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We give some basics about homological algebra of difference representations. We consider both the difference discrete and the difference rational case. We define the corresponding cohomology theories and show the existence of spectral sequences relating these cohomology theories with the standard ones.
Using the Kuznetsov formula, we prove several density theorems for exceptional Hecke and Laplacian eigenvalues of Maaß cusp forms of weight or for the congruence subgroups , , and . These improve and extend upon results of Sarnak and Huxley, who prove similar but slightly weaker results via the Selberg trace formula.
We examine the set of -orbits in the set of irreducible components of affine Deligne–Lusztig varieties for a hyperspecial subgroup and minuscule coweight . Our description implies in particular that its number of elements is bounded by the dimension of a suitable weight space in the Weyl module associated with of the dual group.
For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Moreover, if the map is an automorphism, there exists a Zariski dense set of such points with pairwise disjoint orbits.
We prove a version of weakly functorial big Cohen–Macaulay algebras that suffices to establish Hochster and Huneke’s vanishing conjecture for maps of Tor in mixed characteristic. As a corollary, we prove an analog of Boutot’s theorem that direct summands of regular rings are pseudorational in mixed characteristic. Our proof uses perfectoid spaces and is inspired by the recent breakthroughs on the direct summand conjecture by André and Bhatt.
Let be an inner form of a general linear group over a nonarchimedean locally compact field of residue characteristic , let be an algebraically closed field of characteristic different from and let be the category of smooth representations of over . In this paper, we prove that a block (indecomposable summand) of is equivalent to a level- block (a block in which every simple object has nonzero invariant vectors for the pro--radical of a maximal compact open subgroup) of , where is a direct product of groups of the same type of .
We study the algebraic dynamics of endomorphisms of projective spaces with coefficients in a -adic field whose reduction in positive characteristic is the Frobenius. In particular, we prove a version of the dynamical Manin–Mumford conjecture and the dynamical Mordell–Lang conjecture for the coherent backward orbits of such endomorphisms. We also give a new proof of a dynamical version of the Tate–Voloch conjecture in this case. Our method is based on the theory of perfectoid spaces introduced by P. Scholze. In the appendix, we prove that under some technical condition on the field of definition, a dynamical system for a polarized lift of Frobenius on a projective variety can be embedded into a dynamical system for some endomorphism of a projective space.
Let be polynomials of degree such that no is conjugate to or to , where is the Chebyshev polynomial of degree . We let be their coordinatewise action on , i.e., is given by . We prove a dynamical version of the Pink–Zilber conjecture for subvarieties of with respect to the dynamical system , if .
A pseudolength function defined on an arbitrary group is a map obeying , the symmetry property , and the triangle inequality for all . We consider pseudolength functions which saturate the triangle inequality whenever , or equivalently those that are homogeneous in the sense that for all . We show that this implies that for all . This leads to a classification of such pseudolength functions as pullbacks from embeddings into a Banach space. We also obtain a quantitative version of our main result which allows for defects in the triangle inequality or the homogeneity property.
We prove that the polynomial invariants of a permutation group are Cohen–Macaulay for any choice of coefficient field if and only if the group is generated by transpositions, double transpositions, and 3-cycles. This unites and generalizes several previously known results. The “if” direction of the argument uses Stanley–Reisner theory and a recent result of Christian Lange in orbifold theory. The “only if” direction uses a local-global result based on a theorem of Raynaud to reduce the problem to an analysis of inertia groups, and a combinatorial argument to identify inertia groups that obstruct Cohen–Macaulayness.
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