We establish integrality and congruence properties for the Eisenstein-Kronecker cocycle of Bergeron, Charollois and García introduced in [arXiv:2107.01992v2 [math.NT]]. As a consequence, we recover the integrality of the critical values of Hecke $L$-functions over imaginary quadratic fields in the split case. Additionally, we construct a $p$-adic measure that interpolates these critical values.
In a previous paper arXiv:2406.06294 [math.NT], the author proved the exact formulae for ranks of partitions modulo each prime $p\geq 5$. In this paper, for $p=5$ and $7$, we prove special vanishing properties of the Kloosterman sums appearing in the exact formulae. These vanishing properties imply a new proof of Dyson's rank conjectures. Specifically, we give a new proof of Ramanujan's congruences $p(5n+4)\equiv 0\pmod 5$ and $p(7n+5)\equiv 0\pmod 7$.
This addendum devotes to a detailed proof for the inequality (9.14) in our joint work: Arithmetic exponent pairs for algebraic trace functions and applications, with an appendix by Will Sawin, arXiv:1603.07060 [math.NT], which will appear in Algebra and Number Theory. We do not intend to publish this addendum in any journals; arXiv should be a good place for those reader who want to find such details. The proof involves various averages of arithmetic functions.
We generalize the main result of arXiv:1206.6631 [math.NT] to all totally real fields. In other words, for $p>2$ prime, we prove (under a mild Taylor-Wiles hypothesis) that if a modular representation is unramified and $p$-distinguished at all places above $p$, then it arises from a mod $p$ Hilbert modular form of parallel weight one. This (mostly) resolves the weight one part of Serre's conjecture for totally real fields.
In arXiv:1603.03910 [math.NT] we introduced some $C_{n}$ in $Z/2[t]$ defined by a linear recurrence and showed that each $C_{n}$, $n\equiv 0 \bmod{4}$, is a sum of $C_{k}$, $k<n$. Combining this with results from arXiv:1508.07523 [math.NT] we proved that the space $K$, consisting of those odd mod~2 modular forms of level $Γ_{0}(3)$ that are annihilated by the operator $U_{3}+I$, has a basis $m_{i,j}$ "adapted to $T_{7}$ and $T_{13}$" in the sense of Nicolas and Serre. (And so the "completed shallow Hecke algebra" attached to $K$ is a power series ring in $T_{7}$ and $T_{13}$.) This note derives analogous results in level $Γ_{0}(5)$. Now $U_{3}+I$ is replaced by $U_{5}+I$, and the operators $T_{7}$ and $T_{13}$ by $T_{3}$ and $T_{7}$. In place of level $Γ_{0}(3)$ results from 1508.07523, we use level $Γ_{0}(5)$ results from arXiv:1603.07085 [math.NT]. A linear recurrence again plays the key role. Now $C_{n+6} = C_{n+5} + (t^{6}+t^{5}+t^{2}+t)C_{n}+t^{n}(t^{2}+t)$, $C_{0}=0$, $C_{1}=C_{2}=1$, $C_{3}=t$, $C_{4}=t^{2}$, $C_{5}=t^{4}+t^{2}+t$, and we prove that each $C_{n}$, $n\equiv 0$ or $2\bmod{6}$ is a sum of $C_{k}$, $k<n$.
We investigate generalised polynomials (i.e. polynomial-like expressions involving the use of the floor function) which take the value $0$ on all integers except for a set of density $0$. Our main result is that the set of integers where a sparse generalised polynomial takes non-zero value cannot contain a translate of an IP set. We also study some explicit constructions, and show that the characteristic functions of the Fibonacci and Tribonacci numbers are given by generalised polynomails. Finally, we show that any sufficiently sparse $\{0,1\}$-valued sequence is given by a generalised polynomial. (This paper is essentially the first half of our earlier submission arXiv:1610.03900 [math.NT]. Because the material in arXiv:1610.03900 [math.NT] touches upon many different subjects, we believe it is preferable to split it into two independent papers.)
