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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

We investigate the relations between the rings ${\bf E}$, ${\bf G}$ and ${\bf D}$ of values taken at algebraic points by arithmetic Gevrey series of order either $-1$ ($E$-functions), $0$ (analytic continuations of $G$-functions) or $1$ (renormalization of divergent series solutions at $\infty$ of $E$-operators) respectively. We prove in particular that any element of ${\bf G}$ can be written as multivariate polynomial with algebraic coefficients in elements of ${\bf E}$ and ${\bf D}$, and is the limit at infinity of some $E$-function along some direction. This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously $E$-functions and arithmetic Gevrey series of order 1. Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of Andr\'e and Beukers for $E$-functions) and the conjecture ${\bf D}\cap{\bf E}=\overline{\mathbb Q}$ (but not necessarily all these conjectures at the same time), we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers' linear independence theorem for values of $E$-functions, the transcendance of the values of the Gamma function and its derivatives at all non-integral algebraic numbers, the transcendance of Gompertz constant as well as the fact that Euler's constant is not in ${\bf E}$., Comment: 18 pages

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

In this paper we study dynamical systems of intervals moving on incommensurable metric graphs. They can be generally viewed as congruent intervals moving around some graph with unit velocity and propagating on all respective incident edges whenever sliding through any vertex. In particular, we analyse the phenomenon of saturation: a state of the system when the entire graph is covered by these moving intervals. Our main contributions are the following: (1) we prove the existence of the finite moment of permanent saturation for any incommensurable metric graph and any positive length of these intervals; (2) we present an upper bound for the moment of permanent saturation. To show the validity of our results, we reduce the system of moving intervals to the system of dispersing moving points for the analysis of which we primarily use methods of discrepancy theory and number theory, specifically Kronecker sequences and the celebrated Three Gap Theorem.

We provide several extensions of the modular method which were motivated by the problem of completing previous work to prove that, for any integer $n \geq 2$, the equation \[ x^{13} + y^{13} = 3 z^n \] has no non-trivial solutions. In particular, we present four elimination techniques which are based on: (1) establishing reducibility of certain residual Galois representations over a totally real field; (2) generalizing image of inertia arguments to the setting of abelian surfaces; (3) establishing congruences of Hilbert modular forms without the use of often impractical Sturm bounds; and (4) a unit sieve argument which combines information from classical descent and the modular method. The extensions are of broader applicability and provide further evidence that it is possible to obtain a complete resolution of a family of generalized Fermat equations by remaining within the framework of the modular method. As a further illustration of this, we complete a theorem of Anni-Siksek to show that, for $\ell, m\ge 5$, the only solutions to the equation $x^{2\ell} + y^{2m} = z^{13}$ are the trivial ones., Several modifications after the referees' comments. To appear in Publicacions Matem\`atiques

We study how often exceptional configurations of irreducible polynomials over finite fields occur in the context of prime number races and Chebyshev's bias. In particular, we show that three types of biases, which we call "complete bias", "lower order bias" and "reversed bias", occur with probability going to zero among the family of all squarefree monic polynomials of a given degree in $\mathbb{F}_q[x]$ as $q$, a power of a fixed prime, goes to infinity. The bounds given improve on a previous result of Kowalski, who studied a similar question along particular $1$-parameter families of reducible polynomials. The tools used are the large sieve for Frobenius developed by Kowalski, an improvement of it due to Perret-Gentil and considerations from the theory of linear recurrence sequences and arithmetic geometry., Comment: 22 pages. Comments welcome

In this paper we prove that there are finitely many modular curves that admit a smooth plane model. Moreover, if the degree of the model is greater than or equal to 19, no such curve exists. For modular curves of Shimura type we show that none can admit a smooth plane model of degree 5, 6 or 7. Further, if a modular curve of Shimura type admits a smooth plane model of degree 8 we show that it must be a twist of one of four curves.

