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This work concerns the weight of irreducible polynomials over a finite field, i. e. the number of non-zero coefficients of these polynomials. We introduce a polynomial analog of the Vinogradov's method developed by Gallagher and Vaughan, which leads to upper bounds for associated exponential sums. This allows us to study the distribution in arithmetic progressions of the weight of irreducible polynomials and to provide an asymptotic estimate (with an error term) for the number of irreducible polynomials of a given degree whose weight is close to the expected value., Ce travail concerne le poids des polynômes irreductibles sur un corps fini, c'est-à-dire le nombre de coefficients non nuls de ces polynômes. Nous introduisons un analogue polynomial de la méthode de Vinogradov développée par Gallagher et Vaughan afin de majorer les sommes d'exponentielles associées. Cela nous permet d'une part d'étudier la répartition dans les progressions arithmétiques du poids des polynômes irréductibles et d'autre part de donner une estimation asymptotique (avec terme d'erreur) du nombre de polynômes irréductibles de degré fixé ayant un poids donné proche de la valeur moyenne.

In 2008 or earlier, Michel Mend\`es France asked for an instance of a real number $x$ such that both $x$ and $1/x$ are simply normal to a given integer base $b$. We give a positive answer to this question by constructing a number $x$ such that both $x$ and its reciprocal $1/x$ are continued fraction normal as well as normal to all integer bases greater than or equal to $2$. Moreover, $x$ and $1/x$ are both computable., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1704.03622

A conjecture of Min\'a\v{c} and T\^an predicts that for any n>2, any prime p and any field k, the Massey product of n Galois cohomology classes in H^1(k,Z/pZ) must vanish if it is defined. We establish this conjecture when k is a number field., Comment: 33 pages; improved exposition, final version

Combining McShane's identity on a hyperbolic punctured torus with Schmutz's work on the Markov Uniqueness Conjecture (MUC), we find that MUC is equivalent to the identity \begin{equation} \sum_{n=1}^\infty \, \left( 3- L_n \right) \, = \, 4 - \varphi - \sqrt 2 \end{equation} where $L_n$ is the $n$th Lagrange number and $\varphi=\frac{1+\sqrt5}2$ is the golden ratio., Comment: 5 pages, 2 figures, comments welcome!

We study the local optimality of periodic point sets in $\mathbb{R}^n$ for energy minimization in the Gaussian core model, that is, for radial pair potential functions $f_c(r)=e^{-c r}$ with $c>0$. By considering suitable parameter spaces for $m$-periodic sets, we can locally rigorously analyze the energy of point sets, within the family of periodic sets having the same point density. We derive a characterization of periodic point sets being $f_c$-critical for all $c$ in terms of weighted spherical $2$-designs contained in the set. Especially for $2$-periodic sets like the family $\mathsf{D}^+_n$ we obtain expressions for the hessian of the energy function, allowing to certify $f_c$-optimality in certain cases. For odd integers $n\geq 9$ we can hereby in particular show that $\mathsf{D}^+_n$ is locally $f_c$-optimal among periodic sets for all sufficiently large~$c$., 27 pages, 2 figures

In this note, we prove a general identity between a $q$-multisum $B_N(q)$ and a sum of $N^2$ products of quotients of theta functions. The $q$-multisum $B_N(q)$ recently arose in the computation of a probability involving modules over finite chain rings., Comment: 7 pages, to appear in the Electronic Journal of Combinatorics

Given a hyperelliptic curve $C$ of genus $g$ over a number field $K$ and a Weierstrass model $\mathscr{C}$ of $C$ over the ring of integers ${\mathcal O}_K$ (i.e. the hyperelliptic involution of $C$ extends to $\mathscr{C}$ and the quotient is a smooth model of ${\mathbb P}^1_K$ over ${\mathcal O}_K$), we give necessary and sometimes sufficient conditions for $\mathscr{C}$ to be defined by a global Weierstrass equation. In particular, if $C$ has everywhere good reduction, we prove that it is defined by a global Weierstrass equation with invertible discriminant if the class number $h_K$ is prime to $2(2g+1)$, confirming a conjecture of M. Sadek., Minor typos. To appear in Trans. AMS

