Showing papers on "Ring of integers published in 2020"

Expand-and-Randomize: An Algebraic Approach to Secure Computation

Abstract: We consider the secure computation problem in a minimal model, where Alice and Bob each holds an input and wish to securely compute a function of their inputs at Carol without revealing any additional information about the inputs. For this minimal secure computation problem, we propose a novel coding scheme built from two steps. First, the function to be computed is expanded such that it can be recovered while additional information might be leaked. Second, a randomization step is applied to the expanded function such that the leaked information is protected. We implement this expand-and-randomize coding scheme with two algebraic structures - the finite field and the modulo ring of integers, where the expansion step is realized with the addition operation and the randomization step is realized with the multiplication operation over the respective algebraic structures.

There are no universal ternary quadratic forms over biquadratic fields

Abstract: We study totally positive definite quadratic forms over the ring of integers ; we prove several new results about their properties.

Computing Tropical Varieties Over Fields with Valuation

Abstract: We show how the tropical variety of an ideal $$I\unlhd K[x_1,\ldots ,x_n]$$ over a field K with non-trivial discrete valuation can always be traced back to the tropical variety of an ideal $$\pi ^{-1}I\unlhd R\llbracket t\rrbracket [x_1,\ldots ,x_n]$$ over some dense subring R in its ring of integers. We show that this connection is compatible with the Grobner polyhedra covering them. Combined with previous works, we thus obtain a framework for computing tropical varieties over general fields with valuations, which relies on the existing theory of standard bases if $$\pi ^{-1}I$$ is generated by elements in $$R[t,x_1,\ldots ,x_n]$$ .

On ternary Diophantine equations of signature $ p,p,2 $ over number fields

Abstract: Let $K$ be a totally real number field with narrow class number one and $O_K$ be its ring of integers. We prove that there is a constant $B_K$ depending only on $K$ such that for any prime exponent $p>B_K$ the Fermat type equation $x^p+y^p=z^2$ with $x,y,z\in O_K$ does not have certain type of solutions. Our main tools in the proof are modularity, level lowering, and image of inertia comparisons.

Generalised representations as q-connections in integral $p$-adic Hodge theory

Abstract: We relate various approaches to coefficient systems in relative integral $p$-adic Hodge theory, working in the geometric context over the ring of integers of a perfectoid field. These include small generalised representations over $A_{\text{inf}}$ inspired by Faltings, modules with q-connection in the sense of q-de Rham cohomology, crystals on the prismatic site of Bhatt--Scholze, and q-deformations of Higgs bundles.

On indefinite and potentially universal quadratic forms over number fields

Abstract: A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary integral quadratic forms over $k$. A number field $k$ admits a ternary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is even. In this case, there are infinitely many classes of such ternary integral quadratic forms over $k$. An integral quadratic form over a number field $k$ with more than one variables represents all integers of $k$ over the ring of integers of a finite extension of $k$ if and only if this quadratic form represents $1$ over the ring of integers of a finite extension of $k$.

Supersingular main conjectures, Sylvester's conjecture and Goldfeld's conjecture

Abstract: We prove a $p$-converse theorem for elliptic curves $E/\mathbb{Q}$ with complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ in which $p$ is ramified. Namely, letting $r_p = \mathrm{corank}_{\mathbb{Z}_p}\mathrm{Sel}_{p^{\infty}}(E/\mathbb{Q})$, we show that $r_p \le 1 \implies \mathrm{rank}_{\mathbb{Z}}E(\mathbb{Q}) = \mathrm{ord}_{s = 1}L(E/\mathbb{Q},s) = r_p$ and $\#\mathrm{Sha}(E/\mathbb{Q}) < \infty$. In particular, this has applications to two classical problems of arithmetic. First, it resolves Sylvester's conjecture on rational sums of cubes, showing that for all primes $\ell \equiv 4,7,8 \pmod{9}$, there exists $(x,y) \in \mathbb{Q}^{\oplus 2}$ such that $x^3 + y^3 = \ell$. Second, combined with work of Smith, it resolves the congruent number problem in 100\% of cases and establishes Goldfeld's conjecture on ranks of quadratic twists for the congruent number family. The method for showing the above $p$-converse theorem relies on new interplays between Iwasawa theory for imaginary quadratic fields at nonsplit primes and relative $p$-adic Hodge theory. In particular, we show that a certain de Rham period $q_{\mathrm{dR}}$ introduced by the author in previous work can be used to construct analytic continuations of ordinary Serre-Tate expansions on the Igusa tower to the infinite-level overconvergent locus. Using this coordinate, one can construct 1-variable measures for Hecke characters and CM-newforms satisfying a new type of interpolation property. Moreover, one can relate the Iwasawa module of elliptic units to these anticyclotomic measures via a new "Coleman map", which is roughly the $q_{\mathrm{dR}}$-expansion of the Coleman power series map. Using this, we formulate and prove a new Rubin-type main conjecture for elliptic units, which is eventually related to Heegner points in order to prove the $p$-converse theorem.

