P-adic Valuation Research Articles

SQUARES IN (22 - 1)…(n2 - 1) AND p-ADIC VALUATION

In this paper, we determine all the squares in the sequence [Formula: see text]. From this, one deduces that there are infinitely many squares in this sequence. We also give a formula for the p-adic valuation of the terms in this sequence.

On p-Adic Properties of Central L-Values of Quadratic Twists of an Elliptic Curve

AbstractWe study p-indivisibility of the central values L(1, Ed) of quadratic twists Ed of a semi-stable elliptic curve E of conductor N. A consideration of the conjecture of Birch and Swinnerton-Dyer shows that the set of quadratic discriminants d splits naturally into several families ℱS, indexed by subsets S of the primes dividing N. Let δS = gcdd∈ℱSL(1, Ed)alg, where L(1, Ed)alg denotes the algebraic part of the central L-value, L(1, Ed). Our main theorem relates the p-adic valuations of δS as S varies. As a consequence we present an application to a refined version of a question of Kolyvagin. Finally we explain an intriguing (albeit speculative) relation betweenWaldspurger packets on and congruences of modular forms of integral and half-integral weight. In this context, we formulate a conjecture on congruences of half-integral weight forms and explain its relevance to the problem of p-indivisibility of L-values of quadratic twists.

P-orderings of finite subsets of Dedekind domains

If R is a Dedekind domain, P a prime ideal of R and S?R a finite subset then a P-ordering of S, as introduced by M. Bhargava in (J. Reine Angew. Math. 490:101---127, 1997), is an ordering {a i } i=1 m of the elements of S with the property that, for each 1<i?m, the choice of a i minimizes the P-adic valuation of ? j<i (s?a j ) over elements s?S. If S, S ? are two finite subsets of R of the same cardinality then a bijection ?:S?S ? is a P-ordering equivalence if it preserves P-orderings. In this paper we give upper and lower bounds for the number of distinct P-orderings a finite set can have in terms of its cardinality and give an upper bound on the number of P-ordering equivalence classes of a given cardinality.

Taibleson operators,p-adic parabolic equations and ultrametric diffusion

Taibleson operators,<i>p</i>-adic parabolic equations and ultrametric diffusion

Asymptotic valuations of sequences satisfying first order recurrences

Let t n be a sequence that satisfies a first order homogeneous recurrence t n = Q(n)t n-1 , where Q is a polynomial with integer coefficients. We describe the asymptotic behavior of the p-adic valuation of t n .

On irrationality measures of [formula omitted

We prove that the number τ = ∑ l = 0 ∞ d l / ∏ j = 1 l ( 1 + d j r + d 2 j s ) , where d ∈ Z , | d | > 1 , and r , s ∈ Q , s ≠ 0 , are such that 1 + d j r + d 2 j s ≠ 0 for any j ∈ Z + , has an irrationality measure 7/3 or 7/2 depending on whether r = − d − h − s d h for some h ∈ N or r 2 ⩽ 4 s . More generally, irrationality measures are given for τ in both the archimedean and p-adic valuations, and also when d , r , s are certain algebraic numbers. For example, we give an effective irrationality measure 7/3 for B d ( d ) , where B q ( z ) is a q-analogue of the Bessel function, and we get effective irrationality measures 7/3 and 7/2 for the p-adic numbers τ p + and τ p − , respectively, where τ p ± = ∑ l = 0 ∞ p l 2 / ∏ j = 1 l ( 1 ± p j ) 2 .

Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X0(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X0(N), if N=pM with p prime, p∤M and the genus of X0(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.

P-ADIC ORDER BOUNDED GROUP VALUATIONS ON ABELIAN GROUPS

AbstractFor a fixed integer e and prime p we construct the p-adic order bounded group valuations for a given abelian group G. These valuations give Hopf orders inside the group ring KG where K is an extension of $\mathbb{Q} _{p}$ with ramification index e. The orders are given explicitly when G is a p-group of order p or p2. An example is given when G is not abelian.

P-adic valuations and k-regular sequences

A sequence is said to be k- automatic if the n th term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial f ( x ) ∈ Q p [ x ] , we consider the sequence { v p ( f ( n ) ) } n = 0 ∞ , where v p is the p-adic valuation. We show that this sequence is p-regular if and only if f ( x ) factors into a product of polynomials, one of which has no roots in Z p , the other which factors into linear polynomials over Q . This answers a question of Allouche and Shallit.

P-Adic estimates of hamming weights in abelian codes over galois rings

A generalization of McEliece's theorem on the p-adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with a technique known as trace-averaging introduced here. The original theorem of McEliece concerned cyclic codes over prime fields. Delsarte and McEliece later extended this to Abelian codes over finite fields. Calderbank, Li, and Poonen extended McEliece's original theorem to cover cyclic codes over the rings Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> , Wilson strengthened their results and extended them to cyclic codes over Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> , and Katz strengthened Wilson's results and extended them to Abelian codes over Zopf <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup> . It is natural to ask whether there is a single analogue of McEliece's theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers. In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more

Trace funcions and Galois invariant $p$-adic measures

Let p be a prime number, Qp the field of p-adic numbers, Qp a fixed algebraic closure of Qp, and Cp the completion of Qp with respect to the p-adic valuation. We study trace functions associated to p-adic measures defined on compact subsets of Cp which are invariant under the action of the Galois group G = Galcont(Cp/Qp).

