We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic. Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial $p(n)$ with at least one irrational coefficient (except for the constant one) and integer $m\geq 2$, the sequence $\lfloor p(n) \rfloor \bmod{m}$ is never automatic. We also prove that the conjecture is equivalent to the claim that the set of powers of an integer $k\geq 2$ is not given by a generalised polynomial.
We prove a number of p-adic congruences for the coefficients of powers of a multivariate polynomial f(x) with coefficients in a ring R of characteristic zero. If the Hasse--Witt operation is invertible, our congruences yield p-adic limit formulas which conjecturally describe the Gauss--Manin connection and the Frobenius operator on the unit-root crystal attached to f(x). As a second application, we associate with f(x) formal group laws over R. Under certain assumptions these formal group laws are coordinalizations of the Artin--Mazur functors. (This is a final version which we send for a publication.)
We study the distribution of 2-Selmer ranks in the family of quadratic twists of an elliptic curve E over an arbitrary number field K. Under the assumption that Gal(K(E[2])/K) = S_3 we show that the density (counted in a non-standard way) of twists with Selmer rank r exists for all positive integers r, and is given via an equilibrium distribution, depending only on a single parameter (the `disparity'), of a certain Markov process that is itself independent of E and K. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual F_p-representations of the absolute Galois group of K by characters of order p.
This monograph is intended to be considered as my habilitation (D.Sc.) thesis; because of that and as everything has already appeared in English, it is performed exclusively in Russian. The monograph comprises a detailed introduction and seven chapters that represent part of my work influenced by Apéry's proof from 1978 of the irrationality of $ζ(2)$ and $ζ(3)$, the values of Riemann's zeta function. Chapter 1 is about "at least one of the four numbers $ζ(5)$, $ζ(7)$, $ζ(9)$ and $ζ(11)$ is irrational" (based in part on arXiv:math.NT/0206176). Chapter 2 explains a connection between the generalized multiple integrals introduced by Beukers in his proof of Apéry's result and the very-well-poised hypergeometric series; it is based on arXiv:math.CA/0206177. Chapter 3 surveys some arithmetic and hypergeometric $q$-analogies and establishes the irrationality measure $μ(ζ_q(2))<3.518876$ for a $q$-analogue of $ζ(2)$; it closely follows the text in Sb. Math. 193 (2002), 1151--1172, but also incorporates the sharper analysis of the hypergeometric construction by Smet and Van Assche (arXiv:0809.2501 [math.CA]) to produce the improvement upon the 2002 result. Chapter 4 is devoted to the measure $μ(ζ(2))<5.095412$ and is based on arXiv:1310.1526 [math.NT]; Chapter 5 is establishing the estimate $||(3/2)^k||>0.5803^k$ for the distance from $(3/2)^k$ to the nearest integer, with the English version published in J. Théor. Nombres Bordeaux 19 (2007), 313--325. Chapter 6 reproduces the solution (from arXiv:math.CA/0311195) to the problem of Asmus Schmidt about generalized Apéry's numbers. Finally, Chapter 7 is about expressing the special $L$-values as periods (in the sense of Kontsevich and Zagier), in particular, as values of hypergeometric functions; it is based on the publication in Springer Proc. Math. Stat. 43 (2013), 381--395.
We study a modular function $Λ_{k,\ell}$ which is one of generalized $λ$ functions. We show $Λ_{k,\ell}$ and the modular invariant function $j$ generate the modular function field with respect to the modular subgroup $Γ_1(N)$. Further we prove that $Λ_{k,\ell}$ is integral over $\mathbf Z[j]$. From these results, we obtain that the value of $Λ_{k,\ell}$ at an imaginary quadratic point is an algebraic integer and generates a ray class field over the Hilbert class field.
