Showing papers on "Ring of integers published in 2014"
TL;DR: In this paper, a method to compute rational models for the Hilbert modular surfaces Y−(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in ℚ(D) via moduli space of elliptic K3 surfaces with a Shioda-Inose structure, is presented.
Abstract: We outline a method to compute rational models for the Hilbert modular surfaces Y−(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in ℚ(D), via moduli spaces of elliptic K3 surfaces with a Shioda–Inose structure. In particular, we compute equations for all thirty fundamental discriminants D with 1
65 citations
05 Jan 2014
TL;DR: A very simple and efficient output-sensitive algorithm for matrix multiplication that works over any field and shows the randomized communication complexity of the problem is closely related to the sizes of matching vector families, which have been studied in the design of locally decodable codes.
Abstract: Hopcroft's problem in d dimensions asks: given n points and n hyperplanes in Rd, does any point lie on any hyperplane? Equivalently, if we are given two sets of n vectors each in Rd+1, is there a pair of vectors (one from each set) that are orthogonal? This problem has a long history and a multitude of applications. It is widely believed that for large d, the problem is subject to the curse of dimensionality: all known algorithms need at least f(d) · n2-1/O(d) time for fast-growing functions f, and at the present time there is little hope that a n2-e · poly(d) time algorithm will be found.We consider Hopcroft's problem over finite fields and integers modulo composites, leading to both surprising algorithms and hardness reductions. The algorithms arise from studying the communication problem of determining whether two lists of vectors (one list held by Alice, one by Bob) contain an orthogonal pair of vectors over a discrete structure (one from each list). We show the randomized communication complexity of the problem is closely related to the sizes of matching vector families, which have been studied in the design of locally decodable codes. Letting HOPCROFTR denote Hopcroft's problem over a ring R, we give randomized algorithms and almost matching lower bounds (modulo a breakthrough in SAT algorithms) for HOPCROFTR, when R is the ring of integers modulo m or a finite field.Building on the ideas developed here, we give a very simple and efficient output-sensitive algorithm for matrix multiplication that works over any field.
64 citations
TL;DR: In this paper, the problem of associating to any representation an admissible A[GL_n(E)]-module in a manner compatible with the local Langlands correspondence at characteristic zero points of Spec A was considered.
Abstract: Let E be a nonarchimedean local field with residue characteristic l, and suppose we have an n-dimensional representation of the absolute Galois group G_E of E over a reduced complete Noetherian local ring A with finite residue field k of characteristic p different from l. We consider the problem of associating to any such representation an admissible A[GL_n(E)]-module in a manner compatible with the local Langlands correspondence at characteristic zero points of Spec A. In particular we give a set of conditions that uniquely characterise such an A[GL_n(E)]-module if it exists, and show that such an A[GL_n(E)]-module always exists when A is the ring of integers of a finite extension of Q_p. We also use these results to define a "modified mod p local Langlands correspondence" that is more compatible with specialization of Galois representations than the mod p local Langlands correspondence of Vigneras.
53 citations
TL;DR: In this paper, the authors give sketches of classical undecidability results in number theory, like Godel's first Incompleteness Theorem (that the first order theory of the integers in the language of rings is undecidable), Julia Robinson's extensions of this result to arbitrary number fields and rings of integers in them, as well as to the ring of totally real integers.
Abstract: In these lecture notes we give sketches of classical undecidability results in number theory, like Godel’s first Incompleteness Theorem (that the first order theory of the integers in the language of rings is undecidable), Julia Robinson’s extensions of this result to arbitrary number fields and rings of integers in them, as well as to the ring of totally real integers, and Matiyasevich’s negative solution of Hilbert’s 10th problem, i.e., the undecidability of the existential first-order theory of the integers. As Hilbert’s 10th problem is still open for the rationals (i.e., the question whether the existential theory of the field of rational numbers is decidable) we also present a sketch of the fact that there is a universal definition of the ring of integers inside the field of rationals. In terms of complexity this is the simplest definition known so far. If one had an existential definition instead then Hilbert’s 10th problem over the rationals would reduce to that over the integers (and hence be, as expected, unsolvable), but, modulo a widely believed in conjecture in number theory, we also indicate why there should be no such existential definition. We conclude with a list of nice open questions in the area.
47 citations
05 Jan 2014
TL;DR: It is shown that the problem of minimizing a quadratic polynomial with integer coefficients over the integer points in a general two-dimensional rational polyhedron is solvable in time bounded by a polynometric in the input size.