For a primitive Dirichlet character χ \chi modulo q q , we define M ( χ ) = max t | ∑ n ≤ t χ ( n ) | M(\chi )=\max _{t } |\sum _{n \leq t} \chi (n)| . In this paper, we study this quantity for characters of a fixed odd order g ≥ 3 g\geq 3 . Our main result provides a further improvement of the classical Pólya-Vinogradov inequality in this case. More specifically, we show that for any such character χ \chi we have M ( χ ) ≪ ε q ( log ⁡ q ) 1 − δ g ( log ⁡ log ⁡ q ) − 1 / 4 + ε , \begin{equation*} M(\chi )\ll _{\varepsilon } \sqrt {q}(\log q)^{1-\delta _g}(\log \log q)^{-1/4+\varepsilon }, \end{equation*} where δ g ≔ 1 − g π sin ⁡ ( π / g ) \delta _g ≔1-\frac {g}{\pi }\sin (\pi /g) . This improves upon the works of Granville and Soundararajan [J. Amer. Math. Soc. 20 (2007), pp. 357–384] and of Goldmakher [Algebra Number Theory 6 (2012), pp. 123–163]. Furthermore, assuming the Generalized Riemann Hypothesis (GRH) we prove that M ( χ ) ≪ q ( log 2 ⁡ q ) 1 − δ g ( log 3 ⁡ q ) − 1 4 ( log 4 ⁡ q ) O ( 1 ) , \begin{equation*} M(\chi ) \ll \sqrt {q} \left (\log _2 q\right )^{1-\delta _g} \left (\log _3 q\right )^{-\frac {1}{4}}\left (\log _4 q\right )^{O(1)}, \end{equation*} where log j \log _j is the j j -th iterated logarithm. We also show unconditionally that this bound is best possible (up to a power of log 4 ⁡ q \log _4 q ). One of the key ingredients in the proof of the upper bounds is a new Halász-type inequality for logarithmic mean values of completely multiplicative functions, which might be of independent interest.

A power series being given as the solution of a linear differential equation with appropriate initial conditions, minimization consists in finding a non-trivial linear differential equation of minimal order having this power series as a solution. This problem exists in both homogeneous and inhomogeneous variants; it is distinct from, but related to, the classical problem of factorization of differential operators. Recently, minimization has found applications in Transcendental Number Theory, more specifically in the computation of non-zero algebraic points where Siegel's $E$-functions take algebraic values. We present algorithms and implementations for these questions, and discuss examples and experiments., Comment: 48 pages

A classical problem due to Abel is to determine if a differential equation $y'=\eta y$ admits a non-trivial solution $y$ algebraic over $\mathbb C(x)$ when $\eta$ is a given algebraic function over $\mathbb C(x)$. Risch designed an algorithm that, given $\eta$, determines whether there exists an algebraic solution or not. In this paper, we adopt a different point of view when $\eta$ admits a Puiseux expansion with rational coefficients at some point in $\mathbb C\cup \{\infty\}$, which can be assumed to be 0 without loss of generality. We prove the following arithmetic characterization: there exists a non-trivial algebraic solution of $y'=\eta y$ if and only if the coefficients of the Puiseux expansion of $x\eta(x)$ at $0$ satisfy Gauss congruences for almost all prime numbers. We then apply our criterion to hypergeometric series: we completely determine the equations $y'=\eta y$ with an algebraic solution when $x\eta(x)$ is an algebraic hypergeometric series with rational parameters, and this enables us to prove a prediction Golyshev made using the theory of motives. We also present three other applications, in particular to diagonals of rational fractions and to directed two-dimensional walks., Comment: In this version we correct some important typos