Hematopoietic stem cells (HSCs), still represent a certain mystery in biology, have a unique property of dividing into equal cells and repopulating the hematopoietic tissue. This potential enables their use in transplantation treatments. The quality of the HSC grafts for transplantation is evaluated by flow cytometric determination of the CD34+cells, which enables optimal timing of the first apheresis and the acquisition of maximal yield of the peripheral blood stem cells (PBSCs). To identify a more efficient method for evaluating CD34+cells, we compared the following alternative methods with the reference method: hematopoietic progenitor cells (HPC) enumeration (using the Sysmex XE-2100 analyser), detection of CD133+cells, and quantification of aldehyde dehydrogenase activity in the PBSCs. 266 aphereses (84 patients) were evaluated. In the preapheretic blood, the new methods produced data that were in agreement with the reference method. The ROC curves have shown that for the first-day apheresis target, the optimal predictive cut-off value was 0.032 cells/mL for the HPC method (sensitivity 73.4%, specificity 69.3%). HPC method exhibited a definite practical superiority as compared to other methods tested. HPC enumeration could serve as a supplementary method for the optimal timing of the first apheresis; it is simple, rapid, and cheap.
We continue our investigation of the distribution of the fractional parts of $a γ$, where $a$ is a fixed non-zero real number and $γ$ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. We establish some connections to pair correlation functions and the distribution of primes in short intervals. We also discuss analogous results for a more general L-function. This is a sequel to the paper math.NT/0405459.
In math.NT/9907019 we proposed an analog of the classical Riemann hypothesis for characteristic p valued L-series based on the work of Wan, Diaz-Vargas, Thakur, Poonen, and Sheats for the zeta function $ζ_{\Fr[θ]}(s)$. During the writing of math.NT/9907019, we made two assumptions that have subsequently proved to be incorrect. The first assumption is that we can ignore the trivial zeroes of characteristic p L-series in formulating our conjectures. Instead, we show here how the trivial zeroes influence nearby zeroes and so lead to counter-examples of the original Riemann hypothesis analog. We then sketch an approach to handling such ``near-trivial'' zeroes via Hensel's and Krasner's Lemmas (whereas classically one uses Gamma-factors). Moreover, we show that $ζ_{\Fr[θ]}(s)$ is not representative of general L-series as, surprisingly, all its zeroes are near-trivial, much as the Artin-Weil zeta-function of $\mathbb{P}^1/\Fr$ is not representative of general complex L-functions of curves over finite fields. Consequently, the ``critical zeroes'' (= all zeroes not effected by the trivial zeroes) of characteristic p L-series now appear to be quite mysterious. The second assumption made while writing math.NT/9907019 is that certain Taylor expansions of classical L-series of number fields would exhibit complicated behavior with respect to their zeroes. We present a simple argument that this is not so, and, at the same time, give a characterization of functional equations.
For the solution $\{u_n\}_{n=0}^\infty$ to the polynomial recursion $(n+1)^5u_{n+1}-3(2n+1)(3n^2+3n+1)(15n^2+15n+4)u_n -3n^3(3n-1)(3n+1)u_{n-1}=0$, where $n=1,2,...$, with the initial data $u_0=1$, $u_1=12$, we prove that all $u_n$ are integers. The numbers $u_n$, $n=0,1,2,...$, are denominators of rational approximations to $ζ(4)$ (see math.NT/0201024). We use Andrews's generalization of Whipple's transformation of a terminating ${}_7F_6(1)$-series and the method from math.NT/0311114.
Let $\mathbb F_2^n$ be the finite field of cardinality $2 ^{n}$. For all large $n$, any subset $A\subset \mathbb F_2^n\times \mathbb F_2 ^n$ of cardinality \begin{equation*} \abs{A} \gtrsim 4^n \log\log n (\log n) ^{-1} \end{equation*} must contain three points $ \{(x,y) ,(x+d,y) ,(x,y+d)\}$ for $x,y,d\in \mathbb F_2^n$ and $d\neq0$. Our argument is an elaboration of an argument of Shkredov \cite {math.NT/0405406}, building upon the finite field analog of Ben Green \cite {math.NT/0409420}. The interest in our result is in the exponent on $ \log n$, which is larger than has been obtained previously.
This paper is the sequel of our paper "Arithmetic height functions over finitely generated fields" (cf. math.NT/9809016). In this paper, we define the canonical height of subvarieties of an abelian variety over a finitely generated field over Q. We also prove that the canonical height of a subvariety is zero if and only if it is a translation of an abelian subvariety by a torsion point.
