American Journal of Mathematics最新文献

abstract:

Let ${Fgn}$ be the moduli space of $n$-pointed $K3$ surfaces of genus $g$ with at worst rational double points. We establish an isomorphism between the ring of pluricanonical forms on ${Fgn}$ and the ring of certain orthogonal modular forms, and give applications to the birational type of ${Fgn}$. We prove that the Kodaira dimension of ${Fgn}$ stabilizes to $19$ when $n$ is sufficiently large. Then we use explicit Borcherds products to find a lower bound of $n$ where ${Fgn}$ has nonnegative Kodaira dimension, and compare this with an upper bound where ${Fgn}$ is unirational or uniruled using Mukai models of $K3$ surfaces in $gleq 20$. This reveals the exact transition point of Kodaira dimension in some~$g$.

abstract:

We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, from certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.

abstract:

In this paper we give a complete and explicit description of the Howe correspondence of unipotent characters for a finite reductive dual pair of a symplectic group and an even orthogonal group in terms of the Lusztig parametrization. That is, the conjecture by Aubert-Michel-Rouquier is confirmed.

abstract:

The aim of this work is to study homogeneous stable solutions to the thin (or fractional) one-phase free boundary problem.

The problem of classifying stable (or minimal) homogeneous solutions in dimensions $ngeq3$ is completely open. In this context, axially symmetric solutions are expected to play the same role as Simons' cone in the classical theory of minimal surfaces, but even in this simpler case the problem is open.

The goal of this paper is twofold. On the one hand, our first main contribution is to find, for the first time, the stability condition for the thin one-phase problem. Quite surprisingly, this requires the use of ``large solutions'' for the fractional Laplacian, which blow up on the free boundary.

On the other hand, using our new stability condition, we show that any axially symmetric homogeneous stable solution in dimensions $nle 5$ is one-dimensional, emph{independently} of the parameter $sin (0,1)$.

abstract:

For any fixed relatively prime positive integers $a$, $b$ and $c$ with $min{a,b,c}>1$, we prove that the equation $a^x+b^y=c^z$ has at most two solutions in positive integers $x$, $y$ and $z$, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat's equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M.~A. Bennett, On some exponential equations of S. S. Pillai, Canad. J. Math. 53 (2001), no.~2, 897--922] which asserts that Pillai's type equation $a^x-b^y=c$ has at most two solutions in positive integers $x$ and $y$ for any fixed positive integers $a$, $b$ and $c$ with $min{a,b}>1$.

abstract:

We construct $p$-adic $L$-functions for torsion classes for $GL(n+1)timesGL(n)$ and along the way prove new congruences between special values of Rankin-Selberg $L$-functions for $GL(n+1)timesGL(n)$ over arbitrary number fields. This allows us to control the behavior of $p$-adic $L$-functions under Tate twists and to prove the existence of non-abelian $p$-adic $L$-functions for Hida families on $GL(n!+!1)linebreaktimesGL(n)$. As an application, we establish generic non-vanishing results for central $L$-values: We give sufficient local conditions for twisted central Rankin-Selberg $L$-values to be generically non-zero.

abstract:

We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision $varkappa$, the expansion holds with an $mathrm{O}(N^{-varkappa-1})$ error in $N$-dependent neighborhoods of the exterior region as the degree $N$ tends to infinity. The main ingredient is the derivation and analysis of Riemann-Hilbert hierarchies---sequences of scalar Riemann-Hilbert problems---which allows us to express all higher order correction terms in closed form. Indeed, the expansion may be understood as a Neumann series involving an explicit operator. The expansion theorem leads to a semiclassical asymptotic expansion of the corresponding hard edge probability wave function in terms of distributions supported on $partialmathscr{D}$.

abstract:

In this article we provide the first examples of property (T) $mathrm{II}_1$ factors $mathcal{N}$ with trivial fundamental group, $mathcal{F}(mathcal{N})=1$. Our examples arise as group factors $mathcal{N}=mathcal{L}(G)$ where $G$ belong to two distinct families of property (T) groups previously studied in the literature: the groups introduced by Valette in [Geom. Dedicata 112 (2005), 183--196] and the ones introduced recently in [Anal. PDE 16 (2023), 433--476] using the Belegradek-Osin Rips construction from [Groups Geom. Dyn. 2 (2008), 1--12]. In particular, our results provide a continuum of explicit pairwise nonisomorphic property (T) factors.

abstract:

We generalize the theory of linked partition ideals due to Andrews using finite automata in formal language theory and apply it to prove three Rogers--Ramanujan type identities for modulus 14 that were posed by Nandi through a vertex operator theoretic construction of the level 4 standard modules of the affine Lie algebra $A^{(2)}_{2}$.

abstract:

We study random walks on a $d$-dimensional torus by affine expanding maps whose linear parts commute. Assuming an irrationality condition on their translation parts, we prove that the Haar measure is the unique stationary measure. We deduce that if $Ksubsetmathbb{R}^d$ is an attractor of a finite iterated function system of $ngeq 2$ maps of the form $xmapsto D^{-1}x+t_i$ ($i=1,dotsc,n$), where $D$ is an expanding $dtimes d$ integer matrix, and is the same for all the maps, under an irrationality condition on the translation parts $t_i$, almost every point in $K$ (w.r.t. any Bernoulli measure) has an equidistributed orbit under the map $xmapsto Dx$ (multiplication mod $mathbb{Z}^d$). In the one-dimensional case, this conclusion amounts to normality to base $D$. Thus for example, almost every point in an irrational dilation of the middle-thirds Cantor set is normal to base $3$.