Abstract: We show that the problem of minimizing a quadratic polynomial with integer coefficients over the integer points in a general two-dimensional rational polyhedron is solvable in time bounded by a polynomial in the input size.
22 citations
TL;DR: In this article, the Deligne formal model of the Drinfeld p-adic halfplane relative to a local field F represents a moduli problem of polarized OF -modules with an action of the ring of integers in a quadratic extension E of F.
Abstract: We show that the Deligne formal model of the Drinfeld p-adic halfplane relative to a local field F represents a moduli problem of polarized OF -modules with an action of the ring of integers in a quadratic extension E of F . The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of SL2(F ) and SU(C)(F ) for a two-dimensional split hermitian space C for E/F .
22 citations
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TL;DR: In this article, it was shown that the valuation of a non-zero rational function is piecewise linear on a skeleton in a strictly semistable non-archimedean field.
Abstract: Let $K$ be a complete, algebraically closed non-archimedean field with ring of integers $K^\circ$ and let $X$ be a $K$-variety. We associate to the data of a strictly semistable $K^\circ$-model $\mathscr X$ of $X$ plus a suitable horizontal divisor $H$ a skeleton $S(\mathscr X,H)$ in the analytification of $X$. This generalizes Berkovich's original construction by admitting unbounded faces in the directions of the components of H. It also generalizes constructions by Tyomkin and Baker--Payne--Rabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a non-zero rational function is piecewise linear on $S(\mathscr X, H)$. For such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton. Moreover, we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a well-known result by Sturmfels--Tevelev. We show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization. We also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one.
21 citations
TL;DR: In this paper, a characterization of the catenary degree of a Krull monoid with finite class groups is given. But the exact value of the k-catenary degree is not known for any further class groups.
Abstract: Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf c (H)$ of $H$ is the smallest integer $N$ with the following property: for each $a \in H$ and each two factorizations $z, z'$ of $a$, there exist factorizations $z = z_0, ..., z_k = z'$ of $a$ such that, for each $i \in [1, k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases, suppose that $|G| \ge 3$. Then the catenary degree depends only on the class group $G$ and we have $\mathsf c (H) \in [3, \mathsf D (G)]$, where $\mathsf D (G)$ denotes the Davenport constant of $G$. It is well-known when $\mathsf c (H) \in \{3,4, \mathsf D (G)\}$ holds true. Based on a characterization of the catenary degree determined in the first paper (The catenary degree of Krull monoids I), we determine the class groups satisfying $\mathsf c (H)= \mathsf D (G)-1$. Apart from the mentioned extremal cases the precise value of $\mathsf c (H)$ is known for no further class groups.
20 citations
TL;DR: In this article, the existence of closed subschemes of a quasi-projective morphism over an affine base was shown to be a special case of a moving lemma pertaining to horizontal 1-cycles on a regular scheme X projective and flat over S. This condition is satisfied if R is the ring of integers of a number field.
Abstract: Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X/S containing a given closed subscheme C, and intersecting properly a closed set F.
Assume now that the base S is the spectrum of a ring R such that for any finite morphism Z -> S, Pic(Z) is a torsion group. This condition is satisfied if R is the ring of integers of a number field, or the ring of functions of a smooth affine curve over a finite field. We prove in this context a moving lemma pertaining to horizontal 1-cycles on a regular scheme X quasi-projective and flat over S. We also show the existence of a finite surjective S-morphism to the projective space P_S^d for any scheme X projective over S when X/S has all its fibers of a fixed dimension d.
19 citations
TL;DR: It is shown that, if h∈K[X] maps every element of OK of degree n to an algebraic integer, then h(X) is integral-valued over OK, that is, h(OK)⊂OK, and that the integral closure of the ring of polynomials in Q[X].
17 citations
TL;DR: In this paper, the exterior powers of a π -divisible O-module scheme of dimension at most 1 over a local field were shown to exist and commute with field extensions.
TL;DR: In this paper, the wild quotient singularities of the quotient Y/H were desingularized to obtain the regular model of X/K, and the most precise information on the resolution of these singularities was obtained when the special fiber Yk/k is ordinary.