Towards Hilbert's Tenth Problem for rings of integers through Iwasawa theory and Heegner points

Abstract: For a positive proportion of primes p and q, we prove that $${\mathbb {Z}}$$ is Diophantine in the ring of integers of $${\mathbb {Q}}(\root 3 \of {p},\sqrt{-q})$$ . This provides a new and explicit infinite family of number fields K such that Hilbert’s tenth problem for $$O_K$$ is unsolvable. Our methods use Iwasawa theory and congruences of Heegner points in order to obtain suitable rank stability properties for elliptic curves.

The unit equation has no solutions in number fields of degree prime to $3$ where $3$ splits completely

Abstract: Let $K$ be a number field with ring of integers $\mathcal O_{K}$. We prove that if $3$ does not divide $ [K:\mathbb Q]$ and $3$ splits completely in $K$, then the unit equation has no solutions in $K$. In other words, there are no $x, y \in \mathcal O_{K}^{\times}$ with $x + y = 1$. Our elementary $p$-adic proof is inspired by the Skolem-Chabauty-Coleman method applied to the restriction of scalars of the projective line minus three points. Applying this result to a problem in arithmetic dynamics, we show that if $f \in \mathcal O_{K}[x]$ has a finite cyclic orbit in $\mathcal O_{K}$ of length $n$ then $n \in \{1, 2, 4\}$.

The Schinzel hypothesis for polynomials

Abstract: The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is replaced by a polynomial ring and prove the Schinzel hypothesis for a wide class of them: polynomials in at least one variable over the integers, polynomials in several variables over an arbitrary field, etc. We achieve this goal by developing a version over rings of the Hilbert specialization property. A polynomial Goldbach conjecture is deduced, along with a result on spectra of rational functions.

Models and Integral Differentials of Hyperelliptic Curves

Abstract: Let $C:y^2=f(x)$ be a hyperelliptic curve of genus $g\geq 2$, defined over a discretely valued complete field $K$, with ring of integers $O_K$. Under certain conditions on $C$, mild when residue characteristic is not $2$, we explicitly construct the minimal regular model with normal crossings $\mathcal{C}/O_K$ of $C$. In the same setting we determine a basis of integral differentials of $C$, that is an $O_K$-basis for the global sections of the relative dualising sheaf $\omega_{\mathcal{C}/O_K}$.

The endomorphism ring of projectives and the Bernstein centre

Abstract: Let $F$ be a local non-archimedean field and $\mathcal{O}_F$ its ring of integers. Let $\Omega$ be a Bernstein component of the category of smooth representations of $GL_n(F)$, let $(J, \lambda)$ be a Bushnell-Kutzko $\Omega$-type, and let $\mathfrak{Z}_{\Omega}$ be the centre of the Bernstein component $\Omega$. This paper contains two major results. Let $\sigma$ be a direct summand of $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} \lambda$. We will begin by computing $\mathrm{c\text{--} Ind}_{GL_n(\mathcal{O}_F)}^{GL_n(F)} \sigma\otimes_{\mathfrak{Z}_{\Omega}}\kappa(\mathfrak{m})$, where $\kappa(\mathfrak{m})$ is the residue field at maximal ideal $\mathfrak{m}$ of $\mathfrak{Z}_{\Omega}$, and the maximal ideal $\mathfrak{m}$ belongs to a Zariski-dense set in $\mathrm{Spec}\: \mathfrak{Z}_{\Omega}$. This result allows us to deduce that the endomorphism ring $\mathrm{End}_{GL_n(F)}(\mathrm{c\text{--} Ind}_{GL_n(\mathcal{O}_F)}^{GL_n(F)} \sigma)$ is isomorphic to $\mathfrak{Z}_{\Omega}$, when $\sigma$ appears with multiplicity one in $\mathrm{Ind}_J^{GL_n(\mathcal{O}_F)} \lambda$.

