Showing papers on "Ring of integers published in 2004"

Néron models, Lie algebras, and reduction of curves of genus one

Abstract: Let K be a discrete valuation field. Let OK denote the ring of integers of K , and let k be the residue field of OK , of characteristic p ≥ 0. Let S := SpecOK . Let X K be a smooth geometrically connected projective curve of genus 1 over K . Denote by EK the Jacobian of X K . Let X/S and E/S be the minimal regular models of X K and EK , respectively. In this article, we investigate the possible relationships between the special fibers Xk and Ek . In doing so, we are led to study the geometry of the Picard functor Pic X/S when X/S is not necessarily cohomologically flat. As an application of this study, we are able to prove in full generality a theorem of Gordon on the equivalence between the Artin-Tate and Birch-SwinnertonDyer conjectures. Recall that when k is algebraically closed, the special fibers of elliptic curves are classified according to their Kodaira type, which is denoted by a symbol T ∈ {In, I∗n, n ∈ Z≥0, II, II∗, III, III∗, IV, IV∗}. Given a type T and a positive integer m, we denote by mT the new type obtained from T by multiplying all the multiplicities of T by m. When k is algebraically closed, the relationships between the type of a curve of genus 1 and the type of its Jacobian can be summarized as follows.

Les variétés sur le corps à un élément

Abstract: We propose a definition of varieties over the field with one element. These have extensions of scalars to the ring of integers which are varieties in the usual sense. We show that toric varieties can be defined over the field with one element. We also discuss zeta functions for such objects. We give a motivic interpretation of the image of the J-homomorphism defined by Adams. ~ ~ ~ ~

Euclidean Rings of Algebraic Integers

Abstract: Let K be a nit e Galois extension of the eld of rational numbers with unit rank greater than 3. We prove that the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally.

Asymptotic variation of L functions of one-variable exponential sums

Abstract: Let d>2 and let p be a prime coprime to d. Let Z_pbar be the ring of integers of Q_pbar. Suppose f(x) is a degree-d polynomial over Qbar and Z_pbar. Let P be a prime ideal over p in the ring of integers of Q(f), where Q(f) is the number field generated by coefficients of f in Qbar. Let A^d be the dimension-d affine space over Qbar, identified with the space of coefficients of degree-d monic polynomials. Let NP(f mod P) denote the p-adic Newton polygon of L(f mod P;T). Let HP(A^d) denote the p-adic Hodge polygon of A^d. We prove that there is a Zariski dense open subset U defined over Q in A^d such that for every geometric point f(x) in U(Qbar) we have lim_{p-->oo} NP(f mod P) = HP(A^d), where P is any prime ideal in the ring of integers of Q(f) lying over p. This proves a conjecture of Daqing Wan.