Abstract: Let K be a complete discrete valuation field with ring of integers OK and algebraically closed residue field k of characteristic p > 0. Let X/K be a smooth proper geometrically connected curve of genus g > 0, withX(K) 6= ∅ if g = 1. Assume thatX/K does not have good reduction, and that it obtains good reduction over a Galois extension L/K of degree p. Let Y/OL be the smooth model of XL/L. Let H := Gal(L/K). In this article, we provide information on the regular model of X/K obtained by desingularizing the wild quotient singularities of the quotient Y/H . The most precise information on the resolution of these quotient singularities is obtained when the special fiber Yk/k is ordinary. As a corollary, we are able to produce for each odd prime p an infinite class of wild quotient singularities having pairwise distinct resolution graphs. The information on the regular model of X/K also allows us to gather insight into the p-part of the component group of the Neron model of the Jacobian of X .
TL;DR: In this article, the authors explicitly compute the matrix coefficients of depth-zero supercuspidal representations of nonarchimedean local fields with integer ring o, maximal ideal p =$o, and residue field k = o/p of cardinality q.
Abstract: Let F be a nonarchimedean local field with integer ring o, maximal ideal p =$o, and residue field k = o/p of cardinality q. The supercuspidal representations of GL2(F) are precisely those irreducible admissible representations which do not arise as constituents of parabolic induction. They are characterized by having matrix coefficients which are compactly supported modulo the center. In this paper we explicitly compute the matrix coefficients of depth-zero supercuspidal representations. These are the supercuspidals with the smallest possible conductor exponent, namely 2. First discovered by Mautner [1964, Section 9], they arise by compact induction from the (q 1)-dimensional representations of GL2(o) inflated from the cuspidal series of the finite group GL2(k). In the first section, we show that the matrix coefficients of any supercuspidal representation are expressible in terms of those of the finite-dimensional inducing representation. Thus, the task at hand essentially reduces to a computation of the matrix coefficients of the cuspidal representations of GL2(k). The latter is achieved in Theorem 2.7 using the explicit model from [Piatetski-Shapiro 1983]. With global applications in mind, in Section 3 we single out the case where the test vector in the supercuspidal matrix coefficient is a unit new vector. The resulting function, given in (3-6) and Theorem 3.2, may be used to define an integral operator on the global automorphic spectrum of GL2 which isolates those cuspidal
TL;DR: In this article, the irreducible constituents of the restriction of unramified principal series representations of GL(3) to GL(O) are described, where O is the ring of integers of a nonarchimedean local field.
TL;DR: In this paper, it was shown that digit expansion of shift operators becomes an orthonormal basis for the space of continuous functions on F q [ [ [ T ], including a closed-form expression for expansion coefficients, and this is also true for p-adic integers, excluding the coefficient formula.
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TL;DR: In this article, the normal zeta functions of the Heisenberg groups of the form H(R) were explicitly computed and shown to satisfy functional equations upon the inversion of the prime.
Abstract: We compute explicitly the normal zeta functions of the Heisenberg groups $H(R)$, where $R$ is a compact discrete valuation ring of characteristic zero. These zeta functions occur as Euler factors of normal zeta functions of Heisenberg groups of the form $H(\mathcal{O}_K)$, where $\mathcal{O}_K$ is the ring of integers of an arbitrary number field~$K$, at the rational primes which are non-split in~$K$. We show that these local zeta functions satisfy functional equations upon the inversion of the prime.
TL;DR: In this paper, it was shown that a hyperelliptic curve has a Weierstrass model over the ring of integers of a number field with height effectively bounded only in terms of the genus of the curve.
Abstract: Let $C$ be a hyperelliptic curve of genus $g\geq 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\geq 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$.
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TL;DR: In this article, a semi-orthogonal decomposition of coherent sheaves on Grassmannians over the ring of integers has been derived, with components being full subcategories of the derived category of representations.
Abstract: In this paper we study the derived categories of coherent sheaves on Grassmannians $\operatorname{Gr}(k,n),$ defined over the ring of integers. We prove that the category $D^b(\operatorname{Gr}(k,n))$ has a semi-orthogonal decomposition, with components being full subcategories of the derived category of representations of $GL_k.$ This in particular implies existence of a full exceptional collection, which is a refinement of Kapranov's collection \cite{Kap}, which was constructed over a field of characteristic zero.
We also describe the right dual semi-orthogonal decomposition which has a similar form, and its components are full subcategories of the derived category of representations of $GL_{n-k}.$ The resulting equivalences between the components of the two decompositions are given by a version of Koszul duality for strict polynomial functors.
We also construct a tilting vector bundle on $\operatorname{Gr}(k,n).$ We show that its endomorphism algebra has two natural structures of a split quasi-hereditary algebra over $\mathbb{Z},$ and we identify the objects of $D^b(\operatorname{Gr}(k,n)),$ which correspond to the standard and costandard modules in both structures.