Integral p-adic Hodge theory of formal schemes in low ramification

Abstract: We prove that for any proper smooth formal scheme $\frak X$ over $\mathcal O_K$, where $\mathcal O_K$ is the ring of integers in a complete discretely valued nonarchimedean extension $K$ of $\mathbb Q_p$ with perfect residue field $k$ and ramification degree $e$, the $i$-th Breuil-Kisin cohomology group and its Hodge-Tate specialization admit nice decompositions when $ie

A Verified Implementation of the Berlekamp-Zassenhaus Factorization Algorithm.

Abstract: We formally verify the Berlekamp–Zassenhaus algorithm for factoring square-free integer polynomials in Isabelle/HOL. We further adapt an existing formalization of Yun’s square-free factorization algorithm to integer polynomials, and thus provide an efficient and certified factorization algorithm for arbitrary univariate polynomials. The algorithm first performs factorization in the prime field $$\mathrm {GF}(p){}$$ and then performs computations in the ring of integers modulo $$p^k$$, where both p and k are determined at runtime. Since a natural modeling of these structures via dependent types is not possible in Isabelle/HOL, we formalize the whole algorithm using locales and local type definitions. Through experiments we verify that our algorithm factors polynomials of degree up to 500 within seconds.

Twisted-PHS: Using the Product Formula to Solve Approx-SVP in Ideal Lattices

Abstract: Approx-Svp is a well-known hard problem on lattices, which asks to find short vectors on a given lattice, but its variant restricted to ideal lattices (which correspond to ideals of the ring of integers \(\mathcal {O}_{K}\) of a number field K) is still not fully understood. For a long time, the best known algorithm to solve this problem on ideal lattices was the same as for arbitrary lattice. But recently, a series of works tends to show that solving this problem could be easier in ideal lattices than in arbitrary ones, in particular in the quantum setting.

Bases of Minimal Vectors in Lagrangian Lattices

Abstract: Motivated by the ring of integers of cyclic number fields of prime degree, we introduce the notion of Lagrangian lattices. Furthermore, given an arbitrary non-trivial lattice $\mathcal{L}$ we construct a family of full-rank sub-lattices $\{\mathcal{L}_{\alpha}\}$ of $\mathcal{L}$ such that whenever $\mathcal{L}$ is Lagrangian it can be easily checked whether or not $\mathcal{L}_{\alpha}$ has a basis of minimal vectors. In this case, a basis of minimal vectors of $\mathcal{L}_{\alpha}$ is given.

Low-Rank Parity-Check Codes over the Ring of Integers Modulo a Prime Power

Abstract: We define and analyze low-rank parity-check (LRPC) codes over extension rings of the finite chain ring ${{\mathbb{Z}}_{{p^r}}}$, where p is a prime and r is a positive integer. LRPC codes have originally been proposed by Gaborit et al. (2013) over finite fields for cryptographic applications. The adaption to finite rings is inspired by a recent paper by Kamche et al. (2019), which constructed Gabidulin codes over finite principle ideal rings with applications to space-time codes and network coding. We give a decoding algorithm based on simple linear-algebraic operations. Further, we derive an upper bound on the failure probability of the decoder. The upper bound is valid for errors whose rank is equal to the free rank.

Spectrum of the zero-divisor graph on the ring of integers modulo n

Abstract: For a commutative ring R with non-zero identity, let Z∗(R) denote the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple undirected graph with all non-zero zero-divisors as vertices and two distinct vertices x, y ∈ Z∗(R) are adjacent if and only if xy = 0. In this paper, the adjacency matrix and spectrum of Γ(Zpk ) are investigated. Also, the implicit computation of the spectrum of Γ(Zn) is described.

On the left zero divisor graph of 3 × 3 matrices over ℤ2

Abstract: In this article left zero divisor graph of the ring R, where R is the set of all 3 × 3 matrices over the ring of integers modulo 2 is constructed We present algorithm to compute the vertices and e