The paper is a continuation of another paper (https://philpapers.org/rec/PENFLT-2) published as Part I. Now, the case of “n=3” is inferred as a corollary from the Kochen and Specker theorem (1967): the eventual solutions of Fermat’s equation for “n=3” would correspond to an admissible disjunctive division of qubit into two absolutely independent parts therefore versus the contextuality of any qubit, implied by the Kochen – Specker theorem. Incommensurability (implied by the absence of hidden variables) is considered as dual to quantum contextuality. The relevant mathematical structure is Hilbert arithmetic in a wide sense (https://dx.doi.org/10.2139/ssrn.3656179), in the framework of which Hilbert arithmetic in a narrow sense and the qubit Hilbert space are dual to each other. A few cases involving set theory are possible: (1) only within the case “n=3” and implicitly, within any next level of “n” in Fermat’s equation; (2) the identification of the case “n=3” and the general case utilizing the axiom of choice rather than the axiom of induction. If the former is the case, the application of set theory and arithmetic can remain disjunctively divided: set theory, “locally”, within any level; and arithmetic, “globally”, to all levels. If the latter is the case, the proof is thoroughly within set theory. Thus, the relevance of Yablo’s paradox to the statement of Fermat’s last theorem is avoided in both cases. The idea of “arithmetic mechanics” is sketched: it might deduce the basic physical dimensions of mechanics (mass, time, distance) from the axioms of arithmetic after a relevant generalization, Furthermore, a future Part III of the paper is suggested: FLT by mediation of Hilbert arithmetic in a wide sense can be considered as another expression of Gleason’s theorem in quantum mechanics: the exclusions about (n = 1, 2) in both theorems as well as the validity for all the rest values of “n” can be unified after the theory of quantum information. The availability (respectively, non-availability) of solutions of Fermat’s equation can be proved as equivalent to the non-availability (respectively, availability) of a single probabilistic measure as to Gleason’s theorem.

We study the harmonically weighted one-level density of low-lying zeros of $L$-functions in the family of holomorpic newforms of fixed even weight $k$ and prime level $N$ tending to infinity. For this family, Iwaniec, Luo and Sarnak proved that the Katz--Sarnak prediction for the one-level density holds unconditionally when the support of the Fourier transform of the implied test function is contained in $(-\tfrac32,\tfrac32)$. In this paper, we extend this admissible support to $(-\Theta_k,\Theta_k)$, where $\Theta_2 = 1.866\dots$ and $\Theta_k$ tends monotonically to $2$ as $k$ tends to infinity. This is asymptotically as good as the best known GRH result. The main novelty in our analysis is the use of zero-density estimates for Dirichlet $L$-functions., Comment: 12 pages

In algebraic geometry, theorems of K\"uronya and Lozovanu characterize the ampleness and the nefness of a Cartier divisor on a projective variety in terms of the shapes of its associated Okounkov bodies. We prove the analogous result in the context of Arakelov geometry, showing that the arithmetic ampleness and nefness of an adelic $\mathbb{R}$-Cartier divisor $\overline{D}$ are determined by arithmetic Okounkov bodies in the sense of Boucksom and Chen. Our main results generalize to arbitrary projective varieties criteria for the positivity of toric metrized $\mathbb{R}$-divisors on toric varieties established by Burgos Gil, Moriwaki, Philippon and Sombra. As an application, we show that the absolute minimum of $\overline{D}$ coincides with the infimum of the Boucksom--Chen concave transform, and we prove a converse to the arithmetic Hilbert-Samuel theorem under mild positivity assumptions. We also establish new criteria for the existence of generic nets of small points and subvarieties., Comment: 30 pages

Given a K3 surface $X$ over a number field $K$ with potentially good reduction everywhere, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline{K}}$ has infinitely many rational curves or $X$ has infinitely many unirational specializations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field $K$ with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of $K$., Final version. To appear in Forum of Mathematics, Pi

We give a formula for the cyclotomic valuation of $q$-Pochhammer symbols in terms of (generalized) Dwork maps. We also obtain a criterion for the $q$-integrality of basic hypergeometric series in terms of certain step functions, which generalize Christol step functions. This provides suitable $q$-analogs of two results proved by Christol: a formula for the $p$-adic valuation of Pochhammer symbols and a criterion for the $N$-integrality of hypergeometric series.