The Ramanujan differential operator, a certain CM elliptic curve and Kummer congruences

Let τ be a point in the upper half-plane such that the elliptic curve corresponding to τ can be defined over Q ,a nd letf be a modular form on the full modular group with rational Fourier coefficients. By applying the Ramanujan differential operator D to f ,w e obtain a family of modular forms D l f . In this paper we study the behavior of D l (f )(τ ) modulo the powers of a prime p> 3. We show that for p ≡ 1 mod 3 the quantities D l (f )(τ ), suitably normalized, satisfy Kummer-type congruences, and that for p ≡ 2m od 3 the p-adic valuations of D l (f )(τ ) grow arbitrarily large. We prove these congruences by making a connection with a certain elliptic curve whose reduction modulo p is ordinary if p ≡ 1 mod 3 and supersingular otherwise.

Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups

Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups

A relation between p-adic L-functions and the Tamagawa number conjecture for Hecke characters

We prove that the submodule in K-theory which gives the exact value % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaiikaiaabw % hacaqGWbGaaeiiaiaabshacaqGVbGaaeiiamrr1ngBPrwtHrhAYaqe % guuDJXwAKbstHrhAGq1DVbacfaGae8hjHO1aa0baaSqaaiaacIcaca % WGWbGaaiykaaqaaiaacQcaaaGccaGGPaaaaa!4A5B! $$({\text{up to }}\mathbb{Z}_{(p)}^* )$$ of the L-function by the Beilinson regulator map at non-critical values for Hecke characters of imaginary quadratic fields K with cl (K) = 1(p-local Tamagawa number conjecture) satisfies that the length of its coimage under the local Soule regulator map is the p-adic valuation of certain special values of p-adic L-functions associated to the Hecke characters. This result yields immediately, up to Jannsen’s conjecture, an upper bound for % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4iaiaadI % eadaqhaaWcbaGaamyzaiaadshaaeaacaaIYaaaaOGaaiikamrr1ngB % PrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae8NdX-0aaSbaaS % qaaiaadUeaaeqaaOGaai4waiaaigdacaGGVaGaam4uaiaac2facaGG % SaGaaGjbVlaadAfadaWgaaWcbaGaamiCaaqabaGccaGGOaGaamyBai % aacMcacaGGPaaaaa!5288! $$\# H_{et}^2 (\mathcal{O}_K [1/S],\;V_p (m))$$ in terms of the valuation of these p-adic L-functions, where V p denotes the p-adic realization of a Hecke motive.

On Ultrametricity, Data Coding, and Computation

The triangular inequality is a defining property of a metric space, while the stronger ultrametric inequality is a defining property of an ultrametric space. Ultrametric distance is defined from p-adic valuation. It is known that ultrametricity is a natural property of spaces in the sparse limit. The implications of this are discussed in this article. Experimental results are presented which quantify how ultrametric a given metric space is. We explore the practical meaningfulness of this property of a space being ultrametric. In particular, we examine the computational implications of widely prevalent and perhaps ubiquitous ultrametricity.

INTEGRAL BASES OVER p-ADIC FIELDS

Let p be a prime number, <TEX>$Q_{p}$</TEX> the field of p-adic numbers, K a finite extension of <TEX>$Q_{p}$</TEX>, <TEX>$\bar{K}}$</TEX> a fixed algebraic closure of K and <TEX>$C_{p}$</TEX> the completion of K with respect to the p-adic valuation. Let E be a closed subfield of <TEX>$C_{p}$</TEX>, containing K. Given elements <TEX>$t_1$</TEX>...,<TEX>$t_{r}$</TEX> <TEX>$\in$</TEX> E for which the field K(<TEX>$t_1$</TEX>...,<TEX>$t_{r}$</TEX>) is dense in E, we construct integral bases of E over K.

Existence of certain fundamental discriminants and class numbers of real quadratic fields

Let D be the fundamental discriminant of the quadratic field Q( √ D), h(D) its class number, and χD := ( D · ) the usual Kronecker character. Let p be prime, Zp the ring of p-adic integers, and λp(Q( √ D)) the Iwasawa λ-invariant of the cyclotomic Zp-extension of Q( √ D). Let Rp(D) denote the p-adic regulator of Q( √ D), and | · |p denote the usual multiplicative p-adic valuation normalized so that |p|p = 1 p . In [9], by applying Sturm’s theorem on the congruence of modular forms to Cohen’s half integral weight modular forms, Ono proved the following theorem.

P-Adic interpolation and approximation of a continuous function by linear combinations of shifts of p-adic valuations

In this work, questions about interpolation and approximation of a continuous function f: Z p→ R (f: Z p→ Q p) by functions of the form ∑ k=1 N λ k | x− x k | p are discussed. The theorem about uniform approximation of a continuous function f: Z p→ R is proved. Nonexistence of such approximation for a Q p -valued function is shown.

DISTRIBUTION OF BINOMIAL COEFFICIENTS AND DIGITAL FUNCTIONS

The distribution of binomial coefficients in residue classes modulo prime powers and with respect to the p-adic valuation is studied. For this purpose, general asymptotic results for arithmetic functions depending on blocks of digits with respect to q-ary expansions are established.

Quantum cyclotomic orders of 3-manifolds

This paper provides a topological interpretation for number theoretic properties of quantum invariants of 3-manifolds. In particular, it is shown that the p-adic valuation of the quantum SO(3)-invariant of a 3-manifold M, for odd primes p, is bounded below by a linear function of the mod p first betti number of M. Sharper bounds using more delicate topological invariants are given as well.