In this paper, we study the cohomology of the ramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ by using the Bruhat-Tits stratification on its special fiber. As such, we apply the same method that we developped for the unramified case in two previous papers. More precisely, we first investigate the cohomology of a given closed Bruhat-Tits stratum. It is isomorphic to a generalized Deligne-Lusztig variety which is in general not smooth, and is associated to a finite group of symplectic similitudes. We determine the weights of the Frobenius and most of the unipotent representations occuring in its cohomology. This computation involves the spectral sequence associated to a stratification by classical Deligne-Lusztig varieties, which are parabolically induced from Coxeter varieties of smaller symplectic groups. In particular, all the unipotent representations contribute to only two cuspidal series. Then, we introduce the analytical tubes of the closed Bruhat-Tits strata, which give an open cover of the generic fiber of the Rapoport-Zink space. Using the associated \v{C}ech spectral sequence, we prove that certain cohomology groups of the Rapoport-Zink space at hyperspecial level fail to be admissible if $n$ is large enough. Eventually, when $n=2$ in the split case, when $n=3$ and when $n=4$ in the non-split case, we give a complete description of the cohomology of the supersingular locus of the associated Shimura variety at hyperspecial level, in terms of automorphic representations. In particular, certain automorphic representations occur with a multiplicity depending on $p$. Thus, our computations in the ramified case recover all the main features of the unramified case, despite new technical difficulties due to the closed Bruhat-Tits strata not being smooth., Comment: 54 pages. The previous version contained some mistakes. They are now fixed in this new version. The last section and the main theorem have been modified accordingly. Comments are welcome!

For a regular non-critical p-refinement of a cohomological cuspidal automorphic representation of GL(2) over a totally real number field, we prove a functional equation of its attached p-adic L-function. We obtain it from the interpolation formula between p-adic and complex L-functions and the functional equation of L-functions. We use this functional equation to prove the trivial zero conjecture at the central critical point.On the other hand, we develop a theory of overconvergent modular symbols with values in p-adic distributions on the projective line of the field of p-adic rational numbers inspired by Stevens's overconvergent modular symbols and the idea of Colmez with the hope that one can obtain some functional equations of p-adic L-functions involving the transformation mapping z to its inverse on the projective line of p-adic rational numbers., Pour un p-raffinement non-critique et régulier d'une représentation automorphe cuspidale cohomologique de GL(2) sur un corps de nombres totalement réel, nous prouvons une équation fonctionnelle de sa fonction L p-adique attachée. Nous obtenons cela grâce à la formule d'interpolation entre les fonctions L p-adiques et complexes et l'équation fonctionnelle des fonctions L. Nous utilisons cette équation fonctionnelle pour prouver la conjecture des zéros triviaux au point critique central.D'autre part, nous développons une théorie des symboles modulaires surconvergents à valeurs dans des distributions p-adiques sur la droite projective du corps des nombres rationnels p-adiques inspirée par les symboles modulaires surconvergents de Stevens et une idée de Colmez dans l'espoir d'obtenir certaines équations fonctionnelles des fonctions L p-adiques faisant intervenir la transformation de z en son inverse sur la droite projective des nombres rationnels p-adiques.

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 E be a field of characteristic p. In a previous paper of ours, we defined and studied super-H\"older vectors in certain E-linear representations of Z_p. In the present paper, we define and study super-H\"older vectors in certain E-linear representations of a general p-adic Lie group. We then consider certain p-adic Lie extensions K_\infty / K of a p-adic field K, and compute the super-H\"older vectors in the tilt of K_\infty. We show that these super-H\"older vectors are the perfection of the field of norms of K_\infty / K. By specializing to the case of a Lubin-Tate extension, we are able to recover E((Y)) inside the Y-adic completion of its perfection, seen as a valued E-vector space endowed with the action of O_K^\times given by the endomorphisms of the corresponding Lubin-Tate group., Comment: v3: minor corrections

In this thesis we investigate the sequence of Mahler measures of a family of bivariate regular exact polynomials, called Pd := P0≤i+j≤d xiyj , unbounded in both degree and the genus of the algebraic curve. We obtain a closed formula for the Mahler measure of Pd in termsof special values of the Bloch–Wigner dilogarithm. We approximate m(Pd), for 1 ≤ d ≤ 1000,with arbitrary precision using SageMath. Using 3 different methods we prove that the limitof the sequence of the Mahler measure of this family converges to 92π2 ζ(3). Moreover, we compute the asymptotic expansion of the Mahler measure of Pd which implies that the rate of the convergence is O(log dd2 ). We also prove a generalization of the theorem of the Boyd-Lawton which asserts that the multivariate Mahler measures can be approximated using the lower dimensional Mahler measures. Finally, we prove that the Mahler measure of Pd, for arbitrary d can be written as a linear combination of L-functions associated with an odd primitive Dirichlet character. In addition, we compute explicitly the representation of the Mahler measure of Pd in terms of L-functions, for 1 ≤ d ≤ 6., Dans cette thèse, nous étudions la suite de mesures de Mahler d’une famille de polynômes à deux variables exacts et réguliers, que nous notons Pd := P0≤i+j≤d xiyj . Elle n’est bornée ni en volume, ni en genre de la courbe algébrique sous-jacente. Nous obtenons une expression pour la mesure de Mahler de Pd comme somme finie de valeurs spéciales du dilogarithme de Bloch-Wigner. Nous utilisons SageMath pour approximer m(Pd) pour 1 ≤ d ≤ 1000. En recourant à trois méthodes différentes, nous prouvons que la limite de la suite de mesures de Mahler de cette famille converge vers 92π2 ζ(3). De plus, nous calculons le développement asymptotique de la mesure de Mahler de Pd et prouvons que sa vitesse de convergence est de O(log dd2 ). Nous démontrons également une généralisation du théorème de Boyd-Lawton, affirmant que les mesures de Mahler multivariées peuvent être approximéess en utilisant les mesures de Mahler de dimension inférieure. Enfin, nous prouvons que la mesure de Mahler de Pd pour d arbitraire peut être écrite comme une combinaison linéaire de fonctions L associées à un caractère de Dirichlet primitif impair. Nous calculons finalement explicitement la représentation de la mesure de Mahler de Pd en termes de fonctions L, pour 1 ≤ d ≤ 6.

