2-adic Valuation Research Articles

On thep-part of the Birch–Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of

AbstractWe study infinite families of quadratic and cubic twists of the elliptic curveE=X0(27). For the family of quadratic twists, we establish a lower bound for the 2-adic valuation of the algebraic part of the value of the complexL-series ats=1, and, for the family of cubic twists, we establish a lower bound for the 3-adic valuation of the algebraic part of the sameL-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

Sums of three squares under congruence condition modulo a prime

Let p be an odd prime. We show that the integral points on the sphere with radius n are equidistributed modulo p as n⟶∞ where n is not of the shape 4l(8m+7) and its 2-adic valuation is bounded. In particular if n is sufficiently large and if n satisfies a congruence equation α12+α22+α32≡n(modp) where p2|n if all αi≡0(modp), then there are integers xi with xi≡αi(modp)(i=1,2,3) satisfying x12+x22+x32=n. The similar result holds also in the case modulo 8.

The 2-adic valuations of differences of Stirling numbers of the second kind

Let m, n, k and c be positive integers, ν2(k) be the 2-adic valuation of k and S(n,k) be the Stirling numbers of the second kind. We show that if 2≤m≤n and c is odd, then ν2(S(c2n+1,2m−1)−S(c2n,2m−1))=n+1 except when n=m=2 and c=1, in which case ν2(S(8,3)−S(4,3))=6. This solves a conjecture of Lengyel proposed in 2009.

Divisibility by 2 of Stirling numbers of the second kind and their differences

Let n, k, a and c be positive integers and b be a nonnegative integer. Let ν2(k) and s2(k) be the 2-adic valuation of k and the sum of binary digits of k, respectively. Let S(n,k) be the Stirling number of the second kind. It is shown that ν2(S(c2n,b2n+1+a))⩾s2(a)−1, where 0<a<2n+1 and 2∤c. Furthermore, one gets that ν2(S(c2n,(c−1)2n+a))=s2(a)−1, where n⩾2, 1⩽a⩽2n and 2∤c. Finally, it is proved that if 3⩽k⩽2n and k is not a power of 2 minus 1, then ν2(S(a2n,k)−S(b2n,k))=n+ν2(a−b)−⌈log2k⌉+s2(k)+δ(k), where δ(4)=2, δ(k)=1 if k>4 is a power of 2, and δ(k)=0 otherwise. This confirms a conjecture of Lengyel raised in 2009 except when k is a power of 2 minus 1.

PROOF OF A CONJECTURE OF FARHI

We verify a conjecture of Farhi concerning the growth of a curious arithmetic function defined by 2-adic valuation. We also prove a stronger version suggested by Zhong and Tan.

On the Parity of Certain Coefficients for a $q$-Analogue of the Catalan Numbers

The 2-adic valuation (highest power of 2) dividing the well-known Catalan numbers, $C_n$, has been completely determined by Alter and Kubota and further studied combinatorially by Deutsch and Sagan. In particular, it is well known that $C_n$ is odd if and only if $n = 2^k-1$ for some $k \geq 0$. The polynomial $F_n^{ch}(321;q) = \sum_{\sigma \in Av_n(321)} q^{ch(\sigma)}$, where $Av_n(321)$ is the set of permutations in $S_n$ that avoid 321 and $ch$ is the charge statistic, is a $q$-analogue of the Catalan numbers since specializing $q=1$ gives $C_n$. We prove that the coefficient of $q^i$ in $F_{2^k-1}^{ch}(321;q)$ is even if $i \geq 1$, giving a refinement of the "if" direction of the $C_n$ parity result. Furthermore, we use a bijection between the charge statistic and the major index to prove a conjecture of Dokos, Dwyer, Johnson, Sagan and Selsor regarding powers of 2 and the major index. In addition, Sagan and Savage have recently defined a notion of $st$-Wilf equivalence for any permutation statistic $st$ and any two sets of permutations $\Pi$ and $\Pi'$. We say $\Pi$ and $\Pi'$ are $st$-Wilf equivalent if $\sum_{\sigma \in Av_n(\Pi)} q^{st(\sigma)} = \sum_{\sigma \in Av_n(\Pi')} q^{st(\sigma)}$. In this paper we show how one can characterize the charge-Wilf equivalence classes for subsets of $S_3$.

