Algebra & Number Theory最新文献

We prove that the Galois equidistribution of torsion points of the algebraic torus ${\mathbb{𝔾}}_m^d$ extends to the singular test functions of the form $\log <!--FUNCTION\; APPLICATION-->\left|P\right|$, where $P$ is a Laurent polynomial having algebraic coefficients that vanishes on the unit real $d$-torus in a set whose Zariski closure in ${\mathbb{𝔾}}_m^d$ has codimension at least $2$. Our result includes a power-saving quantitative estimate of the decay rate of the equidistribution. It refines an ergodic theorem of Lind, Schmidt, and Verbitskiy, of which it also supplies a purely Diophantine proof. As an application, we confirm Ih’s integrality finiteness conjecture on torsion points for a class of atoral divisors of ${\mathbb{𝔾}}_m^d$.

Given a smooth projective toric variety $X$ of Picard rank 2, we resolve the diagonal sheaf on $X�\times �X$ by a linear complex of length $\dim <!--FUNCTION\; APPLICATION-->X$ consisting of finite direct sums of line bundles. As applications, we prove a new case of a conjecture of Berkesch, Ermana and Smith that predicts a version of Hilbert’s syzygy theorem for virtual resolutions, and we obtain a Horrocks-type splitting criterion for vector bundles over smooth projective toric varieties of Picard rank 2, extending a result of Eisenbud, Erman and Schreyer. We also apply our results to give a new proof, in the case of smooth projective toric varieties of Picard rank 2, of a conjecture of Orlov concerning the Rouquier dimension of derived categories.

We explain how logarithmic structures select principal components in an intersection of schemes. These manifest in Chow homology and can be understood using strict transforms under logarithmic blowups. Our motivation comes from Gromov–Witten theory. The toric contact cycles in the moduli space of curves parameterize curves that admit a map to a fixed toric variety with prescribed contact orders. We show that they are intersections of virtual strict transforms of double ramification cycles in blowups of the moduli space of curves. We supply a calculation scheme for the virtual strict transforms, and deduce that toric contact cycles lie in the tautological ring of the moduli space of curves. This is a higher-dimensional analogue of a result of Faber and Pandharipande. The operational Chow rings of Artin fans play a basic role, and are shown to be isomorphic to rings of piecewise polynomials on associated cone complexes. The ingredients in our analysis are Fulton’s blowup formula, Aluffi’s formulas for Segre classes of monomial schemes, piecewise polynomials, and degeneration methods. A model calculation in toric intersection theory is treated without logarithmic methods and may be read independently.

Spectral moment formulae of various shapes have proven very successful in studying the statistics of central $L$-values. We establish, in a completely explicit fashion, such formulae for the family of $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(3)�\times \mathrm{GL}<!--FUNCTION\; APPLICATION-->(2)$ Rankin–Selberg $L$-functions using the period integral method. Our argument does not rely on either the Kuznetsov or Voronoi formulae. We also prove the essential analytic properties and derive explicit formulae for the integral transform of our moment formulae. We hope that our method will provide deeper insights into moments of $L$-functions for higher-rank groups.

We associate with a generalised deep hole of the Leech lattice vertex operator algebra a generalised hole diagram. We show that this Dynkin diagram determines the generalised deep hole up to conjugacy and that there are exactly $70$ such diagrams. In an earlier work we proved a bijection between the generalised deep holes and the strongly rational, holomorphic vertex operator algebras of central charge $24$ with nontrivial weight-$1$ space. Hence, we obtain a new, geometric classification of these vertex operator algebras, generalising the classification of the Niemeier lattices by their hole diagrams.

We prove that the double (or canonical unramified) wavefront set of an irreducible depth-0 supercuspidal representation of a reductive $p$-adic group is a singleton provided $p�>3(h�-1)$, where $h$ is the Coxeter number. We deduce that the geometric wavefront set is also a singleton in this case, proving a conjecture of Mœglin and Waldspurger. When the group is inner to split and the representation belongs to Lusztig’s category of unipotent representations, we give an explicit formula for the double and geometric wavefront sets. As a consequence, we show that the nilpotent part of the Deligne–Langlands–Lusztig parameter of a unipotent supercuspidal representation is precisely the image of its geometric wavefront set under Spaltenstein’s duality map.

We study affine Deligne–Lusztig varieties $X_{w(b)}$ when the finite part of the element $w$ in the Iwahori–Weyl group is a partial $\sigma$-Coxeter element. We show that such $w$ is a cordial element and $X_{w(b)}\ne \varnothing$ if and only if $b$ satisfies a certain Hodge–Newton indecomposability condition. Our main result is that for such $w$ and $b$, $X_{w(b)}$ has a simple geometric structure: the $\sigma$-centralizer of $b$ acts transitively on the set of irreducible components of $X_{w(b)}$; and each irreducible component is an iterated fibration over a classical Deligne–Lusztig variety of Coxeter type, and the iterated fibers are either ${\mathbb{𝔸}}^1$ or ${\mathbb{𝔾}}_m$.

We realize two families of combinatorial symmetric functions via the complex character theory of the finite general linear group ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n({\mathbb{𝔽}}_q)$: chromatic quasisymmetric functions and vertical strip LLT polynomials. The associated ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n({\mathbb{𝔽}}_q)$ characters are elementary in nature and can be obtained by induction from certain well-behaved characters of the unipotent upper triangular groups ${\mathrm{UT}<!--FUNCTION\; APPLICATION-->}_n({\mathbb{𝔽}}_q)$. The proof of these results also gives a general Hopf algebraic approach to computing the induction map. Additional results include a connection between the relevant ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n({\mathbb{𝔽}}_q)$ characters and Hessenberg varieties and a reinterpretation of known theorems and conjectures about the relevant symmetric functions in terms of ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n({\mathbb{𝔽}}_q)$.

We generalize an inequality for the determinant of a real matrix proved by A. Schinzel, to more general exterior products of vectors in Euclidean space. We apply this inequality to the logarithmic embedding of $S$-units contained in a number field $k$. This leads to a bound for the exterior product of $S$-units expressed as a product of heights. Using a volume formula of P. McMullen we show that our inequality is sharp up to a constant that depends only on the rank of the $S$-unit group but not on the field $k$. Our inequality is related to a conjecture of F. Rodriguez Villegas.

We prove an asymptotic formula for primes of the shape $f(a,b^2)$ with $a$, $b$ integers and of the shape $f(a,p^2)$ with $p$ prime. Here $f$ is a binary quadratic form with integer coefficients, irreducible over $\mathbb{Q}$ and has no local obstructions. This refines the seminal work of Friedlander and Iwaniec on primes of the form $x^2�+y^4$ and of Heath-Brown and Li on primes of the form $a^2�+p^4$, as well as earlier work of the author with Lam and Schindler on primes of the form $f(a,p)$ with $f$ a positive definite form.