International audience; There are two types of integer gcd algorithms: those which compute the sequence of remainders of Euclid's algorithm and those which build different sequences. The former are more difficult to validate and analyse, whereas the latter are simpler and more efficient. When one wants the euclidean remainders (for instance if one wants to compute continued fractions), only the former can be used. Our main focus is the subquadratic time Thull-Yap GCD algorithm, and in fact on its core computing a half gcd (TYHGCD). This algorithm is tricky due to the difficulty in correcting the remainder sequence that comes back from a recursive call. The aim of this work is to revise TYHGCD in order to implement it using GMP. We clarify some points of the algorithm, in particular the stopping conditions that are always difficult to set correctly. We add a base case to speed up the whole algorithm, using Jebelean's quadratic algorithm with a stopping condition. We give our own modified version and add the proofs where needed. We insist on the test phase for this algorithm, giving families of hard cases for all branches, some of which are rarely activated. We give some details on our implementation in GMP using low-level functions, adding some remarks on the use of fast multiplications techniques. We pay attention to the data structure needed to store partial quotients, enabling to navigate rapidly back and forth in the sequence of Euclidean remainders. Benchmarks are provided. Some comments are made on Lichtblau's algorithm, which is close in spirit to the Thull-Yap algorithm.

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

In this thesis, we study the Hodge-Tate structure of the proétale cohomology of Shimura varieties. This document is divided in four main issues. First, we construct an integral model of the perfectoid modular curve. Using this formal scheme, we prove some vanishing results for the coherent cohomology of the perfectoid modular curve, we also provide a description of the dual completed cohomology as an inverse limit of integral modular forms of weight 2 by normalized traces. Secondly, we construct the overconvergent Eichler-Shimura map for the first coherent cohomology group, complementing the work of Andreatta-Iovita-Stevens. More precisely, we construct a map from the overconvergent cohomology with compact support of Boxer-Pilloni to the locally analytic modular symbols of Ash-Stevens. We reinterpret the construction of these maps in terms of the Hodge-Tate period map and the perfectoid modular curve. Thirdly, in a joint work with Joaquín Rodrigues Jacinto, we develop the classical theory of locally analytic representations of p-adic Lie groups in the context of condensed mathematics. Inspired from foundational works of Lazard, Schneider-Teitelbaum and Emerton, we define a notion of solid locally analytic representation for a compact p-adic Lie group. We prove that the category of solid locally analytic representations can be described as modules over algebras of analytic distributions. As an application, we prove a cohomological comparison theorem between solid group cohomology, solid group cohomology of the (derived) locally analytic vectors, and Lie algebra cohomology. Finally, we generalize the work of Lue Pan to arbitrary Shimura varieties. We construct a geometric Sen operator for a particular class of proetale modules over the structural sheaf which we call relative locally analytic. We prove that this Sen operator is related with the p-adic Simpson correspondence, and that it computes proétale cohomology. We apply this theory to Shimura varieties, obtaining that the computation of proétale cohomology can be translated in terms of Lie algebra cohomology over the flag variety via the Hodge-Tate period map. In particular, we prove that the Cp-extension of scalars of the locally analytic completed cohomology can be described as the analytic cohomology of the infinite-at-p level Shimura variety, of the locally analytic sections of the structural sheaf. This implies a rational version of the Calegari-Emerton conjectures for any Shimura variety without the hypothesis of the infinite-at-p level Shimura variety to be perfectoid. Then, we study the isotypic components of the locally analytic completed cohomology for the action of a Borel subalgebra. Using the interpretation as Lie algebra cohomology over the flag variety, we construct overconvergent BGG maps generalizing the previous work for the modular curve. In addition, we give a local proof of the classical Hodge-Tate decompositions for Shimura varieties, using the dual BGG resolution and the Hodge-Tate period map., Dans ce manuscrit, nous étudions la structure de