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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!

Let IK be a complete ultrametric field and let A be a unital commutative ultrametric Banach IK-algebra. Suppose that the multiplicative spectrum admits a partition in two open closed subsets. Then there exist unique idempotents u, v ϵ A such that ϕ (u) = 1, ϕ (v) = 0 ∀ ϕ ϵ U, ϕ (u) = 0 ϕ (v) = 1 ∀ ϕ ϵ V . Suppose that IK is algebraically closed. If an element x ϵ A has an empty annulus r < |ξ − a| < s in its spectrum sp(x), then there exist unique idempotents u, v such that ϕ (u) = 1; ϕ (v) = 0 whenever ϕ (x − a) ≤ r and ϕ (u) = 0; ϕ (v) = 1 whenever ϕ (x − a) ≥ s.

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

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 $m$.

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 $\frac{1}{2}$. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. 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. Let $n > 5040$ be $n = r \times q$, where $q$ denotes the largest prime factor of $n$. If $n > 5040$ is the smallest number such that Robin inequality does not hold, then we show the following inequality is also satisfied: $\sqrt[q]{e} + \frac{\log \log r}{\log \log n} > 2$.

We study the inequities in the distribution of Frobenius elements in Galois extensions of the rational numbers with Galois groups that are either dihedral $D_{2n}$ or (generalized) quaternion $\mathbb H_{2n}$ of two-power order. In the spirit of recent work of Fiorilli and Jouve arXiv:2001.05428, we study, under natural hypotheses, some families of such extensions, in a horizontal aspect, where the degree is fixed, and in a vertical aspect, where the degree goes to infinity. Our main contribution uncovers in families of extensions a phenomenon, for which Ng gave numerical evidence : real zeros of Artin L-functions sometimes have a radical influence on the distribution of Frobenius elements., Comment: 41 pages, minor corrections were made. To appear in Algebra & Number Theory Issue 15-4

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. Since then, this is called as the Robin inequality. It is known that the Robin inequality is satisfied for many classes of numbers. We show more classes of numbers for which the Robin inequality is always satisfied.

We define the resultant of two power series with coefficients in the ring of integers of a $p$-adic field. In order to do this, we prove a universal version of the Weierstrass preparation theorem., v2: added a section on a universal Hensel factorization theorem

Let \begin{document}$ f : [0,1)\rightarrow [0,1) $\end{document} be a \begin{document}$ 2 $\end{document} -interval piecewise affine increasing map which is injective but not surjective. Such a map \begin{document}$ f $\end{document} has a rotation number and can be parametrized by three real numbers. We make fully explicit the dynamics of \begin{document}$ f $\end{document} thanks to two specific functions \begin{document}$ {\boldsymbol{\delta}} $\end{document} and \begin{document}$ \phi $\end{document} depending on these parameters whose definitions involve Hecke-Mahler series. As an application, we show that the rotation number of \begin{document}$ f $\end{document} is rational, whenever the three parameters are all algebraic numbers, extending thus the main result of [ 16 ] dealing with the particular case of \begin{document}$ 2 $\end{document} -interval piecewise affine contractions with constant slope.

Let k k be a number field and X X a smooth, geometrically integral quasi-projective variety over k k . For any linear algebraic group G G over k k and any G G -torsor g : Z → X g: Z \to X , we observe that if the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S S for all twists of Z Z by elements in H^1_{\text {\'{e}t}}(k,G), then the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S S for X X . As an application, we show that any homogeneous space of the form G / H G/H with G G a connected linear algebraic group over k k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some compactness assumptions when k k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G / H G/H with G G semisimple simply connected and H H finite, using the theory of torsors and descent.

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.

$G$-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel's $G$-functions, satisfy a condition of moderate growth called Galochkin condition, encoded by a $p$-adic quantity, the size. Previous works of Chudnovsky, Andr\'e and Dwork have provided inequalities between the size of a $G$ -operator and certain computable constants depending among others on its solutions. First, we recall Andr\'e's idea to attach a notion of size to differential modules and detail his results on the behavior of the size relatively to the standard algebraic operations on the modules. This is the corner stone to prove a quantitative version of Andr\'e's generalization of Chudnovsky's Theorem: for $f(z)=\sum_{\alpha, k,\ell} c_{\alpha, k,\ell} z^{\alpha} \log(z)^k f_{\alpha, k,\ell}(z)$, where $f_{\alpha, k,\ell}(z)$ are $G$-functions, we can determine an upper bound on the size of the minimal operator $L$ over $\overline{\mathbb{Q}}(z)$ of $f(z)$ in terms of quantities depending on the $f_{\alpha, k,\ell}(z)$, the rationals $\alpha$ and the integers $k$. We give two applications of this result: we estimate the size of a product of two $G$-operators in function of the size of each operator; we also compute a constant appearing in a Diophantine problem encountered by the author., Comment: to appear in manuscripta mathematica