International audience; Let D be the open unit disk $|x| < R$ of a complete ultrametric algebraically closed field $K$. We define the growth order $ρ(f)$, the growth type $σ(f)$ and the cotype $ψ(f)$ of an analytic function in $D$ and we show that, denoting by $ q(f, r)$ the number of zeros of f in the disk $|x| ≤ r$ and putting $|f |(r) = sup |x|≤r |f (x)|$, the infimum $θ(f)$ of the s such that$ lim r→R − q(f, r)(R − r) s = 0 $ satisfies $θ(f) − 1 ≤ ρ(f) ≤ θ(f) $ and the infimum of the $s$ such that $ lim r→R − log(|f |(r))(R − r) s = 0 $ is equal to $ρ(f)$. Moreover, if $ ψ(f) < +∞$, then $\theta(f)=\rho(f)$ and $\sigma(f)=0$. In zero residue characteristic, then $\rho(f')=\rho(f), \\sigma(f')=\sigma(f),$ $ \psi(f')=\psi(f)$. Suppose $\K$ has zero characteristic.Consider two unbounded analytic functions $f, \ g$ in $D$. If $\rho(f)\neq \rho(g)$, then $\dsp{f\over g}$ has at most two perfectly branched values and if $\rho(f)=\rho(g)$ but $\sigma(f)\neq \sigma(g)$, then $\dsp{f\over g}$ has at most three perfectly branched values; moreover, if $2\sigma(g) < \sigma(f)$, then $\dsp{f\over g}$ has at most two perfectly branched values.

Robin criterion states that the Riemann Hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. Let $q_{1} = 2, q_{2} = 3, \ldots, q_{m}$ denote the first $m$ consecutive primes, then an integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$ with $a_{1} \geq a_{2} \geq \cdots \geq a_{m} \geq 0$ is called an Hardy-Ramanujan integer. If the Riemann Hypothesis is false, then there are infinitely many Hardy-Ramanujan integers $n > 5040$ such that Robin inequality does not hold and $n < (4.48311)^{m} \times N_{m}$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$.

Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues--Fontaine curve. In particular, we define a category of $\ell$-adic sheaves on the stack $\mathrm{Bun}_G$ of $G$-bundles on the Fargues--Fontaine curve, prove a geometric Satake equivalence over the Fargues--Fontaine curve, and study the stack of $L$-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define $L$-parameters associated with irreducible smooth representations of $G(E)$, a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of $L$-parameters on the category of $\ell$-adic sheaves on $\mathrm{Bun}_G$., Comment: 350 pages, v2: very minor corrections, submitted version. Comments welcome!

Motivated by questions of Fouvry and Rudnick on the distribution of Gaussian primes, we develop a very general setting in which one can study inequities in the distribution of analogues of primes through analytic properties of infinitely many $L$-functions. In particular, we give a heuristic argument for the following claim : for more than half of the prime numbers that can be written as a sum of two square, the odd square is the square of a positive integer congruent to $1 \bmod 4$., Comment: 25 pages, 4 figures

The Riemann hypothesis has been considered to be the most important unsolved problem in pure mathematics. The David Hilbert's list of 23 unsolved problems contains the Riemann hypothesis. Besides, it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis could be true.

This paper is concerned with trinomials of the form $z^{n}+az^{m}+b$, where $0, Comment: submitted version, comments very welcome!

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. The Riemann hypothesis belongs to the David Hilbert's list of 23 unsolved problems and it is one of the Clay Mathematics Institute's Millennium Prize Problems. The Robin criterion states that the Riemann hypothesis is true if and only if the inequality $\sigma(n)< e^{\gamma } \times n \times \log \log n$ holds for all natural numbers $n> 5040$, where $\sigma(x)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Nicolas criterion states that the Riemann hypothesis is true if and only if the inequality $\prod_{q \leq q_{n}} \frac{q}{q-1} > e^{\gamma} \times \log\theta(q_{n})$ is satisfied for all primes $q_{n}> 2$, where $\theta(x)$ is the Chebyshev function. Using both inequalities, we show that the Riemann hypothesis is most likely true.

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. Many consider it to be the most important unsolved problem in pure mathematics. The Robin's inequality consists in $\sigma(n) < e^{\gamma } \times n \times \ln \ln n$ where $\sigma(n)$ is the divisor function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. The Robin's inequality is true for every natural number $n > 5040$ if and only if the Riemann hypothesis is true. Given a natural number $n = q_{1}^{a_{1}} \times q_{2}^{a_{2}} \times \cdots \times q_{m}^{a_{m}}$ such that $n > 5040$, $q_{1}, q_{2}, \cdots, q_{m}$ are prime numbers and $a_{1}, a_{2}, \cdots, a_{m}$ are positive integers, then the Robin's inequality is true for $n$ when $q_{1}^{\alpha} \times q_{2}^{\alpha} \times \cdots \times q_{m}^{\alpha} \leq n$, where $\alpha = (\ln n')^{\beta}$, $\beta = (\frac{\pi^{2}}{6} - 1)$ and $n'$ is the squarefree kernel of $n$.

