{\infty } e(\text {Hilb}^{d}(A)^{G})q^{d}. \end{aligned}$$We prove the reciprocal \(Z_{A,G}^{-1}\) is a modular form of weight \(\frac{1}{2}e(A/G)\) for the congruence subgroup \(\Gamma _{0}(|G|)\) and give explicit expressions in terms of eta products. Refined formulas for the \(\chi _{y}\)-genera of \(\text {Hilb}(A)^{G}\) are also given. For the group generated by the standard involution \(\tau : A \rightarrow A\), our formulas arise from the enumerative geometry of the orbifold Kummer surface \([A/\tau ]\). We prove that a virtual count of curves in the stack is governed by \(\chi _{y}(\text {Hilb}(A)^{\tau })\). Moreover, the coefficients of \(Z_{A, \tau }\) are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin.

Isogenies of certain K3 surfaces of rank 18

We construct geometric isogenies between three types of two-parameter families of K3 surfaces of Picard rank 18. One is the family of Kummer surfaces associated with Jacobians of genus-two curves admitting an elliptic involution, another is the family of Kummer surfaces associated with the product of two non-isogeneous elliptic curves, and the third is the twisted Legendre pencil. The isogenies imply the existence of algebraic correspondences between these K3 surfaces and prove that the associated four-dimensional Galois representations are isomorphic. We also apply our result to several subfamilies of Picard rank 19. The result generalizes work of van Geemen and Top.

A special configuration of 12 conics and generalized Kummer surfaces

A generalized Kummer surface X obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $$9\mathbf{A}_{2}$$ -configuration of $$(-2)$$ -curves. Such a configuration plays the role of the $$16\mathbf{A}_{1}$$ -configurations for usual Kummer surfaces. In this paper we construct 9 other such $$9\mathbf{A}_{2}$$ -configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $$9\mathbf{A}_{2}$$ -configurations are obtained by taking the pullback of a certain configuration of 12 conics which are in special position with respect to the branch curve, plus some singular quartic curves. We then construct some automorphisms of the K3 surface sending one configuration to another. We also give various models of X and of the generic fiber of its natural elliptic pencil.

Quantum Information in the Protein Codes, 3-Manifolds and the Kummer Surface

Every protein consists of a linear sequence over an alphabet of 20 letters/amino acids. The sequence unfolds in the 3-dimensional space through secondary (local foldings), tertiary (bonds) and quaternary (disjoint multiple) structures. The mere existence of the genetic code for the 20 letters of the linear chain could be predicted with the (informationally complete) irreducible characters of the finite group Gn:=Zn⋊2O (with n=5 or 7 and 2O the binary octahedral group) in our previous two papers. It turns out that some quaternary structures of protein complexes display n-fold symmetries. We propose an approach of secondary structures based on free group theory. Our results are compared to other approaches of predicting secondary structures of proteins in terms of α helices, β sheets and coils, or more refined techniques. It is shown that the secondary structure of proteins shows similarities to the structure of some hyperbolic 3-manifolds. The hyperbolic 3-manifold of smallest volume—Gieseking manifold—some other 3 manifolds and the oriented hypercartographic group are singled out as tentative models of such secondary structures. For the quaternary structure, there are links to the Kummer surface.

Moduli spaces of abstract and embedded Kummer varieties

In this paper, we investigate the construction of two moduli stacks of Kummer varieties. The first one is the stack [Formula: see text] of abstract Kummer varieties and the second one is the stack [Formula: see text] of embedded Kummer varieties. We will prove that [Formula: see text] is a Deligne-Mumford stack and its coarse moduli space is isomorphic to [Formula: see text], the coarse moduli space of principally polarized abelian varieties of dimension [Formula: see text]. On the other hand, we give a modular family [Formula: see text] of embedded Kummer varieties embedded in [Formula: see text], meaning that every geometric fiber of this family is an embedded Kummer variety and every isomorphic class of such varieties appears at least once as the class of a fiber. As a consequence, we construct the coarse moduli space [Formula: see text] of embedded Kummer surfaces and prove that it is obtained from [Formula: see text] by contracting the locus swept by a particular linear equivalence class of curves. We conjecture that this is a general fact: [Formula: see text] could be obtained from [Formula: see text] via a contraction for all [Formula: see text].

F-theory models with 3 to 8 U(1) factors on K3 surfaces

In this study, we construct four-dimensional F-theory models with 3 to 8 U(1) factors on products of K3 surfaces. We provide explicit Weierstrass equations of elliptic K3 surfaces with Mordell–Weil ranks of 3 to 8. We utilize the method of quadratic base change to glue pairs of rational elliptic surfaces together to yield the aforementioned types of K3 surfaces. The moduli of elliptic K3 surfaces constructed in the study include Kummer surfaces of specific complex structures. We show that the tadpole cancels in F-theory compactifications with flux when these Kummer surfaces are paired with appropriately selected attractive K3 surfaces. We determine the matter spectra on F-theory on the pairs.