We show that the unresolved examples of Christol's conjecture $ \, _3F_{2}\left([2/9,5/9,8/9],[2/3,1],x\right)$ and $_3F_{2}\left([1/9,4/9,7/9],[1/3,1],x\right)$, are indeed diagonals of rational functions. We also show that other $\, _3F_2$ and $\, _4F_3$ unresolved examples of Christol's conjecture are diagonals of rational functions. Finally we give two arguments that show that it is likely that the $\, _3F_2([1/9, 4/9, 5/9], \, [1/3,1], \, 27 \cdot x)$ function is a diagonal of a rational function., 13 pages

International audience; We propose a new proof of the Hermite-Lindeman Theorem in an ultrametric field by using classical properties of analytic functions. The proof remains valid in zero residue characteristic. Definitions and notations: The Archimedean absolute value of C is denoted by |. | ∞. Let IK be an algebraically closed complete ultrametric field of characteristic 0 and residue characteristic p. We denote by |. | the ultrametric absolute value of IK and we denote by Ω an algebraic closure of Q in IK. Given a ∈ IK and r > 0, we denote by d(a, r −) the disk {x ∈ IK |x−a| < r}, we denote by A(d(a, r −)) the IK-Banach algebra of power series f (x) = ∞ n=0 a n (x − a) n converging in d(a, r −) and for all s ∈]0, r[, we put |f |(s) = sup n∈IN |a n |r n. Given a ∈ Ω, we call denominator of a any strictly positive integer n such that na is integral over Z and we denote by den(a) the smallest denominator of a. Next, considering the conjugates a 2 , ..., a n of a over Q and putting a 1 = a, we put |a| = max{|a 1 | ∞ , ..., |a n | ∞ }. Next, we put s(a) = Log(max(|a|, den(a)).

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.

Let $${ \rm I\!K}$$ be an ultrametric complete valued field and $${ \rm I\!E}$$ be an ultrametric space. We examine some Banach algebras $$S$$ of bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ with the use of ultrafilters, particularly the relation of stickness. We recall and deepen results obtained in a previous paper by N. Mainetti and the third author concerning the whole algebra $$\cal A$$ of all bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ . Every maximal ideal of finite codimension of $$\cal A$$ is of codimension $$1$$ and we show that this property also holds for every algebra $$S$$ , provided $${ \rm I\!K}$$ is perfect. If $$S$$ admits the uniform norm on $${ \rm I\!E}$$ as its spectral norm, then every maximal ideal is the kernel of only one multiplicative semi-norm, the Shilov boundary is equal to the whole multiplicative spectrum and the Banaschewski compactification of $${ \rm I\!E}$$ is homeomorphic to the multiplicative spectrum of $$S$$ .

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 estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope., 23 pages. Submitted for publication

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 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.

We state conjectures on the asymptotic behavior of the Masur-Veech volumes of strata in the moduli spaces of meromorphic quadratic differentials and on the asymptotics of their area Siegel-Veech constants as the genus tends to infinity., 10 pages

We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

We prove the equidistribution of the Hodge locus for certain non-isotrivial, polarized variations of Hodge structure of weight $2$ with $h^{2,0}=1$ over complex, quasi-projective curves. Given some norm condition, we also give an asymptotic on the growth of the Hodge locus. In particular, this implies the equidistribution of elliptic fibrations in quasi-polarized, non-isotrivial families of $K3$ surfaces., Comment: 33 pages. To appear in Journal f\"ur die reine und angewandte Mathematik (Crelles Journal)

