The aim of the present paper is to develop the theory of 𝒟-modules in positive characteristic. In Sections 2, 3, and 4, we study higher-level generalizations of differential modules in positive characteristic. These objects may be regarded as ring-theoretic counterparts of vector bundles on an algebraic curve equipped with an action of the ring of (logarithmic) differential operators of finite level introduced by P. Berthelot and C. Montagnon. The well-known existence assertion for a cyclic vector of a differential module is generalized to higher level. In Sections 5, 6, and 7, we introduce and discuss (dormant) opers of level 𝑁 > 0 on a pointed smooth curve whose structure group is either GL_𝑛 or PGL_𝑛. Some of the results in Sections 3 and 4 are applied to prove a duality theorem between dormant PGL_𝑛-opers of level 𝑁 and dormant PGL_{𝑝^𝑁−𝑛}-opers of level 𝑁. Finally, in the case where the underlying curve is a 3-pointed projective line, we establish a bijective correspondence between dormant PGL₂-opers of level 𝑁 and certain tamely ramified coverings. These assertions are building blocks to establish the enumerative geometry of higher-level dormant opers.

We introduce a refinement of bounded cohomology and prove that the suitable comparison homomorphisms vanish for an amenable group. We investigate in this context Thompson's group 𝐹 and provide further evidence towards its amenability. We show that the space of 1-bounded cocycles of degree two is essentially as big as the space of Lipschitz functions on the underlying group. We also explain that such classes define metrics on the group.

In this paper we study the Hilbert series of triangle groups Δ(𝑝, 𝑞, 𝑟). The 11 groups in this series are, conjecturally, the only cocompact triangle groups that admit matrix models over totally real fields.

We provide evidence for this conjecture, along with explicit integral models for every group in the Hilbert series. The most remarkable among them, Δ(14, 21, 42), is the only known triangle group with a split invariant quaternion algebra.

Using this special group, we construct the first example of a compact Kobayashi geodesic curve 𝑉 on a Hilbert modular variety (aside from those that reside on proper Shimura subvarieties). For comparison, there are no compact Kobayashi geodesic curves in the moduli space ℳ_𝑔.

For an algebraic Hecke character defined on a CM field 𝐹 of degree 2𝑑, Katz constructed a 𝑝-adic 𝐿-function of 𝑑 + 1 + 𝛿_𝐹,𝑝 variables in his innovative paper published in 1978, where 𝛿_𝐹,𝑝 denotes the Leopoldt defect for 𝐹 and 𝑝. We shall generalise the result of Katz under several technical conditions (containing the absolute unramifiedness of 𝐹 at 𝑝), and construct a 𝑝-adic Artin 𝐿-function of 𝑑 + 1 + 𝛿_𝐹,𝑝 variables, which interpolates critical values of the Artin 𝐿-function associated to a 𝑝-unramified Artin representation of the absolute Galois group 𝐺_𝐹 of 𝐹. Our construction is an analogue over a CM field of Greenberg's construction over a totally real field, but there appear new difficulties which do not matter in Greenberg's case.

Though Mahler equations have been introduced nearly one century ago, the study of their solutions is still a fruitful topic for research. In particular, the Galois theory of Mahler equations has been the subject of many recent papers. Nevertheless, long is the way to a complete understanding of relations between solutions of Mahler equations. One step along this way is the study of singularities. Mahler equations with a regular singularity at 0 have rather “nice” solutions: they can be expressed with the help of Puiseux series and solutions of equations with constant coefficients. In a previous paper, the authors described an algorithm to determine whether an equation is regular singular at 0 or not. Exploiting information from the Frobenius method and Newton polygons, we improve this algorithm by significantly reducing its complexity, by providing some simple criterion for an equation to be regular singular at 0, and by extending its scope to equations with Puiseux coefficients.

Let 𝐺 = exp 𝔤 be an exponential solvable Lie group with Lie algebra 𝔤 and 𝔤^* the dual vector space of 𝔤. We take two (real) polarizations 𝔥₁, 𝔥₂ of 𝔤 at 𝑓 ∈ 𝔤^* which satisfy the Pukanszky condition and define a unitary character 𝜒_𝑓 of 𝐻_𝑗 = exp(𝔥_𝑗) (𝑗 = 1, 2) by the formula 𝜒_𝑓(exp 𝑋) = 𝑒^{𝑖𝑓(𝑋)} (𝑋 ∈ 𝔥_𝑗). Then, it is well known that the two monomial representations 𝜋_𝑗 = ind^𝐺_𝐻𝑗 𝜒_𝑓 (𝑗 = 1, 2) are mutually equivalent and we even have a natural candidate of the intertwining operator between them. In order to verify that it is a true intertwining operator, the principal obstacle is the convergence of the integral in question. A property which assures this convergence, is the closedness of the simple product set 𝐻₂ 𝐻₁ in 𝐺. In this paper, we establish this property, generalizing then a previous proof in the particular case when one of the polarizations is of Vergne type.

The concept of concrete regularity structure gives the algebraic backbone of the operations involved in the local expansions used in the regularity structure approach to singular stochastic partial differential equations. The spaces and the details of the structures depend on each equation. We introduce here a parameter-dependent universal algebraic regularity structure that can host all the regularity structures used in the study of singular stochastic partial differential equations. This is done by using the correspondence between the notions of model on a regularity structure and the notion of paracontrolled system. We prove that the iterated paraproducts that form the fundamental bricks of paracontrolled systems have some local expansion properties that are governed by this universal structure.

Rate distortion dimension describes the theoretical limit of lossy data compression methods as the distortion bound goes to zero. It was originally introduced in the context of information theory, and recently it was discovered that it has an intimate connection to Gromov's theory of mean dimension of dynamical systems. This paper studies the behavior of rate distortion dimension of ℝ^𝑑-actions under ergodic decomposition. Our main theorems provide natural convexity and concavity of upper and lower rate distortion dimensions under convex combination of invariant probability measures. We also present examples which clarify the validity and limitations of the theorems.

We give an explicit dimension formula for paramodular forms of degree two of prime level with plus or minus sign of the Atkin–Lehner involution of weight det^𝑘 Sym(𝑗) with 𝑘 ≥ 3, as well as a dimension formula for algebraic modular forms of any weight associated with the binary quaternion hermitian maximal lattices in non-principal genus of prime discriminant with fixed sign of the involution. These two formulas are essentially equivalent by a recent result of N. Dummigan, A. Pacetti. G. Rama and G. Tornaría on correspondence between algebraic modular forms and paramodular forms with signs. So we give the formula by calculating the latter. When 𝑝 is odd, our formula for the latter is based on a class number formula of some quinary lattices by T. Asai and its interpretation to the type number of quaternion hermitian forms given in our previous works. On paramodular forms, we also give a dimensional bias between plus and minus eigenspaces, some list of palindromic Hilbert series, numerical examples for small 𝑝 and 𝑘, and the complete list of primes 𝑝 such that there is no paramodular cusp form of level 𝑝 of weight 3 with plus sign. This last result has geometric meaning on moduli of Kummer surface with (1, 𝑝) polarization.