Algebra & Number Theory最新文献

For a $d$-dimensional regular proper variety $X$ over the function field of a smooth variety $B$ over a field $k$ and for $i�\ge 0$, we define a subgroup ${\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^i{(X)}^{(0)}$ of ${\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^i(X)$ and construct a “refined height pairing”

in the category of abelian groups up to isogeny. For $i�=1,d$, ${\mathrm{CH}<!--FUNCTION\; APPLICATION-->}^i{(X)}^{(0)}$ is the group of cycles numerically equivalent to $0$. This pairing relates to pairings defined by P. Schneider and A. Beilinson if $B$ is a curve, to a refined height defined by

We study the conjugacy classes of the classical affine groups. We derive generating functions for the number of classes analogous to formulas of Wall and the authors for the classical groups. We use these to get good upper bounds for the number of classes. These naturally come up as difficult cases in the study of the noncoprime $k(GV�)$ problem of Brauer.

The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category $C$ to $C$ extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer spectra. We give conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to prove that Balmer spectra associated to finite-dimensional cosemisimple quasitriangular Hopf algebras (in particular, group algebras in characteristic dividing the order of the group) coincide with the Balmer spectra associated to their Drinfeld doubles, and that the thick ideals of both categories are in bijection. An analogous theorem is proven for certain Benson–Witherspoon smash coproduct Hopf algebras, which are not quasitriangular in general.

We develop a version of Sen theory for equivariant vector bundles on the Fargues–Fontaine curve. We show that every equivariant vector bundle canonically descends to a locally analytic vector bundle. A comparison with the theory of $(\phi ,\Gamma )$-modules in the cyclotomic case then recovers the Cherbonnier–Colmez decompletion theorem. Next, we focus on the subcategory of de Rham locally analytic vector bundles. Using the $p$-adic monodromy theorem, we show that each locally analytic vector bundle $\mathcal{ℰ}$ has a canonical differential equation for which the space of solutions has full rank. As a consequence, $\mathcal{ℰ}$ and its sheaf of solutions $\mathrm{Sol}<!--FUNCTION\; APPLICATION-->(\mathcal{ℰ})$ are in a natural correspondence, which gives a geometric interpretation of a result of Berger on $(\phi ,\Gamma )$-modules. In particular, if $V$ is a de Rham Galois representation, its associated filtered $(\phi ,N,G_K)$-module is realized as the space of global solutions to the differential equation. A key to our approach is a vanishing result for the higher locally analytic vectors of representations satisfying the Tate–Sen formalism, which is also of independent interest.

By using results on poles of $L$-functions and theta correspondence, we give a bound on $b$ for $(\chi ,b)$-factors of the global Arthur parameter of a cuspidal automorphic representation $\pi$ of a classical group or a metaplectic group where $\chi$ is a conjugate self-dual automorphic character and $b$ is an integer which is the dimension of an irreducible representation of ${\mathrm{SL}<!--FUNCTION\; APPLICATION-->}_2(ℂ)$. We derive a more precise relation when $\pi$ lies in a generic global $A$-packet.

This paper is an extension of Kim et al. (2020a), and we prove equidistribution theorems for families of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur’s invariant trace formula in terms of Shintani zeta functions in a uniform way. Several applications, including the vertical Sato–Tate theorem and low-lying zeros for standard $L$-functions of holomorphic Siegel cusp forms, are discussed. We also show that the “nongenuine forms”, which come from nontrivial endoscopic contributions by Langlands functoriality classified by Arthur, are negligible.

We prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge type at primes of good reduction. We make use of the global Serre–Tate coordinates of Chai as well as recent results of D’Addezio about the monodromy groups of isocrystals. The new ingredients in this paper are a general monodromy theorem for Hecke-stable subvarieties for Shimura varieties of Hodge type, and a rigidity result for the formal completions of ordinary Hecke orbits. Along the way, we show that classical Serre–Tate coordinates can be described using unipotent formal groups, generalising a result of Howe.

The equation $x^m�=0$ defines a fat point on a line. The algebra of regular functions on the arc space of this scheme is the quotient of $k[x,x^{\prime },x^{(2)},\dots <!--FUNCTION\; APPLICATION-->]$ by all differential consequences of $x^m�=0$. This infinite-dimensional algebra admits a natural filtration by finite-dimensional algebras corresponding to the truncations of arcs. We show that the generating series for their dimensions equals $m∕(1�-�mt)$. We also determine the lexicographic initial ideal of the defining ideal of the arc space. These results are motivated by the nonreduced version of the geometric motivic Poincaré series, multiplicities in differential algebra, and connections between arc spaces and the Rogers–Ramanujan identities. We also prove a recent conjecture put forth by Afsharijoo in the latter context.

We give a streamlined and effective proof of Ozaki’s theorem that any finite $p$-group $\Gamma$ is the Galois group of the $p$-Hilbert class field tower of some number field $F<!--FUNCTION\; APPLICATION-->$. Our work is inspired by Ozaki’s and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field ${k<!--FUNCTION\; APPLICATION-->}_0$ with class number prime to $p$. We construct $F<!--FUNCTION\; APPLICATION-->∕{k<!--FUNCTION\; APPLICATION-->}_0$ by a sequence of $\mathbb{Z}∕p$-extensions ramified only at finite tame primes and also give explicit bounds on $[F<!--FUNCTION\; APPLICATION-->�:{k<!--FUNCTION\; APPLICATION-->}_0]$ and the number of ramified primes of $F<!--FUNCTION\; APPLICATION-->∕{k<!--FUNCTION\; APPLICATION-->}_0$ in terms of $\#\Gamma$.

We calculate certain “wide moments” of central values of Rankin–Selberg $L$-functions $L\left(\pi �\otimes \mathrm{\Omega },\frac{1}{2}\right)$ where $\pi$ is a cuspidal automorphic representation of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_2$ over $\mathbb{Q}$ and $\mathrm{\Omega }$ is a Hecke character (of conductor $1$) of an imaginary quadratic field. This moment calculation is applied to obtain “weak simultaneous” nonvanishing results, which are nonvanishing results for different Rankin–Selberg $L$-functions where the product of the twists is trivial.

The proof relies on relating the wide moments of $L$-functions to the usual moments of automorphic forms evaluated at Heegner points using Waldspurger’s formula. To achieve this, a classical version of Waldspurger’s formula for general weight automorphic forms is derived, which might be of independent interest. A key input is equidistribution of Heegner points (with explicit error terms), together with nonvanishing results for certain period integrals. In particular, we develop a soft technique for obtaining the nonvanishing of triple convolution $L$-functions.