A general hypergeometric construction of linear forms in (odd) zeta values is presented. The construction allows to recover the records of Rhin and Viola for the irrationality measures of $ζ(2)$ and $ζ(3)$, as well as to explain Rivoal's "infinitely-many" result (math.NT/0008051) and to prove that at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$, and $ζ(11)$ is irrational.
In his previous papers (J. reine angew. Math. 544 (2002), 91--110; math.AG/0103203) the author introduced a certain explicit construction of superelliptic jacobians, whose endomorphism ring is the ring of integers in the $p$th cyclotomic field. (Here $p$ is an odd prime.) In the present paper we discuss when these jacobians are mutually non-isogenous. (The case of hyperelliptic jacobians was treated in author's e-print math.NT/0301173 .)
This article develops a new sieve method which by adding an additional axiom to the classical formulation breaks the well-known parity problem and allows one to detect primes in thin, interesting integer sequences. In the accompanying paper [math.NT/9811184] the practicality of the axiom is demonstrated by verifying it (and the other axioms) to produce primes in the sequence a^2+b^4 with the relevant asymptotics.
The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let's mention : (1) the control of the image of the Galois representation modulo $p$, (2) Hida's congruence criterion outside an explicit set of primes $p$, and (3) the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras. We study the arithmetic of the Hilbert modular forms by studying their modulo $p$ Galois representations and our main tool is the action of the inertia groups at the primes above $p$. In order to determine this action, we compute the Hodge-Tate (resp. the Fontaine-Laffaille) weights of the $p$-adic (resp. the modulo $p$) etale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine on the cohomology of the Siegel modular varieties and builds upon the geometric constructions of math.NT/0212071 and math.NT/0212072.
Let A be a complete discrete valuation ring with possibly imperfect residue field, and let $χ$ be a one-dimensional Galois representation over A. I show that the non-logarithmic variant of Kato's Swan conductor is the same for $χ$ and the pullback of $χ$ to the generic residual perfection of A. This implies the conductor from "Conductors and the moduli of residual perfection" (math.NT/0112305) extends the non-logarithmic variant of Kato's.
We compute by a purely local method the (elliptic) twisted by transpose-inverse character χ_{π_Y} of the representation π_Y=I_{(3,1)}(1_3xχ_Y) of G=GL(4,F), where F is a p-adic field, p not 2, and Y is an unramified quadratic extension of F; χ_Y is the nontrivial character of F^\x/N_{Y/F}Y^x. The representation π_Y is normalizedly induced from \pmatrix m_3&\ast 0&m_1\endpmatrix \mapstoχ_Y(m_1), m_i in GL(i,F), on the maximal parabolic subgroup of type (3,1). We show that the twisted character χ_{π_Y} of π_Y is an unstable function: its value at a twisted regular elliptic conjugacy class with norm in C_Y=``GL(2,Y)/F^x'' is minus its value at the other class within the twisted stable conjugacy class. It is zero at the classes without norm in C_Y. Moreover π_Y is the endoscopic lift of the trivial representation of C_Y. We deal only with unramified Y/F, as globally this case occurs almost everywhere. Naturally this computation plays a role in the theory of lifting of C_Y and GSp(2) to GL(4) using the trace formula. Our work extends -- to the context of nontrivial central characters -- the work of math.NT/0606262, where representations of PGL(4,F) are studied. In math.NT/0606262 a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced with Kazhdan in a 3-dimensional case is developed, and the local method of computation introduced by us in the 3-dimensional case is extended. As in math.NT/0606262 we use here the classification of twisted (stable) regular conjugacy classes in GL(4,F).
Jean-Paul Allouche, Jeffrey Shallit, Jonathan Sondow
We discuss the summation of certain series defined by counting blocks of digits in the $B$-ary expansion of an integer. For example, if $s_2(n)$ denotes the sum of the base-2 digits of $n$, we show that $\sum_{n \geq 1} s_2(n)/(2n(2n+1)) = (γ+ \log \frac{4}π)/2$. We recover this previous result of Sondow in math.NT/0508042 and provide several generalizations.
We give a relatively short proof of one of the central cases of the main theorem from the paper "The distribution of integers with a divisor in a given interval", math.NT/0401223. Namely, we determine the order of magnitude of the number of integers <=x with a divisor in (y,2y]. The lower bound uses a different argument than that in the aforementioned paper. As a corollary, we deduce the order of magnitude for the number of distinct products in an N x N multiplication table.