Hodge-Tate de la cohomologie proétale des variétés de Shimura. Cette thèse est divisée dans quatre parties. D’abord, nous construisons un modèle entière de la courbe modulaire perfectoïde. Avec ce schema formel, on montre quelques résultats d’annulation de la cohomologie cohérente en niveau infini, et nous donnons une description du dual de la cohomologie completée en termes de formes modulaires intégrales de poids 2 et de traces normalisées. Dans un second temps, on construit l’application surconvergente d’Eichler-Shimura pour le premier groupe de cohomologie cohérente, il s’agit d’un morphisme de la cohomologie surconvergente à support compact de Boxer-Pilloni vers les symboles modulaires localement analytiques d’Ash- Stevens, qui interpole l’application d’Eichler-Shimura classique. Nous réinterpre ́tons les construc- tions précédentes en termes du morphisme des périodes de Hodge-Tate et de la courbe perfectoïde.Ensuite, dans un travail un commun avec Joaquín Rodrigues Jacinto, nous introduisons le concept de représentation localement analytique solide pour un groupe de Lie p-adique compact G. Nous nous inspirons des travaux de Lazard, Schneider-Teitelbaum et Emerton pour réinterpréter la propriété localement analytique dans la catégorie des représentations solides de G, et nous voyons que les objets obtenus peuvent être décrit en termes de modules sur des algèbres de distributions analytiques. En guise d’une application, nous démontrons quelques théorèmes de comparaison entre la cohomologie solide des groupes et la cohomologie de l’algèbre de Lie des vecteurs localement analytiques derivés. Pour finir, nous généralisons à des variétés de Shimura quelconque les travaux de Lue Pan sur la cohomologie complétée localement analytique des courbes modulaires. Le premier point technique est l’existence d’un opérateur de Sen géométrique qui est lié à la correspondence de Simpson p- adique. On montre que cet opérateur calcule la cohomologie proétale des modules sur le faisceau structural complété dans un sens précis. En appliquant cette théorie dans le cas des variétés de Shimura, nous arrivons à réduire le calcule de la cohomologie proétale de certains faisceaux à celui de la cohomologie de Lie des D-modules sur la variété de drapeaux. En particulier, nous prouvons que l’extension des scalaires à Cp de la cohomologie completée localement analytique se calcule comme la cohomologie des sections localement analytiques du faisceau structural de la variété de Shimura de niveau infini en p sur le site analytique. Comme corollaire, on en déduit une version rationnelle des conjectures de Calegari-Emerton sur l’annulation de la cohomologie completée. Ensuite, nous étudions les composantes isotypiques de la cohomologie completée localement analytique pour l’action d’un Borel. En utilisant le dictionnaire entre cohomologie proétale et cohomologie de Lie des faisceaux sur la variété de drapeaux, on arrive à construire des applications de BGG surconvergentes. De plus, nous donnons une preuve locale de la décomposition de Hodge-Tate avec coefficients, en utilisant la résolution BGG-dual et le morphisme des périodes de Hodge-Tate.

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get for any given $g>0$, any $\varepsilon>0$ and all $q$ large enough, the existence of a curve of genus $g$ over ${\mathbb F}_q$ with at least $1+q+ (2g-\varepsilon) \sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \sqrt{q}$ valid for $g \geq 3$ and odd $q \geq 11$. Finally, explicit constructions of towers of curves improve this result, with a bound of the form $1+q+4 \sqrt{q} -32$ valid for all $g\ge 2$ and for all $q$., Minor revisions

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

Let $p$ be a prime; using modular polynomial $\Phi_p$, T.~Satoh and al\cite{satoh2000canonical,harley2002,vercau} developed several algorithmsto compute the canonical lift of an ordinary elliptic curve $E$ over$\F_{p^n}$ with $j$-invariant not in $\F_{p^2}$. When $p$ is constant, thebest variant has a complexity $\Otilde(n m)$ to lift $E$ to $p$-adicprecision~$m$. As an application, lifting $E$ to precision $m=O(n)$ allowsto recover its cardinality in time $\Otilde(n^2)$. However, taking $p$ intoaccount the complexity is $\Otilde(p^2 n m)$, so Satoh's algorithm can onlybe applied to small~$p$.We propose in this paper two variants of these algorithms, which do notrely on the modular polynomial, for computing the canonical lift of anordinary curve. Our new method yield a complexity of $\Otilde(p n m)$ tolift at precision~$m$, and even $\Otilde(\sqrt{p} nm)$ when we are provideda rational point of $p$-torsion on the curve. This allows to extend Saoth'spoint counting algorithm to larger~$p$.