All the results automatically extend to the case of arbitrary commutative base ring and the category of perfect complexes on the Grassmannian, by extension of scalars (base change).
Similar results over fields of arbitrary characteristic were obtained independently in \cite{BLVdB}, by different methods.
TL;DR: In this article, it was shown that the ramification of a torsion element in a regular proper model over the ring of integers in a number field can be split in an extension of degree p^3.
Abstract: Let k be a number field and F a function field in one variable over k. We prove that the ramification of a \(p\)-torsion element in \(Br\)(\(F\)) on a regular proper model over the ring of integers in \(k\) can be split in an extension of degree \(p^3\). Using this result, we show that Colliot-Thelene’s conjecture on 0-cycles of degree 1 implies finiteness for the \(u\)-invariant of the function field of a curve over a totally imaginary number field and period-index bounds for the Brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers.
TL;DR: In this article, the authors study the canonical morphism $q\colon \mathrm{Pic}^{0}_{X/S}\to J$ which extends the biduality isomorphism on generic fibres and show that it is pro-algebraic in nature.
Abstract: Let $K$ be a local field with algebraically closed residue field and $X_K$ a torsor under an elliptic curve $J_K$ over $K$. Let $X$ be a proper minimal regular model of $X_K$ over the ring of integers of $K$ and $J$ the identity component of the Neron model of $J_K$. We study the canonical morphism $q\colon \mathrm{Pic}^{0}_{X/S}\to J$ which extends the biduality isomorphism on generic fibres. We show that $q$ is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore we interpret our results in relation to Shafarevich's duality theory for torsors under abelian varieties.
TL;DR: An asymptotic formula is given for the average order of e"p over primes p@?x of good reduction, with improved error term O(x^2/log^A @?x) for any positive number A, which previously was set as O( x^2 /log^1^/^8@?X) by [12].
TL;DR: In this article, the density of the set of coprime $m$-tuples of algebraic integers is shown to be 1/\zeta_K(m) where ''zeta'' is the Dedekind zeta function of a number field with ring of integers.
Abstract: Let $K$ be a number field with ring of integers $\mathcal O$. After introducing a suitable notion of density for subsets of $\mathcal O$, generalizing that of natural density for subsets of $\mathbb Z$, we show that the density of the set of coprime $m$-tuples of algebraic integers is ${1/\zeta_K(m)}$, where $\zeta_K$ is the Dedekind zeta function of $K$.
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TL;DR: In this article, the authors studied the distribution of bounded discriminants in number fields and gave an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field.
Abstract: In this paper we study the distribution of orders of bounded discriminants in number fields. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field.
TL;DR: In this article, it was shown that for any fixed Hecke eigenform f ∈ S>>\s 2k>>\s (Γ) and any e>0, the vector space of cuspidal Hilbert modular forms of parallel weight 2k for any number field of degree n over a ring of integers is an orthogonal Hilbert eigenbasis for this space.
Abstract: Let F be a totally real number field of degree n over $\mathbb{Q}$
with ring of integers $\mathcal{O}$
and narrow class number one. Let S
2k
(Γ) be the vector space of cuspidal Hilbert modular forms of parallel weight 2k for $\varGamma=\mathrm{SL}_{2}(\mathcal{O})$
, and let B
2k
be an orthogonal Hecke eigenbasis for this space. For any fixed Hecke eigenform f∈S
2k
(Γ) and any e>0, we prove that $$\# \biggl\{ g \in B_{2k}: L \biggl(f \times g, \frac{1}{2} \biggr)
e0 \biggr\} \gg k^{n- \varepsilon}, $$
where L(f×g,s) is the Rankin–Selberg L–function of f and g.
11 Mar 2014
TL;DR: In this paper, the authors adopt the tableau model for the crystal and obtain the same coefficients using data from each individual tableau; i.e., we do not need to look at the graph structure.
