Let be a field of characteristic . In a previous paper of ours, we defined and studied super-Hölder vectors in certain -linear representations of . In the present paper, we define and study super-Hölder vectors in certain -linear representations of a general -adic Lie group. We then consider certain -adic Lie extensions of a -adic field , and compute the super-Hölder vectors in the tilt of . We show that these super-Hölder vectors are the perfection of the field of norms of . By specializing to the case of a Lubin–Tate extension, we are able to recover inside the -adic completion of its perfection, seen as a valued -vector space endowed with the action of given by the endomorphisms of the corresponding Lubin–Tate group.
We study the variation of linear sections of hypersurfaces in . We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperplane sections of any smooth degree hypersurface in varies maximally for . In the process, we generalize the classical Grauert–Mülich theorem about lines in projective space, both to -planes in projective space and to free rational curves on arbitrary varieties.
We describe separating -invariants of several copies of the algebra of octonions over an algebraically closed field of characteristic two. We also obtain a minimal separating and a minimal generating set for -invariants of several copies of the algebra of octonions in case of a field of odd characteristic.
We study a family of exponential sums that arises in the study of expanding horospheres on . We prove an explicit version of general purity and find optimal bounds for these sums.
We construct scattering diagrams for Chekhov–Shapiro generalized cluster algebras where exchange polynomials are factorized into binomials, generalizing the cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. They turn out to be natural objects arising in Fock and Goncharov’s cluster duality. Analogous features and structures (such as positivity and the cluster complex structure) in the ordinary case also appear in the generalized situation. With the help of these scattering diagrams, we show that generalized cluster variables are theta functions and hence have certain positivity property with respect to the coefficients in the binomial factors.
We show the image of rooted tree maps forms a subspace of the kernel of the evaluation map of multiple -values. To prove this, we define the diamond product as a modified harmonic product and describe its properties. We also show that -conjugate rooted tree maps are their antipodes.
We investigate the distribution of the maximum of character sums over the family of primitive quadratic characters attached to fundamental discriminants . In particular, our work improves results of Montgomery and Vaughan, and gives strong evidence that the Omega result of Bateman and Chowla for quadratic character sums is optimal. We also obtain similar results for real characters with prime discriminants up to , and deduce the interesting consequence that almost all primes with large Legendre symbol sums are congruent to modulo . Our results are motivated by a recent work of Bober, Goldmakher, Granville and Koukoulopoulos, who proved similar results for the family of nonprincipal characters modulo a large prime. However, their method does not seem to generalize to other families of Dirichlet characters. Instead, we use a different and more streamlined approach, which relies mainly on the quadratic large sieve. As an application, we consider a question of Montgomery concerning the positivity of sums of Legendre symbols.
The local structure of terminal Brauer classes on arithmetic surfaces was classified (2021), generalising the classification on geometric surfaces (2005). Part of the interest in these classifications is that it enables the minimal model program to be applied to the noncommutative setting of orders on surfaces. We give étale local structure theorems for terminal orders on arithmetic surfaces, at least when the degree is a prime . This generalises the structure theorem given in the geometric case. They can all be explicitly constructed as algebras of matrices over symbols. From this description one sees that such terminal orders all have global dimension two, thus generalising the fact that terminal (commutative) surfaces are smooth and hence homologically regular.
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