Counting congruence subgroups

Abstract: Let Γ denote the modular group SL(2,Z) and Cn(Γ) the number of congruence subgproups of Γ of index at most n. We prove that lim n→∞ log Cn(Γ) (log n)2/ log log n = 3−2 √ 2 4 . Some extensions of this result for other arithmetic groups are presented as well as a general conjecture. §0. Introduction Let k be an algebraic number field, O its ring of integers, S a finite set of valuations of k (containing all the archimedean ones), and OS = { x ∈ k ∣∣ v(x) ≥ 0, ∀v ∈ S}. Let G be a semisimple, simply connected, connected algebraic group defined over k with a fixed embedding into GLd. Let Γ = G(OS) = G ∩ GLd(OS) be the corresponding S-arithmetic group. We assume that Γ is an infinite group. For every non-zero ideal I of OS let Γ(I) = Ker ( Γ → GLd(OS/I) ) . A subgroup of Γ is called a congruence subgroup if it contains Γ(I) for some I. For n > 0, define Cn(Γ) = # { congruence subgroups of Γ of index at most n } . Theorem 1. There exist two positive real numbers α− and α+ such that for all sufficiently large positive integers n n log n log log nα− ≤ Cn(Γ) ≤ n log n log log nα+ . This theorem is proved in [Lu], although the proof of the lower bound presented there requires the prime number theorem on arithmetic progressions in an interval where its validity depends on the GRH (generalized Riemann hypothesis for arithmetic progressions). The first two authors research is supported in part by the NSF. The third author’s Research is supported in part by OTKA T 034878. All three authors would like to thank Yale University for its hospitality. Typeset by AMS-TEX 1 2 DORIAN GOLDFELD ALEXANDER LUBOTZKY LASZLO PYBER In §2 below, we show that by appealing to a theorem of Linnik [Li1, Li2] on the least prime in an arithmetic progression, the proof can be made unconditional. Following [Lu] we define: α+(Γ) = lim logCn(Γ) λ(n) , α−(Γ) = lim logCn(Γ) λ(n) , where λ(n) = (log n) 2 log log n . It is not difficult to see that α+ and α− are independent of both the choice of the representation of G as a matrix group, as well as independent of the choice of S. Hence α± depend only on G and k. The question whether α+(Γ) = α−(Γ) and the challenge to evaluate them for Γ = SL2(Z) and other groups were presented in [Lu]. It was conjectured by Rademacher that there are only finitely many congruence subgroups of SL2(Z) of genus zero. This counting problem has a long history. Petersson [Pe, 1974] proved that the number of all subgroups of index n and fixed genus goes to infinity exponentially as n → ∞. Dennin [De, 1975] proved that there are only finitely many congruence subgroups of SL2(Z) of given fixed genus and solved Rademacher’s conjecture. It does not seem possible, however, to accurately count all congruence subgroups of index at most n in SL2(Z) by using the theory of Riemann surfaces of fixed genus. Here we prove: Theorem 2. α+(SL2(Z)) = α−(SL2(Z)) = 3−2 √ 2 4 = 0.0428932 . . . We believe that SL2(Z) represents the general case and we expect that α+ = α− for all groups. The proof of the lower bound in Theorem 2 is based on the Bombieri-Vinogradov Theorem [Bo], [Da], [Vi], i.e., the Riemann hypothesis on the average. The upper bound, on the other hand, is proved by first reducing the problem to a counting problem for subgroups of abelian groups and then solving that extremal counting problem. We will, in fact, show a more remarkable result: the answer is independent of O! Theorem 3. Let k be a number field with Galois group g = Gal(k/Q) and with ring of integers O. Let S be a finite set of primes, and OS as above. Assume GRH (generalized Riemann hypothesis) for k and all cyclotomic extensions k(ζ ) with a rational prime and ζ a primitive th root of unity. Then α+(SL2(OS)) = α−(SL2(OS)) = 3 − 2 √ 2 4 . The GRH is needed only for establishing the lower bound. It can be dropped in many cases by appealing to a theorem of Murty and Murty [MM] which generalizes the Bombieri– Vinogradov Theorem cited earlier. COUNTING CONGRUENCE SUBGROUPS 3 Theorem 4. Theorem 3 can be proved unconditionally for k if either (a) g = Gal(k/Q) has an abelian subgroup of index at most 4 (this is true, for example, if k is an abelian extension); (b) d = deg[k : Q] < 42. We conjecture that for every Chevalley group scheme G, the upper and lower limiting constants, α±(G(OS)), depend only on G and not on O. In fact, we have a precise conjecture, for which we need to introduce some additional notation. Let G be a Chevalley group scheme of dimension d = dim(G) and rank = rk(G). Let κ = |Φ+| denote the number of positive roots in the root system of G. Letting R = R(G) = d− 2 = κ , we see that R = +1 2 , (resp. , , −1, 3, 6, 6, 9, 15) if G is of type A (resp. B , C , D , G2, F4, E6, E7, E8). Conjecture. Let k,O, and S be as in Theorem 3, and suppose that G is a simple Chevalley group scheme. Then α+(G(OS)) = α−(G(OS)) = (√ R(R + 1) −R )2 4R2 . The conjecture reflects the belief that “most” subgroups of H = G(Z/mZ) lie between the Borel subgroup B of H and the unipotent radical of B. Our proof covers the case of SL2 and we are quite convinced that this will hold in general. For general G, we do not have such an in depth knowledge of the subgroups of G(Fq) as we do for G = SL2, yet we can still prove: Theorem 5. Let k,O, and S be as in Theorem 3. Let G be a simple Chevalley group scheme of dimension d and rank , and R = R(G) = d− 2 , then: (a) Assuming GRH or the assumptions of Theorem 4; α−(G(OS)) ≥ (√ R(R + 1) −R )2

The endomorphism rings of jacobians of cyclic covers of the projective line

Abstract: Suppose that $K$ is a field of characteristic zero, $K_a$ is its algebraic closure, and that $f(x) \in K[x]$ is an irreducible polynomial of degree $n \ge 5$ , whose Galois group coincides either with the full symmetric group $\Sn$ or with the alternating group $\An$ . Let $p$ be an odd prime, $\Z[\zeta_p]$ the ring of integers in the $p$ th cyclotomic field $\Q(\zeta_p)$ . Suppose that $C$ is the smooth projective model of the affine curve $y^p\,{=}\,f(x)$ and $J(C)$ is the jacobian of $C$ . We prove that the ring $\End(J(C))$ of $K_a$ -endomorphisms of $J(C)$ is canonically isomorphic to $\Z[\zeta_p]$

Z[¿14] is Euclidean

Abstract: We provide the first unconditional proof that the ring Z( √ 14) is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of Q. It is proved that if K is a real quadratic field (modulo the existence of two special primes of K) or if K is a cyclotomic extension of Q then: the ring of integers of K is a Euclidean domain if and only if it is a principal ideal domain. The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when K is a totally real extension of degree at least three. The main changes are a new Motzkin- type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property.

