Algebra & Number Theory最新文献

We give upper bounds on limit multiplicities of certain nontempered representations of unitary groups $U(a,b)$, conditionally on the endoscopic classification of representations. Our result applies to some cohomological representations, and we give applications to the growth of cohomology of cocompact arithmetic subgroups of unitary groups. The representations considered are transfers of products of characters and discrete series on endoscopic groups, and the bounds are obtained using Arthur’s stabilization of the trace formula and the classification established by Mok, and Kaletha, Minguez, Shin and White.

We show how to attach to any rigid analytic variety $V$ over a perfectoid space $P$ a rigid analytic motive over the Fargues–Fontaine curve $\mathcal{𝒳}(P)$ functorially in $V$ and $P$. We combine this construction with the overconvergent relative de Rham cohomology to produce a complex of solid quasicoherent sheaves over $\mathcal{𝒳}(P)$, and we show that its cohomology groups are vector bundles if $V$ is smooth and proper over $P$ or if $V$ is quasicompact and $P$ is a perfectoid field, thus proving and generalizing a conjecture of Scholze. The main ingredients of the proofs are explicit ${\mathbb{𝔹}}^1$-homotopies, the motivic proper base change and the formalism of solid quasicoherent sheaves.

We provide uniform estimates on the number of $\mathbb{Z}∕p^k\mathbb{Z}$-points lying on fibers of flat morphisms between smooth varieties whose fibers have rational singularities, termed (FRS) morphisms. For each individual fiber, the estimates were known by work of Avni and Aizenbud, but we render them uniform over all fibers. The proof technique for individual fibers is based on Hironaka’s resolution of singularities and Denef’s formula, but breaks down in the uniform case. Instead, we use recent results from the theory of motivic integration. Our estimates are moreover equivalent to the (FRS) property, just like in the absolute case by Avni and Aizenbud. In addition, we define new classes of morphisms, called $E$-smooth morphisms ($E�\in �\mathbb{N}$), which refine the (FRS) property, and use the methods we developed to provide uniform number-theoretic estimates as above for their fibers. Similar estimates are given for fibers of $𝜀$-jet flat morphisms, improving previous results by the last two authors.

We define and investigate algebraic torus actions on quiver Grassmannians for nilpotent representations of the equioriented cycle. Examples of such varieties are type $A$ flag varieties, their linear degenerations and finite-dimensional approximations of both the affine flag variety and affine Grassmannian for ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$. We show that these quiver Grassmannians equipped with our specific torus action are GKM-varieties and that their moment graph admits a combinatorial description in terms of the coefficient quiver of the underlying quiver representations. By adapting to our setting results by Gonzales, we are able to prove that moment graph techniques can be applied to construct module bases for the equivariant cohomology of the quiver Grassmannians listed above.

Let $G$ be a linear algebraic group over a field. We show that, under mild assumptions, in a family of primitive generically free $G$-varieties over a base variety $B$, the essential dimension of the geometric fibers may drop on a countable union of Zariski closed subsets of $B$ and stays constant away from this countable union. We give several applications of this result.

Let $K$ be a finite abelian CM-extension of a totally real field $k$ and $T$ a suitable finite set of finite primes of $k$. We determine the Fitting ideal of the minus component of the $T$-ray class group of $K$, except for the $2$-component, assuming the validity of the equivariant Tamagawa number conjecture. As an application, we give a necessary and sufficient condition for the Stickelberger element to lie in that Fitting ideal.

We prove that both the local and global ${\mathbb{𝔸}}^1$-degree of an endomorphism of affine space can be computed in terms of the multivariate Bézoutian. In particular, we show that the Bézoutian bilinear form, the Scheja–Storch form, and the ${\mathbb{𝔸}}^1$-degree for complete intersections are isomorphic. Our global theorem generalizes Cazanave’s theorem in the univariate case, and our local theorem generalizes Kass–Wickelgren’s theorem on EKL forms and the local degree. This result provides an algebraic formula for local and global degrees in motivic homotopy theory.

We study the algebraic dynamics of self-correspondences on a curve. A self-correspondence on a (proper and smooth) curve $C$ over an algebraically closed field is the data of another curve $D$ and two nonconstant separable morphisms ${\pi }_1$ and ${\pi }_2$ from $D$ to $C$. A subset $S$ of $C$ is complete if ${\pi }_1^{-1}(S)�={\pi }_2^{-1}(S)$. We show that self-correspondences are divided into two classes: those that have only finitely many finite complete sets, and those for which $C$ is a union of finite complete sets. The latter ones are called finitary, and happen only when $\mathrm{deg}<!--FUNCTION\; APPLICATION-->{\pi }_1�=\mathrm{deg}<!--FUNCTION\; APPLICATION-->{\pi }_2$ and have a trivial dynamics. For a nonfinitary self-correspondence in characteristic zero, we give a sharp bound for the number of étale finite complete sets.

We give explicit descriptions of rings of differential operators of toric face rings in characteristic $0$. For quotients of normal affine semigroup rings by radical monomial ideals, we also identify which of their differential operators are induced by differential operators on the ambient ring. Lastly, we provide a criterion for the Gorenstein property of a normal affine semigroup ring in terms of its differential operators.

Our main technique is to realize the k-algebras we study in terms of a suitable family of their algebra retracts in a way that is compatible with the characterization of differential operators. This strategy allows us to describe differential operators of any k-algebra realized by retracts in terms of the differential operators on these retracts, without restriction on $\mathrm{char}<!--FUNCTION\; APPLICATION-->(k)$.

For an irreducible complex reflection group $W$ of rank $n$ containing $N$ reflections, we put $g�=2N∕n$ and construct a ${(g�+1)}^n$-dimensional irreducible representation of the Cherednik algebra which is (as a vector space) a quotient of the diagonal coinvariant ring of $W$. We propose that this representation of the Cherednik algebra is the single largest representation bearing this relationship to the diagonal coinvariant ring, and that further corrections to this estimate of the dimension of the diagonal coinvariant ring by ${(g�+1)}^n$ should be orders of magnitude smaller. A crucial ingredient in the construction is the existence of a dot action of a certain product of symmetric groups (the Namikawa–Weyl group) acting on the parameter space of the rational Cherednik algebra and leaving invariant both the finite Hecke algebra and the spherical subalgebra; this fact is a consequence of ideas of Berest and Chalykh on the relationship between the Cherednik algebra and quasiinvariants.