Abstract: In the recent works of Brubaker-Bump-Friedberg, Bump-Nakasuji, and others, the product in the Casselman-Shalika formula is written as a sum over a crystal. The coefficient of each crystal element is defined using the data coming from the whole crystal graph structure. In this paper, we adopt the tableau model for the crystal and obtain the same coefficients using data from each individual tableau; i.e., we do not need to look at the graph structure. We also show how to combine our results with tensor products of crystals to obtain the sum of coefficients for a given weight. The sum is a q-polynomial which exhibits many interesting properties. We use examples to illustrate these properties. 0. Introduction Let F be a p-adic field with a ring of integers oF and residue field of size q. We denote by a uniformizer of oF . Suppose N − is the maximal unipotent subgroup of GLr+1(F ) with maximal torus T , and f ◦ denotes the standard spherical vector corresponding to an unramified character χ of T . Let T (C) be the maximal torus in the dual group GLr+1(C) of GLr+1(F ), and let z ∈ T (C) be the element corresponding to χ via the Satake isomorphism. For a dominant integral weight λ = (λ1 ≥ λ2 ≥ · · · ≥ λr+1), we define ψλ ⎜⎜⎜⎝ 1 x2,1 1 .. . . . xr+1,1 · · · xr+1,r 1 ⎟⎟⎟⎠ = ψ0( 12xr+1,r + · · ·+ rr+1x2,1), where ψ0 is a fixed additive character on F which is trivial on oF but not on p −1. Let χλ(z) be the irreducible character of GLr+1(C) with highest weight λ. Then the Casselman-Shalika formula is
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TL;DR: In this paper, the authors studied the non-existence problem of (nearly) perfect (almost) $m$-ary sequences via their connection to Butson-Hadamard (BH) matrices and (near) conference matrices.
Abstract: In this paper we study the non-existence problem of (nearly) perfect (almost) $m$-ary sequences via their connection to (near) Butson-Hadamard (BH) matrices and (near) conference matrices. Firstly, we apply a result on vanishing sums of roots of unity and a result of Brock on the unsolvability of certain equations over a cyclotomic number field to derive non-existence results for near BH matrices and near conference matrices. Secondly, we refine the idea of Brock in the case of cyclotomic number fields whose ring of integers is not a principal ideal domains and get many new non-existence results.
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TL;DR: In this paper, a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order is given, which enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4/3 log g(X), where g is the genus.
Abstract: We give a detailed description of the arithmetic Fuchsian group of the Bolza surface and the associated quaternion order. This description enables us to show that the corresponding principal congruence covers satisfy the bound sys(X) > 4/3 log g(X) on the systole, where g is the genus. We also exhibit the Bolza group as a congruence subgroup, and calculate out a few examples of "Bolza twins" (using magma). Like the Hurwitz triplets, these correspond to the factoring of certain rational primes in the ring of integers of the invariant trace field of the surface. We exploit random sampling combined with the Reidemeister-Schreier algorithm as implemented in magma to generate these surfaces.
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TL;DR: In this article, the Eichler class number formula was generalized to arbitrary orders in a quaternion algebra over a prime finite field and the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over the prime field were enumerated.
Abstract: Let $F$ be a totally real field with ring of integers $O_F$, and $D$ be a totally definite quaternion algebra over $F$. A well-known formula established by Eichler and then extended by Korner computes the class number of any $O_F$-order in $D$. In this paper we generalize the Eichler class number formula so that it works for arbitrary $\mathbb{Z}$-orders in $D$. The motivation is to count the isomorphism classes of supersingular abelian surfaces in a simple isogeny class over a prime finite field $\mathbb{F}_p$. We give explicit formulas for the number of these isomorphism classes for all primes $p$.
01 Jan 2014
TL;DR: A characterization of all integers in the ring of integers modulo $n such that every positive integer z can be written as the sum of two squares in $\Z_n$.
Abstract: A classical theorem in number theory due to Euler states that a positive integer z can be written as the sum of two squares if and only if all prime factors q of z, with q ≡ 3 (mod 4), occur with even exponent in the prime factorization of z. One can consider a minor variation of this theorem by not allowing the use of zero as a summand in the representation of z as the sum of two squares. Viewing each of these questions in Zn, the ring of integers modulo n, we give a characterization of all integers n ≥ 2 such that every z ∈ Zn can be written as the sum of two squares in Zn.
TL;DR: In this article, it was shown that the results of Calegari and Venkatesh can be extended to the case of nonuniform arithmetic lattices that may contain torsion.
Abstract: Let be a symmetric space of noncompact type. A result of Gelander provides exponential upper bounds in terms of the volume for the torsion homology of the noncompact arithmetic locally symmetric spaces . We show that under suitable assumptions on this result can be extended to the case of nonuniform arithmetic lattices that may contain torsion. Using recent work of Calegari and Venkatesh we deduce from this upper bounds (in terms of the discriminant) for of the ring of integers of totally imaginary number fields . More generally, we obtain such bounds for rings of -integers in F.