International audience; A new Nevanlinna theorem on q p-adic small functions is given. Let f, g, be two meromorphic functions on a complete ultrametric algebraically closed field IK of characteristic 0, or two meromorphic functions in an open disk of IK, that are not quotients of bounded analytic functions by polynomials. If f and g share 7 small meromorphic functions I.M., then f = g. Better results hold when f and g satisfy some property of growth. Particularly , if f and g have finitely many poles or finitely many zeros and share 3 small meromorphic functions I.M., then f = g. 1. Main results Let IK be a complete ultrametric algebraically closed field of characteristic 0. Let us fix a ∈ IK and let R ∈]0, +∞[. We denote by d(a, R −) the disk {x ∈ IK | |x − a| < R}. We denote by A(IK) the IK-algebra of entire functions in IK and by M(IK) the field of meromorphic functions which is its field of fractions. We denote by A(d(a, R −)) the IK-algebra of analytic functions in d(a, R −) i.e. the set of power series converging in the disk d(a, R −) and by M(d(a, R −)) the field of meromorphic functions in d(a, R −) i.e. the field of fractions of A(d(a, R −)). Moreover , we denote by A b (d(a, R −)) the IK-algebra of functions f ∈ A(d(a, R −)) that are bounded in d(a, R −), by M b (d(a, R −)) its field of fractions and we put M u (d(a, R −)) = M(d(a, R −)) \ A b (d(a, R −)). We define N (r, f) ([1], chapter 40 or [3], chapter 2) in the same way as for complex meromorphic functions [2]. Let f be a meromorphic function in all IK (resp. in d(0, R −)) having no zero and no pole at 0. Let (a n) n∈IN be the sequence of poles of f , of respective order s n , with |a n | ≤ |a n+1 | and, given r > 0, (resp. r ∈]0, R[), let q(r) be such that |a q(r) | ≤ r, |a q(r)+1 | > r. We then denote by N (r, f) the counting function of the zeros of f , counting multiplicity, as usual: for all r > 0, we put N (r, f) = q(r) j=0 s j (log |a j | − log(r)). Moreover, we denote by N (r, f) the counting function of the poles of f , ignoring multiplicity 0

We prove that for $1$-motives defined over an algebraically closed subfield of $\C$, viewed as Nori motives, the motivic Galois group is the Mumford-Tate group. In particular, the Hodge realization of the tannakian category of (Nori) motives generated by $1$-motives is fully faithful., slightly expanded version. To appear in Intern. Res. Math. Notices

We prove the weak functoriality of (big) Cohen-Macaulay algebras, which controls the whole skein of "homological conjectures" in commutative algebra [H1][HH2]. Namely, for any local homomorphism $ R\to R'$ of complete local domains, there exists a compatible homomorphism between some Cohen-Macaulay $R$-algebra and some Cohen-Macaulay $R'$-algebra. When $R$ contains a field, this is already known [[3.9]{HH2}]. When $R$ is of mixed characteristic, our strategy of proof is reminiscent of G. Dietz's refined treatment [D] of weak functoriality of Cohen-Macaulay algebras in characteristic $p$; in fact, developing a "tilting argument" due to K. Shimomoto, we combine the perfectoid techniques of [A1][A2] with Dietz's result., Comment: A short erratum to the author's "le lemme d'Abhyankar perfecto\"ide" is added

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

In this paper, we considered two generalisations of the classical Titchmarsh divisor problem: friable variant and short intervals.

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.

International audience; We determine the structure of the graded ring of Siegel modular forms of degree 3. It is generated by 19 modular forms, among which we identify a homogeneous system of parameters with 7 forms of weights 4, 12, 12, 14, 18, 20 and 30. We also give a complete dictionary between the Dixmier-Ohno invariants of ternary quartics and the above generators.

We describe an unexpected connection between bounded height in families of finitely generated subgroups in tori problems and irrationality. Our method allow us to recover effective irrationality measures for some values of algebraic functions

In this work, we give a lower bound and an upper bound for the trace of a Salem number whose Mahler measure is less than 1.3. We use the method of explicit auxiliary functions combined with the recursive algorithm developed in [F3].

{The first version of this text was written and submitted to a journal on April, 12, 2018. This second version was submitted on April, 9, 2019.} We investigate the existence of subsets $A$ and $B$ of $\mathbb{N}:=\{0,1,2,\dots\}$ such that the sumset $A+B:=\{a+b~;a\in A,b\in B\}$ has given asymptotic density. We solve the particular case in which $B$ is a given finite subset of $\mathbb{N}$ and also the case when $B=A$ ; in the later case, we generalize our result to $kA:=\{x_1+\cdots+x_k: x_i\in A, i=1,\dots,k\}$ for an integer $k\geq2.$, Comment: 13 pages

In this paper we study sheaves of logarithmic arithmetic differential operators on a particular semistable model of the projective line. The main result here is that the first cohomology group of these sheaves is non-torsion. We also consider a refinement of the order filtration on the sheaf of level zero (before taking the p-adic completion). The associated graded sheaf, which we explicitly determine, explains to some extent the occurrence of the cohomology classes in degree one., Added acknowledgement of support by the ANR program p-adic Hodge Theory and beyond (Th\'eHopaD)