Let k be a global field, p an odd prime number different from char(k) and S, T disjoint, finite sets of primes of k. Let G_S^T(k)(p)=Gal(k_S^T(p)|k) be the Galois group of the maximal p-extension of k which is unramified outside S and completely split at T. We prove the existence of a finite set of primes S_0, which can be chosen disjoint from any given set M of Dirichlet density zero, such that the cohomology of G_{S\cup S_0}^T(k)(p) coincides with the etale cohomology of the associated marked arithmetic curve. In particular, cd G_{S\cup S_0}^T(k)(p)=2. Furthermore, we can choose S_0 in such a way that k_{S\cup S_0}^T(p) realizes the maximal p-extension k_\p(p) of the local field k_\p for all \p\in S\cup S_0, the cup-product H^1(G_{S\cup S_0}^T(k)(p),\F_p) \otimes H^1(G_{S\cup S_0}^T(k)(p),\F_p) --> H^2(G_{S\cup S_0}^T(k)(p),\F_p) is surjective and the decomposition groups of the primes in S establish a free product inside G_{S\cup S_0}^T(k)(p). This generalizes previous work of the author where similar results were shown in the case T=\emptyset under the restrictive assumption p\nmid Cl(k) and ζ_p\notin k.
We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z=1 case is an identity for the generating function for p-cores due to Klyachko [17] and Garvan, Kim and Stanton [10]. A number of other infinite products are shown to have nonnegative coefficients. In the process a new generalization of the quintuple product identity is derived.
We compute by a purely local method the elliptic, twisted by transpose-inverse, character χ_πof the representation π=I_{(3,1)}(1_3) of PGL(4,F) normalizedly induced from the trivial representation of the maximal parabolic subgroup of type (3,1), where F is a p-adic field. Put C=(GL(2,F)xGL(2,F))'/F^x (F^x embeds diagonally, prime means equal determinants). It is a twisted elliptic endoscopic group of PGL(4). We deduce from the computation that χ_πis an unstable function: its value at one twisted regular elliptic conjugacy class with norm in C is minus its value at the other class within the twisted stable conjugacy class, and zero at the classes without norm in C. Moreover πis the unstable endoscopic lift of the trivial representation of C. Naturally, this computation plays a role in the theory of lifting from C (=``SO(4,F)'') and PGp(2,F) to PGL(4,F) using the trace formula. Our work develops a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced by the first author with Kazhdan in a 3-dimensional case, and it uses the classification of twisted stable and unstable regular conjugacy classes in PGL(4,F). It extends the local method of computation introduced by us in the 3-dimensional case. An extension math.NT/0606263 of our work here to apply to similar representations of GL(4,F) whose central character is nontrivial will appear in Int. J. Number Theory.
There are errors in the proof of the uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments.
We construct relative PEL type embeddings in mixed characteristic (0,2) between hermitian orthogonal Shimura varieties of PEL type. We use this to prove the existence of integral canonical models in unramified mixed characteristic (0,2) of hermitian orthogonal Shimura varieties of PEL type.
Let $F$ be a global function field of characteristic $p>0$, $\mathcal F/F$ a Galois extension with $Gal(\tilde F/F)\simeq \mathbb{Z}_p^{\mathbb N}$ and $E/F$ a non-isotrivial elliptic curve. We study the behaviour of Selmer groups $Sel_E(L)_l$ ($l$ any prime) as $L$ varies through the subextensions of $\mathcal F$ via appropriate versions of Mazur's Control Theorem. In the case $l=p$ we let $\mathcal F=\bigcup \mathcal F_d$ where $\mathcal F_d/F$ is a $\mathbb{Z}_p^d$-extension. With a mild hypothesis on $Sel_E(F)_p$ (essentially a consequence of the Birch and Swinnerton-Dyer conjecture) we prove that $Sel_E(\mathcal F_d)_p$ is a cofinitely generated (in some cases cotorsion) $\mathbb{Z}_p[[Gal(\mathcal F_d/F)]]$-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic $L$-function associated to $E$ in $\mathbb{Z}_p[[Gal(\mathcal F/F)]]$, providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.
Viewing higher local fields as ring objects in the category of iterated pro-ind-objects, a definition of open subgroups in Milnor K-groups of the fields is given. The self-duality of the additive group of a higher local field is proved. By studying norm groups of cohomological objects and using cohomological approach to higher local class field theory the existence theorem is proved.
For an arbitrary field p-torsion and cotorsion of the Milnor groups K_n(F) and K_n^{t}(F)=K_n(F)/\cap_{l\ge1} lK_n(F) are discussed. The work contains further discussions of an analogue of Satz 90 for K_n(F) and K_n^{t}(F) and computation of H^{n+1}_m(F) where F is either the rational function field in one variable F=k(t) or the formal power series F=k((t)).