The difference Galois theory of Mahler equations is an active research area. The present paper aims at developing the analytic aspects of this theory. We first attach a pair of connection matrices to any regular singular Mahler equation. We then show that these connection matrices can be used to produce a Zariski-dense subgroup of the difference Galois group of any regular singular Mahler equation.
We study the Mahler measure of the three-variable Laurent polynomial x + 1/x + y + 1/y + z + 1/z - k where k is a parameter. The zeros of this polynomial define (after desingularization) a family of K3-surfaces. In favorable cases, the K3-surface has Picard number 20, and the Mahler measure is related to its L-function. This was first studied by Marie-Jose Bertin. In this work, we prove several new formulas, extending the earlier work of Bertin.
In the frame of Mahler's method for algebraic independence we show that the algebraic relations over Q linking the values of functions solutions of a system of functional equations come from the algebraic relations between the functions themselves, by specialisation. We deduce some results on the linear independence of the q-regular functions.
We present the first example of a polynomial defining a singular K3-surface whose Mahler measure is expressed in terms of the Mahler measure of the faces of its Newton polyhedron and the L-series of the K3-surface.
This is the first part of a work devoted to the study of linear Mahler systems in several variables from the perspective of transcendence and algebraic independence. We prove two main results concerning systems that are regular singular at the origin. Our interest in Mahler's method comes from the possible applications of these results to old problems involving automata theory and which concern the expansion of both natural numbers and real numbers in integer bases. In particular, problems which involve finite automata and base change. Such applications are studied in Part II of this work.
Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $α$ be a root of $F(x, 1)$ and assume that the field extension $\mathbb Q(α)/\mathbb Q$ is Galois. We prove that, for every sufficiently large prime power $p^k$, the number of solutions to the Diophantine equation of Thue type $$ |F(x, y)| = tp^k $$ in integers $(x, y, t)$ such that $\gcd(x, y) = 1$ and $1 \leq t \leq (p^k)^λ$ does not exceed $24$. Here $λ= λ(d)$ is a certain positive, monotonously increasing function that approaches one as $d$ tends to infinity. We also prove that, for every su…
A total mass is the weighted count of continuous homomorphisms from the absolute Galois group of a local field to a finite group. In the preceding paper, the authors observed that in a particular example, two total masses coming from two different weightings are dual to each other. We discuss the problem how general such a duality holds and relate it to the existence of simultaneous resolution of singularities, using the wild McKay correspondence and the Poincaré duality for stringy invariants. We also exhibit several examples.
After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental i.e. they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift operator $x\mapsto x+h$ ($h\in\mathbb{C}^*$), the $q$-difference operator $x\mapsto qx$ ($q\in\mathbb{C}^*$ not a root of unity), and the Mahler operator $x\mapsto x^p$ ($p\geq 2$ integer). The only restrict…
We show all the possible structures of finite subgroups of the automorphism groups of elliptic curves. Here the automorphism means the biholomorphic transformation. Using the result, we determine every Galois group G at outer Galois point for genus-one curve with singular points. In particular, if G is abelian, then its order is at most nine. However if not so, the order is unbounded. Furthermore, we give every defining equation of the curve with the Galois point in the case where G is abelian.
We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental.
We study the middle convolution of local systems on the punctured affine line in the setting of singular cohomology and in the setting of étale cohomology. We derive a formula to compute the topological monodromy of the middle convolution in the general case and use it to deduce some irreducibility criteria. Then we give a geometric interpretation of the middle convolution in the étale setting. This geometric approach to the convolution and the theory of Hecke characters yields information on the occurring arithmetic determinants. We employ these methods to realize special linear groups regula…
Let $p$ be an irregular prime. Let $K=\Q(ζ)$ be the $p$-cyclotomic field. From Kummer and class field theory, there exist Galois extensions $S/\Q$ of degree $p(p-1)$ such that $S/K$ is a cyclic unramified extension of degree $[S:K]=p$. We give an algebraic construction of the subfields $M$ of $S$ with degree $[M:\Q]=p$ and an explicit formula for the prime decomposition and ramification of the prime number $p$ in the extensions $S/K$, $M/\Q$ and $S/M$. In the last section, we examine the consequences of these results for the Vandiver's conjecture. This article is at elementary level on Classic…
In his paper titled "Torsion points on Fermat Jacobians, roots of circular units and relative singular homology", Anderson determines the homology of the degree $n$ Fermat curve as a Galois module for the action of the absolute Galois group $G_{\mathbb{Q}(ζ_n)}$. In particular, when $n$ is an odd prime $p$, he shows that the action of $G_{\mathbb{Q}(ζ_p)}$ on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial $1-(1-x^p)^p$. If $p$ satisfies Vandiver's conjecture, we prove that the Galois group of this splitting field over $\mathbb{…
We establish a precise relation between Galois theory in its motivic form with the mathematical theory of perturbative renormalization (in the minimal subtraction scheme with dimensional regularization). We identify, through a Riemann-Hilbert correspondence based on the Birkhoff decomposition and the t'Hooft relations, a universal symmetry group (the "cosmic Galois group" suggested by Cartier), which contains the renormalization group and acts on the set of physical theories. This group is closely related to motivic Galois theory. We construct a universal singular frame of geometric nature, in…
Let p be an odd prime. Let K_p = Q(zeta) be the p-cyclotomic field. Let pi be the prime ideal of K_p lying over p. Let G be the Galois group of K_p. Let v be a primitive root mod p. Let sigma be a Q-isomorphism of K_p. Let P(sigma) = sigma^{p-2}v^{-(p-2)}+ ... + sigma v^{-1} +1 in Z[G], where v^n is understood (mod p). We apply Stickelberger relation to odd prime numbers q different of p and to some singular integers A of K_p connected with the p-class group C_p of K_p and prove the pi-adic congruences: 1) pi^{2p-1} | A^{P(σ)} if q = 1 (mod p), 2) pi^{2p-1} || A^{P(σ)} if q = 1 (mod p) and p^{…
We first study hyperplane sections of some singular schemes over a field. We prove a Bertini theorem for the log smoothness of generic hyperplane sections of a large class of log smooth schemes over a log point. We also give an abstract generalization of the usual Bertini theorem for smoothness. We then apply this result to the study of hyperplane sections of regular schemes over a complete discrete valuation ring of characteristic zero with algebraically closed residue field. Lastly, give an exposition of a result of Faltings which states that one can compute the $p$-adic nearby cycles of smo…
We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain `` motivic Galois group'', which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup. The group is obtained through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of th…
In this paper we construct a connection on the trivial G-bundle on the projective line for any simple complex algebraic group G, which is regular outside of the points 0 and infinity, has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point infinity, with slope 1/h, the reciprocal of the Coxeter number of G. This connection, which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic 0 counterpart of a hypothetical family of l-adic representations, which should parametrize a sp…