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

We study Landau's problems in the context of arithmetic combinatorics, combinatorial number theory and general topology as well as abstract algebra, we define gaps and discontinuities of integer topologies at the neighbourhood of infinity.

It is known that the Scholz conjecture on addition chains is true for all integers n with ℓ(2n) = ℓ(n) + 1. There exists infinitely many integers with ℓ(2n) ≤ ℓ(n) and we don't know if the conjecture still holds for them. The conjecture is also proven to hold for integers n with v(n) ≤ 5 and for infinitely many integers with v(n) = 6. There is no specific results on integers with v(n) = 7. In [thurber], an infinite list of integers satisfying ℓ(n) = ℓ(2n) and v(n) = 7 is given. In this paper, we prove that the conjecture holds for all of them.

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

By proposing a numbering of the twenty proteinogenic amino acids deduced from the physicochemical properties of the four coding DNA nucleobases, it is established that this amino acid number, equal to 5x entities, is not arbitrary. Indeed, we demonstrate that many attributes of these twenty amino acids, as a whole, are also 5x in number and that by isolating, since their numbering, the 3x peripheral amino acids from the 2x internal ones, these attributes are divided into ratios of 3/2 as exact value. This is verified both as the physicochemical properties of the 20 amino acids and as the coding configurations of the nucleobases, the source of this numbering., En proposant une numérotation des vingt acides aminés protéinogènes déduite des propriétés physico-chimiques des quatre nucléobases codantes de l'ADN, il est établi que ce nombre d'acides aminés, égal à 5x entités, n'est pas arbitraire. En effet, nous démontrons que de nombreux attributs de ces vingt acides aminés, dans leur ensemble, sont également au nombre de 5x et qu'en isolant, depuis leur numérotation, les 3x acides aminés périphériques des 2x internes, ces attributs sont répartis dans des rapports de 3 /2 comme valeur exacte. Ceci se vérifie à la fois avec les propriétés physico-chimiques des 20 acides aminés et avec les configurations codantes des nucléobases, source de cette numérotation.

International audience; We construct families of elliptic curves defined over number fields and containing torsion groups Z=M1Z x Z=M2Z where (M1;M2) belongs to f(1; 11), (1; 14), (1; 15), (2; 10), (2; 12), (3; 9), (4; 8), (6; 6)g (i.e., when the corresponding modular curve X1(M1;M2) has genus 1). We provide formulae for the curves and give examples of number fields for which the corresponding elliptic curves have non-zero ranks, giving explicit generators using D. Simon's program whenever possible. The reductions of these curves can be used to speed up ECM for factoring numbers with special properties, a typical example being (factors of) Cunningham numbers bn - 1 such that M1 j n. We explain how to find points of potentially large orders on the reduction, if we accept to use quadratic twists.

In this paper proof of the Polignac's Conjecture for gap equal to eight is going to be presented. It will be shown that consecutive primes with gap eight could be obtained through two stage sieve process, and that will be used to prove that infinitely many primes with gap eight exist. The proof represents an simple extension of the recently presented proofs that infinitely many twin, cousin and sexy primes exist. The major contribution of this paper is presentation of all elementary modules that are necessary for the proof of Polgnac's conjecture in general case.

Dans ce texte on reprend les travaux de Kaletha sur les formes intérieures rigides d'un groupe réductif sur un corps local d'un point de vue géométrique. On interprète en particulier la gerbe qu'il a introduite du point de vue de la courbe, en remplaçant celle-ci par une gerbe naturelle vivant au dessus de la courbe. On introduit une catégorie Tannakienne d'isocristauxétendus, généralisant la notion usuelle d'isocristal. Cela nous amèneà définir etétudier un ensemble de Kottwitzétendu d'isocristauxétendus avec structure additionnelles, contenant l'ensemble usuel et dont on montre qu'il est invariant par formes intérieures. Enfin on montre que cet ensemble classifie les fibrés principaux sur la gerbe au dessus de la courbe.