ℤ[] is Euclidean

Abstract: We provide the first unconditional proof that the ring is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of . It is proved that if is a real quadratic field (modulo the existence of two special primes of ) or if is a cyclotomic extension of then: the ring of integers of is a Euclidean domain if and only if it is a principal ideal domain. The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property.

On some geometric representations of GL(n,O)

Abstract: We study a family of complex representations of the group GL(n,O), where O is the ring of integers of a non-archimedean local field F. These representations occur in the restriction of the Grassmann representation of GL(n,F) to its maximal compact subgroup GL(n,O). We compute explicitly the transition matrix between a geometric basis of the Hecke algebra associated with the representation and an algebraic basis which consists of its minimal idempotents. The transition matrix involves combinatorial invariants of lattices of submodules of finite O-modules. The idempotents are p-adic analogs of the multivariable Jacobi polynomials.

A construction of quintic rings

Abstract: We construct a discriminant-preserving map from the set of orbits in the space of quadruples of quinary alternating forms over the integers to the set of isomorphism classes of quintic rings. This map may be regarded as an analogue of the famous map from the set of equivalence classes of integral binary cubic forms to the set of isomorphism classes of cubic rings and may be expected to have similar applications. We show that the ring of integers of every quintic number field lies in the image of the map. These results have been used to establish an upper bound on the number of quintic number fields with bounded discriminant.

Computing weight 2 modular forms of level p 2

Abstract: For a prime p we describe an algorithm for computing the Brandt matrices giving the action of the Hecke operators on the space V of modular forms of weight 2 and level p 2 . For p ≡ 3 mod 4 we define a special Hecke stable subspace Vo of V which contains the space of modular forms with CM by the ring of integers of Q(√-p) and we describe the calculation of the corresponding Brandt matrices.

Binary GCD Like Algorithms for Some Complex Quadratic Rings

Abstract: On the lines of the binary gcd algorithm for rational integers, algorithms for computing the gcd are presented for the ring of integers in \(\mathbb{Q}(\sqrt{d})\) where d ∈ { − 2, − 7, − 11, − 19}. Thus a binary gcd like algorithm is presented for a unique factorization domain which is not Euclidean (case d=-19). Together with the earlier known binary gcd like algorithms for the ring of integers in \(\mathbb{Q}(\sqrt{-1})\) and \(\mathbb{Q}(\sqrt{-3})\), one now has binary gcd like algorithms for all complex quadratic Euclidean domains. The running time of our algorithms is O(n 2) in each ring. While there exists an O(n 2) algorithm for computing the gcd in quadratic number rings by Erich Kaltofen and Heinrich Rolletschek, it has large constants hidden under the big-oh notation and it is not practical for medium sized inputs. On the other hand our algorithms are quite fast and very simple to implement.

Galois cohomology of Witt vectors of algebraic integers

Abstract: Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p\gt 0$ . Let $L/K$ be a finite Galois extension with Galois group $G\,{=}\,G_{L/K}$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. By the normal basis theorem, $L$ is always a projective $K[G]$ -module. But the ring of integers $\mathcal{O}_L$ is a projective $\mathcal{O}_K[G]$ -module if and only $L/K$ is tamely ramified. For wildly ramified extensions, the structure of $\mathcal{O}_L$ as an $\mathcal{O}_K[G]$ -module is very complicated and quite far from understood. We propose that the ring of Witt vectors $W\s(\mathcal{O}_L)$ is a more well-behaved object. Indeed, we show that for a large class of extensions $L/K$ , the pro-abelian group $H^1(G,W\s(\mathcal{O}_L))$ is zero. We conjecture that this is true in general.

The probability that a random monic p-adic polynomial splits into linear factors

Abstract: Let n be a positive integer and let p be a prime. We calculate the probability that a random monic polynomial of degree n with coefficients in the ring Z_p of p-adic integers splits over Z_p into linear factors.

On the representation of integers as linear combinations of consecutive values of a polynomial

Abstract: Let K be a cyclotomic field with ring of integers O K and let f be a polynomial whose values on *Z belong to O K . If the ideal of O K generated by the values of f on Z is O K itself, then every algebraic integer N of K may be written in the following form: N = Σe k f(k) for some integer l, where the e k 's are roots of unity of K. Moreover, there are two effective constants A and B such that the least integer l (for a fixed N) is less than A N + B, where Formula math.