In mathematics, the Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all sufficiently large $n$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. In 1984, Guy Robin proved that the inequality is true for all $n > 5040$ if and only if the Riemann Hypothesis is true. In 2002, Lagarias proved that if the inequality $\sigma(n) \leq H_{n} + exp(H_{n}) \times \log H_{n}$ holds for all $n \geq 1$, then the Riemann Hypothesis is true, where $H_{n}$ is the $n^{th}$ harmonic number. We show certain properties of these both inequalities that leave us to a verified proof of the Riemann Hypothesis. These results are supported by the claim that a numerical computer calculation verifies that the subtraction of \[\log (e^{\gamma} \times q_{m} \times r) + e^{\gamma} \times q_{m} \times r \times \log \log (e^{\gamma} \times q_{m} \times r)\] with \[(q_{m} + 1) \times \log (e^{\gamma} \times (r + 1)) + (q_{m} + 1) \times e^{\gamma} \times (r + 1) \times \log \log (e^{\gamma} \times (r + 1))\] is monotonically increasing as much as $q_{m}$ and $r$ become larger just starting with the initial values of $q_{m} = 47$ and $r = 1$, where $q_{m}$ is a prime number and $r$ is a natural number. In this way, we can confirm that the Riemann Hypothesis is true based on computational mathematics using a simple and naive computer assisted proof.

Let $ (P_{n})_{n\ge 0} $ be the sequence of Padovan numbers defined by $ P_0=0 $, $ P_1 =1=P_2$, and $ P_{n+3}= P_{n+1} +P_n$ for all $ n\ge 0 $. In this paper, we find all Padovan numbers that are concatenations of two distinct repdigits., Comment: Accepted manuscript. 9 pages

We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Ap\'ery-like recurrence relation: these include Ap\'ery, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers' conjectures on the p-adic valuation of Ap\'ery numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the p-Lucas property for almost all primes p., Comment: 25 pages. Version v2 contains cosmetic changes and a modification of the definition of $r$-admissibility

We prove that these Cantor sets are made up of transcendental numbers, apart from their endpoints $0$ and $1$, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field of algebraic numbers for the three numbers $1$, a characteristic Sturmian number, and an arbitrary Sturmian number with the same slope.

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain p-adic estimates of Satake parameters of certain nonregular self-dual automorphic representations of $\mathrm {GL}_n$ .

We prove that, if $x$ and $q\leqslant x^{1/16}$ are two parameters, then for any invertible residue class $a$ modulo $q$ there exists a product of exactly three primes, each one below $x^{1/3}$, that is congruent to $a$ modulo $q$., 6 pages, to submitted to Journal de Th\'{e}orie des Nombres de Bordeaux. Small corrections to previous version

International audience; Let E be a complete ultrametric space, let K be a perfect complete ultra-metric field and let A be a Banach K-algebra which is either a full K-subalgebra of the algebra of continuous functions from E to K owning all characteristic functions of clopens of E, or a full K-subalgebra of the algebra of uniformly continuous functions from E to K owning all characteristic functions of uniformly open subsets of E. We prove that all maximal ideals of finite codimension of A are of codimension 1. Introduction: Let E be a complete metric space provided with an ultrametric distance δ, let K be a perfect complete ultrametric field and let S be a full K-subalgebra of the K-algebra of continuous (resp. uniformly continuous) functions complete with respect to an ultrametric norm. that makes it a Banach K-algebra [3]. In [2], [4], [5], [6] we studied several examples of Banach K-algebras of functions and showed that for each example, each maximal ideal is defined by ultrafilters [1], [7], [8] and that each maximal ideal of finite codimension is of codimension 1: that holds for continuous functions [4] and for all examples of functions we examine in [2], [5], [6]. Thus, we can ask whether this comes from a more general property of Banach IK-algebras of functions, what we will prove here. Here we must assume that the ground field K is perfect, which makes that hypothesis necessary in all theorems.

We consider the higher integral moments for automorphic L-functions in short intervals and give a proof for the conjecture of Conrey et al. under Generalized Riemann Hypothesis for automorphic L-function.