Comments: 43 pages. Most of the manuscript has been professionally proofread.; The binary Goldbach conjecture asserts that every even integer greater than $4$ is the sum of two primes. In this paper, we prove that there exists an integer $K_\alpha$ such that every even integer $x > p_k^2$ can be expressed as the sum of two primes, where $p_k$ is the $k$th prime number and $k > K_\alpha$. To prove this statement, we begin by introducing a type of double sieve of Eratosthenes as follows. Given a positive even integer $x > 4$, we sift from $[1, x]$ all those elements that are congruents to $0$ modulo $p$ or congruents to $x$ modulo $p$, where $p$ is a prime less than $\sqrt{x}$. Therefore, any integer in the interval $[\sqrt{x}, x]$ that remains unsifted is a prime $q$ for which either $x-q = 1$ or $x-q$ is also a prime. Then, we introduce a new way of formulating a sieve, which we call the sequence of $k$-tuples of remainders. By means of this tool, we prove that there exists an integer $K_\alpha > 5$ such that $p_k / 2$ is a lower bound for the sifting function of this sieve, for every even number $x$ that satisfies $p_k^2 < x < p_{k+1}^2$, where $k > K_\alpha$, which implies that $x > p_k^2 \; (k > K_\alpha)$ can be expressed as the sum of two primes.

Let F be a field of characteristic 2 and K a purely inseparable modular extension of F. Our aim in this paper is to give a complete classification of anisotropic semisingular F-quadratic forms φ that have over K a maximal Witt index and a defect index at least equal to the half of the dimension of the quasilinear part. The case of totally singular quadratic forms will be also treated. Our method also allows us to classify the forms φ under the unique hypothesis of maximality of the Witt index over K. This extends a recent result of Sobiech studying the hyperbolicity of nonsingular F-quadratic forms over K [17]. Based on our classifications, we are able to give necessary and sufficient conditions under which an anisotropic semisingular F-quadratic form has a given Witt index over K. We also study the quasi-hyperbolicity of semisingular F-quadratic forms over function fields of certain irreducible polynomials and extend to such forms many results established by the first author in [11].

This thesis is devoted to the central limit theorem (CLT) and its functional version for strictly stationary random fields.In the first part, we deal with strictly stationary random fields that are not necessarily adapted to the natural filtration, and when the asymptotic variance does not necessarily grow linearly. In this part, we give the CLT and its functional version for strictly stationary random fields under the normalization (sn)_{n inZ^d}, where (sn)_{n inZ^d} is a sequence of positive reals which tends to infinity when n tends to infinity.In the second part, we are interested in the extension of Gordin's CLT under a projective L^1 condition. We will see that for random fields indexed by Z^d with d greater than or equal to 2, an additional condition with respect to sequences (d = 1) is necessary in order to obtain the CLT.In the last part, we are interested in the asymptotic behavior of partial sums associated to random fields with values in Banach spaces. In Particular, we obtained the functional version of the CLT for fields of ortho-martingales differences that are ergodic and take values in a 2-smooth or cotype 2 Banach space. Then, using an ortho-martingale approximation, we derive a functional version of the CLT for strictly stationary random fields with values in a L^p space, p in [1,2], under projetive cirteria, Cette thèse est consacrée au théorème central limite (TCL) et à sa version fonctionnelle pour les champs aléatoires strictement stationnaires. Dans un premier temps, nous traiterons des champs aléatoires strictement stationnaires non nécessairement adaptés à la filtration naturelle, et lorsque la variance asymptotique ne croit pas nécessairement linéairement. Dans cette partie, nous donnons le TCL et sa version fonctionnelle pour les champs aléatoires strictement stationnaires sous la normalisation (sn)_{n dans Z^d}, où (sn)_{n dans Z^d} est une suites réels positifs qui tend vers infini quand n tend vers infini. Dans un deuxième temps, nous nous intéresserons à l'extension du TCL de Gordin sous une condition L^1 projective. On verra que dans le cadre des champs aléatoires indexés par Z^d avec d supérieur ou égal à 2, une condition supplémentaire par rapport au cadre des suites (d = 1) est nécessaire pour obtenir le TCL.Dans le dernier chapitre, nous nous intéresserons au comportement asymptotique des sommes partielles associées à des champs aléatoires à valeurs dans des espaces de Banach. En particulier nous établirons la version fonctionnelle du TCL pour les champs de différences d'ortho-martingales ergodiques et à valeurs dans un espace de Banach soit 2-lisse ou de cotype 2. Puis à l'aide d'une approximation ortho-martingale, nous obtiendrons une version fonctionnelle du TCL pour des champs aléatoires strictement stationnaires à valeurs dans un espace L^p, p dans [1,2], sous des critères projectifs