A class of exponential congruences in several variables

Abstract: A problem raised by Selfridge and solved by Pomerance asks to flnd the pairs (a;b) of natural numbers for which 2 a i 2 b divides n a i n b for all integers n. Vajaitu and one of the authors have obtained a generalization which concerns elements fi1;:::;fik and fl in the ring of integers A of a number fleld for which k X i=1 fiifl a i divides k X i=1 fiiz a i for any z 2 A: Here we obtain a further generalization, proving the corresponding flniteness results in a multidimensional setting.

Lubin–Tate Formal Groups over the Ring of Integers of a Multidimensional Local Field

Abstract: Some isomorphism criteria for Lubin–Tate formal groups over the ring of integers of a multidimensional local field are given. Bibliography: 4 titles.

Kernel Groups and Nontrivial Galois Module Structure of Imaginary Quadratic Fields

Abstract: Let $K$ be an algebraic number field with ring of integers $\Cal{O}_{K}$, $p>2$ be a rational prime and $G$ be the cyclic group of order $p $. Let $\Lambda$ denote the order $\Cal{O}_{K}[G].$ Let $Cl(\Lambda)$ denote the locally free class group of $\Lambda$ and $D(\Lambda)$ the kernel group, the subgroup of $Cl(\Lambda)$ consisting of classes that become trivial upon extension of scalars to the maximal order. If $p$ is unramified in $K$, then $D(\Lambda) = T(\Lambda)$, where $T(\Lambda)$ is the Swan subgroup of $Cl(\Lambda).$ This yields upper and lower bounds for $D(\Lambda)$. Let $R(\Lambda)$ denote the subgroup of $Cl(\Lambda)$ consisting of those classes realizable as rings of integers, $\Cal{O}_{L},$ where $L/K$ is a tame Galois extension with Galois group $Gal(L/K) \cong G.$ We show under the hypotheses above that $T(\Lambda)^{(p-1)/2} \subseteq R(\Lambda) \cap D(\Lambda) \subseteq T(\Lambda)$, which yields conditions for when $T(\Lambda)=R(\Lambda) \cap D(\Lambda)$ and bounds on $R(\Lambda) \cap D(\Lambda)$. We carry out the computation for $K=\Bbb{Q}(\sqrt{-d}), d>0, d eq 1$ or $3.$ In this way we exhibit primes $p$ for which these fields have tame Galois field extensions of degree $p$ with nontrivial Galois module structure.

P-adic Fields

Abstract: In this chapter we shall consider fields which are completions of algebraic number fields under discrete valuations. According to Theorem 3.1 every valuation gives rise to a complete field, uniquely determined up to a topological isomorphism. By Theorem 3.3 every discrete valuation v of an algebraic number field K is induced by a prime ideal Þ of its ring of integers. The completion of K under v will be denoted by Kp or K v and called the p-adic field. In the case of K = ℚ we shall not distinguish between the prime p and the prime ideal generated by it, and we shall write ℚp for the field which is the completion of ℚ under the valuation induced by pℤ. The field ℚ p is called the p-adic field.

Semiconjugacy of Quasiperiodic Flows and Finite Index Subgroups of Multiplier Groups

Abstract: It will be shown that if $\phi$ is a quasiperiodic flow on the $n$-torus that is algebraic, if $\psi$ is a flow on the $n$-torus that is smoothly conjugate to a flow generated by a constant vector field, and if $\phi$ is smoothly semiconjugate to $\psi$, then $\psi$ is a quasiperiodic flow that is algebraic, and the multiplier group of $\psi$ is a finite index subgroup of the multiplier group of $\phi$. This will partially establish a conjecture that asserts that a quasiperiodic flow on the $n$-torus is algebraic if and only if its multiplier group is a finite index subgroup of the group of units of the ring of integers in a real algebraic number field of degree $n$.

Automorphisms of Sylow p-Subgroups of Chevalley Groups Defined over Residue Rings of Integers

Abstract: We deal with automorphisms of Sylow p-subgroups SΦ(Zp m) of Chevalley groups of normal types Φ, defined over residue rings Zp m of integers modulo p m, where m≥2 and >3 is a prime. It is shown that in this case all automorphisms of SΦ“Zp m” factor into a product of inner, diagonal, graph, central automorphisms and some explicitly specified automorphism of order p. The results obtained give the answer (under the condition that p>3) to Question 12.42 posed by Levchyuk in [4], which called for furnishing a description of automorphisms of a Sylow p-subgroup of a normal type Chevalley group over a residue ring of integers modulo pm, where m≥2 and p is a prime.

Units in families of totally complex algebraic number fields

Abstract: Multidimensional continued fraction algorithms associated with G L n ( ℤ k ) , where ℤ k is the ring of integers of an imaginary quadratic field K , are introduced and applied to find systems of fundamental units in families of totally complex algebraic number fields of degrees four, six, and eight.