International audience; Discrete logarithm index calculus algorithms are usually more efficient for non-hyperelliptic curves (Diem's method) than for hyperelliptic curves (Gaudry's method). However when the field of definition is not prime, Nagao's algorithm is even faster asymptotically, but is more efficient for hyperelliptic curves than for non-hyperelliptic ones. A natural question is then whether it is possible to adapt Nagao's method and design an index calculus that takes advantage of both the field extension and the non-hyperellipticity. In this work we explain why this is not possible, and why the asymptotic complexity of Nagao's algorithm is optimal using the known decomposition techniques.

Let $K$ be a finite extension of $\mathbf{Q}_p$. We use the theory of $(\varphi,\Gamma)$-modules in the Lubin-Tate setting to construct some corestriction-compatible families of classes in the cohomology of $V$, for certain representations $V$ of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)$. If in addition $V$ is crystalline, we describe these classes explicitly using Bloch-Kato's exponential maps. This allows us to generalize Perrin-Riou's period map to the Lubin-Tate setting., Comment: v2: final version, to appear in Documenta Mathematica. 31 pages

Letn ≥2 be an integer and denote byθnthe real root in (0, 1) of the trinomialGn(X) = −1 +X+Xn. The sequence of Perron numbers(θn−1)n≥2$(\theta _n^{ - 1} )_{n \ge 2} $tends to 1. We prove that the Conjecture of Lehmer is true for{θn−1|n≥2}$\{ \theta _n^{ - 1} |n \ge 2\} $by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the rootsθn,zj,n, ofGn(X) lying in|z|< 1, as a function ofn, jonly. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measuresM(Gn)=M(θn)=M(θn−1)${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$of the trinomialsGnas a function ofnonly, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for{θn−1|n≥2}$\{ \theta _n^{ - 1} |n \ge 2\} $, with a minoration of the house, and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots ofGn, near the unit circle, and limit equidistribution of the conjugates, forntending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.

Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the ”open” unit disk D of IK provided with the Gauss norm. Let Mult(A, ‖. ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Mult m (A, ‖. ‖) be the subset of the ϕ ∈ Mult(A, ‖. ‖) whose kernel is amaximal ideal and let Mult 1(A, ‖. ‖) be the subset of the ϕ ∈ Mult(A, ‖. ‖) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem the question whether Mult 1(A, ‖. ‖) is dense in Mult m (A, ‖. ‖). In a previous paper, it was proved that when IK is spherically complete, the answer is yes. Here we generalize this result to any algebraically closed complete ultrametric field, which particularly applies to ℂ p . On the other hand, we also show that the continuous multiplicative seminorms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of Mult 1(A, ‖. ‖), which suggest that Mult 1(A, ‖. ‖)might be dense in Mult(A, ‖. ‖).

The aim of this article is to give some properties of the so-called Padovan sequence $(T_n)_{n \ge 0}$ defined by $$ T_{n+3}=T_{n+1}+T_n \forall n \in \mathbb{N}, T_1=T_2=T_3=1$$ that is divisibility properties, periods, identities.

Let $K$ be a $p$-adic field. We continue to develop the theory of rigid analytic $p$-divisible groups over $K$. For example, we explain how to find back the category of Banach-Colmez spaces from rigid analytic $p$-divisible groups "in finite level" without perfectoid spaces. We then establish some results about families of rigid analytic $p$-divisible groups. This allows us to prove a "minimality" result in the sense of birationnal geometry for integral models of unramified Rapoport-Zink spaces., Comment: in French

International audience; Let IK be an ultrametric complete field and let E be an open subset of IK of strictly positive codiameter. Let D(E) be the Banach IK-algebra of bounded strictly differentiable functions from E to IK, a notion whose definition is detailed. It is shown that all elements of D(E) have a derivative that is continuous in E. Given a positive number r > 0, all functions that are bounded and are analytic in all open disks of diameter r are strictly differen-tiable. Maximal ideals and continuous multiplicative semi-norms on D(E) are studied by recalling the relation of contiguity on ultrafilters: an equivalence relation. So, the maximal spectrum of D(E) is in bijection with the set of equivalence classes with respect to contiguity. Every prime ideal of D(E) is included in a unique maximal ideal and every prime closed ideal of D(E) is a maximal ideal, hence every continuous multiplicative semi-norm on D(E) has a kernel that is a maximal ideal. If IK is locally compact, every maximal ideal of D(E) is of codimension 1. Every maximal ideal of D(E) is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently , the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. The Shilov boundary of D(E) is equal to the whole set of continuous multiplicative semi-norms. Many results are similar to those concerning algebras of uniformly continuous functions but some specific proofs are required. Introduction and preliminaries

