Algebra & Number Theory最新文献

Much recent literature concerns finiteness properties of infinite-dimensional algebraic varieties equipped with an action of the infinite symmetric group, or of the infinite general linear group. In this paper, we study a common generalisation in which the product of both groups acts on infinite-dimensional spaces, and we show that these spaces are topologically Noetherian with respect to this action.

It is conjectured that the canonical models of varieties (not of general type) are bounded when the Iitaka volume is fixed. We confirm this conjecture when a general fiber of the corresponding Iitaka fibration is in a fixed bounded family of polarized log Calabi–Yau pairs.

We combine the split torsor method and the hyperbola method for toric varieties to count rational points and Campana points of bounded height on certain subvarieties of toric varieties.

Orthogonality is a fundamental theme in representation theory and Fourier analysis. An orthogonality relation for characters of finite abelian groups (now recognized as an orthogonality relation on $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(1)$) was used by Dirichlet to prove infinitely many primes in arithmetic progressions. Asymptotic orthogonality relations for $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(n)$, with $n�\le 3$, and applications to number theory, have been considered by various researchers over the last 45 years. Recently, the authors of the present work have derived an explicit asymptotic orthogonality relation, with a power savings error term, for $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(4,�ℝ)$. Here we extend those results to $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(n,�ℝ)$, $n�\ge 2$.

For $n�\le 5$, our results are contingent on the Ramanujan conjecture at the infinite place, but otherwise are unconditional. In particular, the case $n�=5$ represents a new result. The key new ingredient for the proof of the case $n�=5$ is the theorem of Kim and Shahidi that functorial products of cusp forms on $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(2)�\times \mathrm{GL}<!--FUNCTION\; APPLICATION-->(3)$ are automorphic on $\mathrm{GL}<!--FUNCTION\; APPLICATION-->(6)$. For $n�>5$ (assuming again the Raman

We study algebraic relations among postcritically finite (PCF) parameters in the family $f_c(z)�=z^2�+�c$. It is known that an algebraic curve in ${ℂ}^2$ contains infinitely many PCF pairs $(c_1,c_2)$ if and only if the curve is special (i.e., the curve is a vertical or horizontal line through a PCF parameter, or the curve is the diagonal). Here we extend this result to subvarieties of arbitrary dimension in ${ℂ}^n$ for any $n�\ge 2$. Consequently, we obtain uniform bounds on the number of PCF pairs on nonspecial curves in ${ℂ}^2$ and the number of PCF parameters in real algebraic curves in $ℂ$, depending only on the degree of the curve. We also compute the optimal bound for the general curve of degree $d$. For $d�=1$, we prove that there are only finitely many nonspecial lines in ${ℂ}^2$ containing more than two PCF pairs, and similarly, that there are only finitely many (real) lines in $ℂ�={ℝ}^2$ containing more than two PCF parameters.

Prokhorov and Shokurov introduced the effective adjunction conjecture, also known as the effective basepoint-freeness conjecture, which asserts that the moduli component of an lc-trivial fibration is effectively basepoint-free. Li proposed a variation of this conjecture, known as the $\Gamma$-effective adjunction conjecture, and demonstrated that a weaker version of his conjecture follows from the original Prokhorov–Shokurov conjecture.

In this paper, we prove the equivalence between Prokhorov–Shokurov’s and Li’s effective adjunction conjectures. The key to our proof is establishing uniform rational polytopes for canonical bundle formulas. This relies on recent advancements in the minimal model program theory of algebraically integrable foliations, primarily developed by Ambro–Cascini–Shokurov–Spicer and Chen–Han–Liu–Xie.

Given a family of varieties over the projective line, we study the density of fibres that are everywhere locally soluble in the case that components of higher multiplicity are allowed. We use log geometry to formulate a new sparsity criterion for the existence of everywhere locally soluble fibres and formulate new conjectures that generalise previous work of Loughran and Smeets. These conjectures involve geometric invariants of the associated multiplicity orbifolds on the base of the fibration in the spirit of Campana. We give evidence for the conjectures by providing an assortment of bounds using Chebotarev’s theorem and sieve methods, with most of the evidence involving upper bounds.

We introduce a new language to describe the geometry of affine Deligne–Lusztig varieties in affine flag varieties. This second part of a two-paper series uses this new language, i.e., the double Bruhat graph, to describe certain structure constants of the Iwahori–Hecke algebra. As an application, we describe nonemptiness and dimension of affine Deligne–Lusztig varieties for most elements of the affine Weyl group and arbitrary $\sigma$-conjugacy classes.

The aim of this paper is to study distributional properties of integers without large or small prime factors. Define an integer to be $[y^{\prime },y]$-smooth if all of its prime factors belong to the interval $[y^{\prime },y]$. We identify suitable weights $g_{[y^{\prime },y]}(n)$ for the characteristic function of $[y^{\prime },y]$-smooth numbers that allow us to establish strong asymptotic results on their distribution in short arithmetic progressions. Building on these equidistribution properties, we show that (a $W$-tricked version of) the function $g_{[y^{\prime },y]}(n)�-1$ is orthogonal to nilsequences. Our results apply in the almost optimal range ${(\log <!--FUNCTION\; APPLICATION-->N)}^K�<�y�\le �N$ of the smoothness parameter $y$, where $K�\ge 2$ is sufficiently large, and to any $y^{\prime }�<\min <!--FUNCTION\; APPLICATION-->(\sqrt{y},{(\log <!--FUNCTION\; APPLICATION-->N)}^c)$.

As a first application, we e

We introduce a new language to describe the geometry of affine Deligne–Lusztig varieties in affine flag varieties. This first part of a two-paper series develops the definition and fundamental properties of the double Bruhat graph by studying semi-infinite orbits. This double Bruhat graph was originally introduced by Naito and Watanabe to study periodic $R$-polynomials. We use it to describe the geometry of many affine Deligne–Lusztig varieties, overcoming a previously ubiquitous regularity condition.