We construct a geometrico-symbolic version of the natural extension of the random $\beta $ -transformation introduced by Dajani and Kraaikamp [Random $\beta $ -expansions. Ergod. Th. & Dynam. Sys.23(2) (2003) 461–479]. This construction provides a new proof of the existence of a unique absolutely continuous invariant probability measure for the random $\beta $ -transformation, and an expression for its density. We then prove that this natural extension is a Bernoulli automorphism, generalizing to the random case the result of Smorodinsky [ $\beta $ -automorphisms are Bernoulli shifts. Acta Math. Acad. Sci. Hungar.24 (1973), 273–278] about the greedy transformation.

Let $S=(s_n)_{n\geq 1}$ be a sequence with elements in a commutative monoid $(\mathcal{M},+,0)$. In this paper, we provide an explicit formula for $$\sum_{\la} C(\la) q^{\sum_{n\geq 1} \la_n\cdot s_n}$$ where $\la=(\la_1,\ldots)$ run through some subsets of over-partitions, and $C(\la)$ is a certain product of ``colors'' assigned to the parts of $\la$, and $q^s$ is a formal power of $q$ for $s\in M$. This formula allows us not only to retrieve several known Schmidt-type theorems but also to provide new Schmidt-type theorems for non-periodic sequences $S$. For example, when $(M,+,0)=(\mathbb{Z}_{\geq 0},+,0)$, $s_n=1$ if there exists $i\geq 1$ such $n=\{i(i-1)/2+1\}$ and $s_n=0$ otherwise, we obtain the following statement: for all non-negative integer $m$, the number of partitions such that $\sum_{i\geq 1}\la_{i(i-1)/2+1} =m$ is equal to the number of plane partitions of $m$. Furthermore, we introduce a new family of partitions, the block partitions, generalizing the $k$-elongated partitions. From that family of partitions, we provide a generalization of a Schmidt-type theorem due to Andrews and Paule regarding $k$-elongated partitions and establish a link with the Eulerian polynomials., 18 pages. 2 figures

This thesis is devoted to the study of the lower existence bounds on the genus of hyperelliptic curves over a fixed finite field. More precisely, given an integer N, we ask: what is the minimum number G such that for any g bigger than G there is a genus g hyperelliptic curve having exactly N rational points?The first part of the manuscript deals with so called pointless curves, i.e. when N equals zero. Using some explicit constructions, for each characteristic we establish a new bound that depends linearly on the size of the ground finite field.In the second part one considers the general case. Here, when the characteristic is odd and N is strictly positive, a new quasilinear bound on the genera is obtained by means of implicit constructions.The third part completes the above theoretical results. We propose a new approach to the computer search of hyperelliptic curves over small finite fields of odd characteristic by exploiting the machinery of binary linear codes.This thesis was held under the direction of Professor Alexey Zykin at the University of French Polynesia (2015—2017)., Cette thèse est consacrée à l’étude des bornes inférieures sur le genre des courbes hyperelliptiques sur un corps fini. Plus précisément, étant donné un entier N, nous demandons : quel est le nombre minimum G tel que pour tout g plus grand que G il existe une courbe hyperelliptique de genre g ayantexactement N points rationnels ?La première partie du manuscrit traite des courbes sans points rationnels, c’est-à-dire lorsque N est égal à zéro. En utilisant certaines constructions explicites, nous établissons pour chaque caractéristique une nouvelle borne qui dépend linéairement de la taille du corps fini.Dans la deuxième partie, on considère le cas général. Ici, lorsque la caractéristique est impaire et que N est strictement positif, une nouvelle borne quasi-linéaire sur les genres est obtenue à l’aide de constructions implicites.La troisième partie complète les résultats théoriques ci-dessus. Nous proposons une nouvelle approche de la recherche informatique des courbes hyperelliptiques sur les petits corps finis de caractéristique impaire en exploitant la machinerie des codes linéaires binaires.Cette thèse a été réalisée sous la direction du Professeur Alexey Zykin à l’Université de Polynésie française (2015—2017).