The 2-adic Valuation of Plane Partitions and Totally Symmetric Plane Partitions

This paper confirms a conjecture of Amdeberhan and Moll that the power of 2 dividing the number of plane partitions in an $n$-cube is greater than the power of 2 dividing the number of totally symmetric plane partitions in the same cube when $n$ is even, and less when $n$ is odd.

Nested Recurrence Relations with Conolly-like Solutions

A nondecreasing sequence of positive integers is $(\alpha,\beta)$-Conolly, or Conolly-like for short, if for every positive integer $m$ the number of times that $m$ occurs in the sequence is $\alpha + \beta r_m$, where $r_m$ is $1$ plus the 2-adic valuation of $m$. A recurrence relation is $(\alpha, \beta)$-Conolly if it has an $(\alpha, \beta)$-Conolly solution sequence. We discover that Conolly-like sequences often appear as solutions to nested (or meta-Fibonacci) recurrence relations of the form $A(n) = \sum_{i=1}^k A(n-s_i-\sum_{j=1}^{p_i} A(n-a_{ij}))$ with appropriate initial conditions. For any fixed integers $k$ and $p_1,p_2,\ldots, p_k$ we prove that there are only finitely many pairs $(\alpha, \beta)$ for which $A(n)$ can be $(\alpha, \beta)$-Conolly. For the case where $\alpha =0$ and $\beta =1$, we provide a bijective proof using labelled infinite trees to show that, in addition to the original Conolly recurrence, the recurrence $H(n)=H(n-H(n-2)) + H(n-3-H(n-5))$ also has the Conolly sequence as a solution. When $k=2$ and $p_1=p_2$, we construct an example of an $(\alpha,\beta)$-Conolly recursion for every possible ($\alpha,\beta)$ pair, thereby providing the first examples of nested recursions with $p_i>1$ whose solutions are completely understood. Finally, in the case where $k=2$ and $p_1=p_2$, we provide an if and only if condition for a given nested recurrence $A(n)$ to be $(\alpha,0)$-Conolly by proving a very general ceiling function identity.

NOTES ON A CURIOUS ARITHMETIC FUNCTION

Let v2(n) denote the 2-adic valuation of any positive integer n. Recently, Farhi introduced a curious arithmetic function f defined for any positive integer n by [Formula: see text]. Farhi showed that the inequality [Formula: see text] with c = 4.01055487… holds for all positive integer n and conjectured that one can replace the upper bound cn by 4n in this inequality. In this paper, we show two identities about the product [Formula: see text] and then use it to prove partially Farhi's conjecture. Finally, we propose a conjecture from which the truth of Farhi's conjecture can be deduced. In particular, we confirm the truth of our conjecture for all positive integers n up to 100000 by using Matlab 7.1.

A Binary Tree Representation for the 2-Adic Valuation of a Sequence Arising From a Rational Integral

AbstractWe analyze properties of the 2-adic valuation of an integer sequence that originates from an explicit evaluation of a quartic integral. We present a tree that encodes this valuation.

The 2-adic valuation of a sequence arising from a rational integral

We analyze properties of the 2-adic valuation of an integer sequence that originates from an explicit evaluation of a quartic integral. We also give a combinatorial interpretation of the valuations of this sequence.

Slopes of overconvergent 2-adic modular forms

We explicitly compute all the slopes of the Hecke operator U2 acting on overconvergent 2-adic level 1 cusp forms of weight 0: the nth slope is 1 + 2v((3n)!/n!), where v denotes the 2-adic valuation. We formulate an explicit conjecture about what these slopes should be for weight k forms.

The classification of $p$-divisible groups over $2$-adic discrete valuation rings

Let $\mathscr{O}_K$ be a 2-adic discrete valuation ring with perfect residue field $k$. We classify $p$-divisible groups and $p$-power order finite flat group schemes over $\mathscr{O}_K$ in terms of certain Frobenius module over $\mathfrak{S}:=W(k)[[u]]$. We also show the compatibility with crystalline Dieudonn\'e theory and associated Galois representations. Our approach differs from Lau's generalization of display theory, and we additionally obtain the the compatibility with associated Galois representations.