This note provides truncated formulae with explicit error terms to compute Euler products over primes in arithmetic progressions of rational fractions. It further provides such a formula for the product of terms of the shape $F(1/p, 1/p^s)$ when $F$ is a two-variable polynomial with coefficients in $\mathbb{C}$ and satisfying some restrictive conditions., Comment: 10 pages

Besides various asymptotic results on the concept of sum-product bases in $\mathbb{N}_0$, we consider by probabilistic arguments the existence of thin sets $A,A'$ of integers such that $AA+A=\mathbb{N}_0$ and $A'A'+A'A'=\mathbb{N}_0$.

Let $Sh_K(G,\mu)$ be a Shimura variety of KHT type, as introduced in Harris-Taylor book, associated to some similitude group $G/\mathbb Q$ and a open compact subgroup $K$ of $G(\mathbb A)$. For any irreducible algebraic $\overline{\mathbb Q}_l$-representation $\xi$ of $G$, let $V_\xi$ be the $\mathbb Z_l$-local system on $Sh_K(G,\mu)$. From my paper about p-stabilization, we know that if we allow the local component $K_l$ of $K$ to be small enough, then there must exists some non trivial cohomology classes with coefficient in $V_\xi$. The aim of this paper is then to construct explicitly such torsion classes with the control of $K_l$. As an application we obtain the construction of some new automorphic congruences between tempered and non tempered automorphic representations of the same weight and same level at $l$.

Let C/K: F = 0 be a smooth plane quartic over a complete discrete valuation field K. In a previous paper the authors togetehr with Q. Liu give various characterizations of the reduction (i.e. non-hyperelliptic genus 3 curve, hyperelliptic genus 3 curve or bad) of the stable model of C: in terms of the existence of a special plane quartic model and in terms of the valuations of the Dixmier-Ohno invariants of C. The last one gives in particular an easy computable criterion for the reduction type. However, it does not produce a stable model, even in the case of good reduction. In this paper we give an algorithm to obtain (an approximation of) the stable model when the reduction of the latter is hyperelliptic and the characteristic of the residue field is not 2. This is based on a new criterion giving the reduction type in terms of the valuations of the theta constants of C. Some examples of the computation of these models are given., Comment: 12 pages. arXiv admin note: substantial text overlap with arXiv:1803.05816

We extend field patching to the setting of Berkovich analytic geometry and use it to prove a local-global principle over function fields of analytic curves with respect to completions. In the context of quadratic forms, we combine it with sufficient conditions for local isotropy over a Berkovich curve to obtain applications on the u-invariant. The patching method we adapt was introduced by Harbater and Hartmann in [18], and further developed by these two authors and Krashen in [19]. The results presented in this paper generalize those of [19] on the local-global principle and quadratic forms., Recollement sur les courbes de Berkovich et formes quadratiques. Nous étendons la technique de recollement sur les corps au cadre de la géométrie ana-lytique de Berkovich pour démontrer un principe local-global sur les corps de fonctions de courbes analytiques par rapport à certains de leurs complétés. Dans le contexte des formes quadratiques, nous le combinons avec des conditions suffisantes d'isotropie locale sur une courbe de Berkovich pour obtenir des applications au u-invariant. La méthode de recollement que nous adaptons a été introduite par Harbater et Hartmann dans [18], puis développée par ces deux auteurs et Krashen dans [19]. Dans ce texte, nous présentons des résultats sur le principe local-global et les formes quadratiques qui généralisent ceux de [19].