This is a systematic study of the behaviour of finite coverings of (affine) schemes with regard to two Grothendieck topologies: the canonical topology and the fpqc topology. The history of the problem takes roots in the foundations of Grothendieck topologies, passes through main strides in Commutative Algebra and leads to new Mathematics up to perfectoids and prisms. We first review the canonical topology of affine schemes and show, keeping with Olivier's lost work, that it coincides with the effective descent topology; covering maps are given by universally injective ring maps, which we discuss in detail. We then give a "catalogue raisonn\'e" of examples of finite coverings which separate the canonical, fpqc and fppf topologies. The key result is that finite coverings of regular schemes are coverings for the canonical topology, and even for the fpqc topology (but not necessarily for the fppf topology). We discuss a "weakly functorial" aspect of this result. "Splinters" are those affine Noetherian schemes for which every finite covering is a covering for the canonical topology. We also investigate their mysterious fpqc analogs, and prove that in prime characteristic, they are all regular. This leads us to the problem of descent of regularity by (non-necessarily flat) morphisms $f$ which are coverings for the fpqc topology, which is settled thanks to a recent theorem of Bhatt-Iyengar-Ma., Comment: Final version, accepted in Ann. Sc. Norm. Super. Pisa Cl. Sci

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

We give an introduction to the transalgebraic theory of simply connected log-Riemann surfaces with a finite number of infinite ramification points (transalgebraic curves of genus $0$). We define the base vector space of transcendental functions and establish by elementary means some transcendental properties. We introduce the Ramificant Determinant constructed with transcendental periods and we give a closed-form formula that gives the main applications to transalgebraic curves. We prove an Abel-like Theorem and a Torelli-like Theorem. Transposing to the transalgebraic curve the base vector space of transcendental functions, they generate the structural ring from which the points of the transalgebraic curve can be recovered algebraically, including infinite ramification points., see also arXiv:1512.03776

Formal schemes in algebraic geometry consist of power series with integer degree, this idea can be naturally carried over to power series with rational degree. In this paper formal schemes with rational degree are constructed. These schemes are non noetherian and thus require slight modification of the standard approach. The first part of the paper constructs non noetherian continuous valuation rings from discrete valuation rings, and these rings are reffered to as ‘eka’ (one in Hindi) rings. These new rings are designed to carry the properties of discrete valuation to continuous valuation faithfully. The second part of the paper constructs non noetherian formal schemes with rational degree and shows their admissibility. The corresponding flatness and coherence is proved. Finally line bundles of rational degree are constructed and their Cech cohomology computed.

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].

In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian plus an associated space of Chow-Heegner points. We then apply this condition to prove the finiteness of this set for any modular curves $X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ of genus at least 2 with N prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell-Weil rank is equal to its dimension (and at least 2), which is proven via analytic estimates for orders of vanishing of L-functions of modular forms, thanks to a Kolyvagin-Logachev type result., Comment: 51 pages, comments welcome

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, ‖. ‖).

We give elementary proof of stronger versions of several recent results on intrinsic Diophantine approximation on rational quadric hypersurfaces $X\subset \mathbb{P}^n(\mathbb{R})$. The main tool is a refinement of the simplex lemma, which essentially says that rational points on $X$ which are sufficiently close to each other must lie on a totally isotropic rational subspace of $X$., 15 pages

We study the moments of even order of the generating function $$\prod\limits_{0 \leqslant r \leqslant n} {(1 - e({2^r}x))} $$ of the Thue–Morse sequence and present several conjectures related to these moments.

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.

Let $P$ be a prime polynomial in the variable $Y$ over a finite field and let $f$ be a quadratic irrational in the field of formal Laurant series in the variable $Y^{-1}$. We study the asymptotic properties of the degrees of the coefficients of the continued fraction expansion of quadratic irrationals such as $P^nf$ and prove results that are in sharp contrast to the analogue situation in zero characteristic.

In a previous work, we stated a conjecture, called the Galois Brumer-Stark conjecture, that generalizes the (abelian) Brumer-Stark conjecture to Galois extensions. Other generalizations of the Brumer-Stark conjecture to non-abelian Galois extensions are due to Nickel. Nomura proved that the Brumer-Stark conjecture implies the weak non-abelian Brumer-Stark conjecture of Nickel when the group is monomial. In this paper, we use the methods of Nomura to prove that the Brumer-Stark conjecture implies the Galois Brumer-Stark conjecture for monomial groups in the semi-simple case.