We construct certain entire function (s) which for integer s coincides with the well-known Keiper-Li coe¢cients (see [7], [10]), i.e. (n) = n. This is an even function (s) = (s) and has an in…nitude of complex zeros exhibiting interesting distribution. Extensive tables of more than 3500 complex zeros of (s) with precision of 14 signi…cant digits are included. A detailed analysis of the distribution of these zeros may shed some light on the Riemann hypothesis. It turns out that possible violation of the Riemann hypothesis (if such is the case) would be clearly re ‡ected in the speci…c distribution of these zeros. More speci…cally, they form complex quadruplets if the Riemann hypothesis is true, and aligned real doublets if it is false.

Recent works of Andrews--Newman and Hopkins--Sellers unveil an interesting relation between two partition statistics, the crank and the mex. They state that, for a positive integer $n$, there are as many partitions of $n$ with non-negative crank as partitions of $n$ with odd mex. In this paper, we give a bijective proof of a generalization of this identity provided by Hopkins, Sellers, and Stanton. Our method uses an alternative definition of the Durfee decomposition, whose combinatorial link to the crank was recently studied by Hopkins, Sellers, and Yee., Comment: 12 pages

The analysis of problematic mathematical texts, particularly from India, has required the introduction of a new category of rigorous discourse, apodictic discourse. In this second part, we show that its introduction clarifies the approach to epistemic cultures. We also show that the notion of to the fantasy echo is relevant in Epistemology, as suggested by J.W.Scott. We then continue our earlier analysis of Brahmagupta’s Prop. 12.21-32 on the cyclic quadrilateral and identify discursive strategies that enable him to convey definitions, hypotheses and derivations encoded in the very structure of the propositions stating his new results. We also show that the statements of mathematical formulae in words also follow definite discursive patterns.

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 give, in this paper, all bi-unitary perfect polynomials over the prime field F 2 , with at most four irreducible factors.

The binary Goldbach conjecture asserts that every even integer greater than 4 is the sum of two primes. In this paper, we prove that there exists an integer Kα > 4 such that every even integer x > p 2 k can be expressed as the sum of two primes, where p k is the kth prime number and k > Kα. To prove this statement, we begin by introducing a type of double sieve of Eratosthenes as follows. Given a positive even integer x > 4, we sift from [1, x] all those elements that are congruents to 0 modulo p or congruents to x modulo p, where p is a prime less than √ x. Therefore, any integer in the interval [ √ x, x] that remains unsifted is a prime q for which either x − q = 1 or x − q is also a prime. Then, we introduce a new way of formulating a sieve, which we call the sequence of k-tuples of remainders. By means of this tool, we prove that there exists an integer Kα > 4 such that p k /2 is a lower bound for the sifting function of this sieve, for every even number x that satisfies p 2 k < x < p 2 k+1 , where k > Kα, which implies that x > p 2 k (k > Kα) can be expressed as the sum of two primes. In this version of the paper we give an alternative proof of the crucial lemma that allowed us to obtain this result.

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

The previous two parts of the paper (correspondingly, https://philpapers.org/rec/PENFLT-2 and https://philpapers.org/rec/PENFLT-3) demonstrate that the interpretation of Fermat's last theorem (FLT) in Hilbert arithmetic meant both in a narrow sense and in a wide sense can suggest a proof by induction in Part I and by means of the Kochen-Specker theorem in Part II. The same interpretation can serve also for a proof FLT based on Gleason's theorem and partly similar to that in Part II. The concept of (probabilistic) measure of a subspace of Hilbert space and especially its uniqueness can be unambiguously linked to that of partial algebra or incommensurability, or interpreted as a relation of the two dual branches of Hilbert arithmetic in a wide sense. The investigation of the last relation allows for FLT and Gleason's theorem to be equated in a sense, as two dual counterparts, and the former to be inferred from the latter, as well as vice versa under an additional condition relevant to the Gödel incompleteness of arithmetic to set theory. The qubit Hilbert space itself in turn can be interpreted by the unity of FLT and Gleason's theorem. The proof of such a fundamental result in number theory as FLT by means of Hilbert arithmetic in a wide sense can be generalized to an idea about "quantum number theory". It is able to research mathematically the origin of Peano arithmetic from Hilbert arithmetic by mediation of the "nonstandard bijection" and its two dual branches inherently linking it to information theory. Then, infinitesimal analysis and its revolutionary application to physics can be also re-realized in that wider context, for example, as an exploration of the way for physical quantity of time (respectively, for time derivative in any temporal process considered in physics) to appear at all. Finally, the result admits a philosophical reflection of how any hierarchy arises or changes itself only thanks to its dual and idempotent counterpart.