International audience

We show that Sarnak's conjecture on M\"obius disjointness holds for all subshifts given by bijective substitutions and some other similar dynamical systems, e.g.\ those generated by Rudin-Shapiro type sequences., Comment: 26 pages

Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic genus 3 curve C over a field k of characteristic different from 2, we produce an explicit genus 2 curve X such that Jac(C) is isogenous to the product of Jac(D) and Jac(X). This construction can be seen as a degenerate case of a result by Nils Bruin., Comment: Published version

Let N be a positive integer, be a subset of ℚ and . N is called an α-Korselt number (equivalently α is said an N-Korselt base) if α2p − α1 divides α2N − α1 for every prime divisor p of N. By the Korselt set of N over , we mean the set of all such that N is an α-Korselt number. In this paper we determine explicitly for a given prime number q and an integer l ∈ ℕ \{0, 1}, the set and we establish some connections between the ql -Korselt bases in ℚ and others in ℤ. The case of is studied where we prove that is empty if and only if l = 2. Moreover, we show that each nonzero rational α is an N-Korselt base for infinitely many numbers N = ql where q is a prime number and l ∈ ℕ.

We derive an identity connecting any two Horadam sequences having the same recurrence relation but whose initial terms may be different. Binomial and ordinary sum-mation identities arising from the identity are developed. Illustrative examples are drawn from Fibonacci, Fibonacci-Lucas, Pell, Pell-Lucas, Jacobsthal and Jacobsthal-Lucas sequences and their generalizations. Our new results subsume previously known identities.

Let $\G$ be a semisimple algebraic group over a number field $K$, $\mathcal{S}$ a finite set of places of $K$, $K_\mathcal{S}$ the direct product of the completions $K_v, v \in \mathcal{S}$, and $\OO$ the ring of $\mathcal{S}$-integers of $K$. Let $G = \G(K_\mathcal{S})$, $\Gamma = \G(\OO)$ and $\pi:G \rightarrow G/\Gamma$ the quotient map. We describe the closures of the locally divergent orbits ${T\pi(g)}$ %in $G/\Gamma$ where $T$ is a maximal $K_\mathcal{S}$-split torus in $G$. If $\# S = 2$ then the closure $\overline{T\pi(g)}$ is a finite union of $T$-orbits stratified in terms of parabolic subgroups of $\G \times \G$ and, consequently, $\overline{T\pi(g)}$ is homogeneous (i.e., $\overline{T\pi(g)}= H\pi(g)$ for a subgroup $H$ of $G$) if and only if ${T\pi(g)}$ is closed. On the other hand, if $\# \mathcal{S} > 2$ and $K$ is not a $\mathrm{CM}$-field then $\overline{T\pi(g)}$ is homogeneous for $\G = \mathbf{SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple $K$-ranks for $\G \neq \mathbf{SL}_{n}$. As an application, we prove that $\overline{f(\OO^{n})} = K_{\mathcal{S}}$ for the class of non-rational locally $K$-decomposable homogeneous forms $f \in K_{\mathcal{S}}[x_1, \cdots, x_{n}]$., Comment: arXiv admin note: text overlap with arXiv:1502.02297

Soit \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) la reduction modulo 2 de la fonction \(\Delta \). Une forme modulaire de niveau \(1\), \(f=\sum _{n\geqslant 0} c(n) \,q^n\), a coefficients entiers, est congrue modulo \(2\) a un polynome en \(\Delta \). Soit \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { impair }} 1\) le nombre de coefficients de Fourier impairs de \(f\) d’indice \(\leqslant x\). L’ordre de grandeur de \(W_f(x)\) a ete determine par Serre dans les annees 70. Nous donnons ici un equivalent de \(W_f(x)\). Soit \(p(n)\) la fonction de partition et \(A_0(x)\) (resp. \(A_1(x)\)) le nombre d’entiers \(n \leqslant x\) tels que \(p(n)\) est pair (resp. impair). Dans des articles anterieurs, J.-L. Nicolas a montre que \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) pour \(x\geqslant 3\) et que \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) pour \(x\geqslant 7\). On prouve ici que \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) pour \(x>1\) et que \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) pour \(x\geqslant 2\). Les principaux outils utilises dans la preuve de ces resultats sont la determination de l’ordre de nilpotence d’une forme modulaire de niveau 1 modulo 2, et de la structure de l’espace de ces formes modulaires en tant que module sur l’algebre de Hecke, obtenus dans un travail recent de J.-L. Nicolas et J.-P. Serre.