A topological approach to undefinability in algebraic extensions of $\mathbb{Q}$

Abstract: In this paper we investigate the algebraic extensions $K$ of $\mathbb{Q}$ in which we cannot existentially or universally define the ring of integers $\mathcal{O}_K$. A complete answer to this question would have important consequences. For example, the existence of an existential definition of $\mathbb{Z}$ in $\mathbb{Q}$ would imply that Hilbert's Tenth Problem for $\mathbb{Q}$ is undecidable, resolving one of the biggest open problems in the area. However, a conjecture of Mazur implies that the integers are not existentially definable in the rationals. Although proving that an existential definition of $\mathbb{Z}$ in $\mathbb{Q}$ does not exist appears to be out of reach right now, we show that when we consider all algebraic extensions of $\mathbb{Q}$, this is the generally expected outcome. Namely, we prove that in most algebraic extensions of the rationals, the ring of integers is not existentially definable. To make this precise, we view the set of algebraic extensions of $\mathbb{Q}$ as a topological space homeomorphic to Cantor space. In this light, the set of fields which have an existentially definable ring of integers is a meager set, i.e. is very small. On the other hand, by work of Koenigsmann and Park, it is possible to give a universal definition of the ring of integers in finite extensions of the rationals, i.e. in number fields. Still, we show that their results do not extend to most algebraic infinite extensions: the set of algebraic extensions of $\mathbb{Q}$ in which the ring of integers is universally definable is also a meager set.

Cusp shapes of Hilbert–Blumenthal surfaces

Abstract: We introduce a new fundamental domain $$\mathscr {R}_n$$ for a cusp stabilizer of a Hilbert modular group $$\Gamma $$ over a real quadratic field $$K=\mathbb {Q}(\sqrt{n})$$. This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of $$\mathcal {H}^2\times \mathcal {H}^2$$. The region $$\mathscr {R}_n$$ is the product of $$\mathbb {R}^+$$ with a 3-dimensional tower $$\mathcal {T}_n$$ formed by deformations of lattices in the ring of integers $$\mathbb {Z}_K$$, and makes explicit the cusp cross section’s Sol 3-manifold structure and Anosov diffeomorphism. We include computer generated images and data illustrating various examples.

Non-definability of rings of integers in most algebraic fields

Abstract: We show that the set of algebraic extensions $F$ of $\mathbb{Q}$ in which $\mathbb{Z}$ or the ring of integers $\mathcal{O}_F$ are definable is meager in the set of all algebraic extensions.

There is no Enriques surface over the integers

Abstract: We show that there is no family of Enriques surfaces over the ring of integers. This extends non-existence results of Minkowski for families of finite etale schemes, of Tate and Ogg for families of elliptic curves, and of Fontaine for families of abelian varieties and more general smooth proper schemes with certain restrictions on Hodge numbers. Our main idea is to study the local system of numerical classes of invertible sheaves. Among other things, our result also hinges on the Weil Conjectures, Lang's classification of rational elliptic surfaces in characteristic two, the theory of exceptional Enriques surfaces due to Ekedahl and Shepherd-Barron, some recent results on the base of their versal deformation, Shioda's theory of Mordell--Weil lattices, and an extensive combinatorial study for the pairwise interaction of genus-one fibrations.

Distribution of modular symbols in $\mathbb{H}^3$

Abstract: We introduce a new technique for the study of the distribution of modular symbols, which we apply to congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$ its ring of integers, then for certain congruence subgroups of $\mathrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey asymptotically a normal distribution. These results are specialisations from the more general setting of quotient surfaces of cofinite Kleinian groups, where our methods apply. We avoid the method of moments. Our new insight is to use the behaviour of the smallest eigenvalue of the Laplacian for spaces twisted by modular symbols. Our approach also recovers the first and the second moment of the distribution.

The mod 2 cohomology rings of congruence subgroups in the Bianchi groups

Abstract: We establish a dimension formula involving a number of parameters for the mod 2 cohomology of finite index subgroups in the Bianchi groups (SL $$_2$$ groups over the ring of integers in an imaginary quadratic number field). The proof of our formula involves an analysis of the equivariant spectral sequence, combined with torsion subcomplex reduction. We also provide an algorithm to compute a Ford domain for congruence subgroups in the Bianchi groups from which the parameters in our formula can be explicitly computed.

Mahler's measure and elliptic curves with potential complex multiplication

Abstract: Given an elliptic curve $E$ defined over $\mathbb{Q}$ which has potential complex multiplication by the ring of integers $\mathcal{O}_K$ of an imaginary quadratic field $K$ we construct a polynomial $P_E \in \mathbb{Z}[x,y]$ which is a planar model of $E$ and such that the Mahler measure $m(P_E) \in \mathbb{R}$ is related to the special value of the $L$-function $L(E,s)$ at $s = 2$.