Mathematische Annalen最新文献

Let $\Pi$ be a regular algebraic cuspidal automorphic representation (RACAR) of ${\text{GL}}_3\left(A_Q\right)$ . When $\Pi$ is p-nearly-ordinary for the maximal standard parabolic with Levi ${\text{GL}}_1\times {\text{GL}}_2$ , we construct a p-adic L-function for $\Pi$ . More precisely, we construct a (single) bounded measure $L_p\left(\Pi \right)$ on $Z_p^{\times }$ attached to $\Pi$ , and show it interpolates all the critical values $L(\Pi \times \eta ,-j)$ at p in the left-half of the critical strip for $\Pi$ (for varying $\eta$ and j). This proves conjectures of Coates-Perrin-Riou and Panchishkin in this case. We also prove a corresponding result in the right half of the critical strip, assuming near-ordinarity for the other maximal standard parabolic. Our construction uses the theory of spherical varieties to build a "Betti Euler system", a norm-compatible system of classes in the Betti cohomology of a locally symmetric space for ${\text{GL}}_3$ . We work in arbitrary cohomological weight, allow arbitrary ramification at p along the Levi factor of the standard parabolic, and make no self-duality assumption. We thus give the first constructions of p-adic L-functions for RACARs of ${\text{GL}}_n\left(A_Q\right)$ of 'general type' (i.e. those that do not arise as functorial lifts) for any $n>2$ .

We examine the distribution of zeroes of half-integral weight Hecke cusp forms on the manifold ${\Gamma }_0\left(4\right)H$ near a cusp at infinity. In analogue of the Ghosh-Sarnak conjecture for classical holomorphic Hecke cusp forms, one expects that almost all of the zeroes sufficiently close to this cusp lie on two vertical geodesics $\text{Re}\left(s\right)=-1/2$ and $\text{Re}\left(s\right)=0$ as the weight tends to infinity. We show that, for ${\gg }_{\epsilon }{K}^2/{(logK)}^{3/2+\epsilon }$ of the half-integral weight Hecke cusp forms in the Kohnen plus subspaces with weight bounded by a large parameter K, the number of such "real" zeroes grows almost at the expected rate. We also obtain a weaker lower bound for the number of real zeroes that holds for a positive proportion of forms. One of the key ingredients is the estimation of averaged first and second moments of quadratic twists of modular L-functions.

In this contribution we introduce a novel weak solution concept for two-phase volume-preserving mean curvature flow, having both properties of unconditional global-in-time existence and weak-strong uniqueness. These solutions extend the ones proposed by Hensel and Laux (J Differ Geom 130:209-268, 2025) for the standard mean curvature flow, and consist in evolving varifolds coupled with the phase volumes by a transport equation. First, we show that, in the same setting as in Takasao (Arch Ration Mech Anal 247:52, 2023), any sharp interface limit of solutions to a slightly modified nonlocal Allen-Cahn equation is a varifold solution according to our new definition. Secondly, we crucially introduce a new notion of volume-preserving gradient-flow calibrations, allowing the extended velocity vector field to point in the normal direction on the interface. We show that any sufficiently regular strong solution is calibrated in this sense. Finally, we prove that any classical solution to volume-preserving mean curvature flow, which is then automatically a calibrated flow, is unique in the class of our new varifold solutions.

In this paper, we investigate discrete regularity estimates for a broad class of temporal numerical schemes for parabolic stochastic evolution equations. We provide a characterization of discrete stochastic maximal ${\ell }^p$ -regularity in terms of its continuous counterpart, thereby establishing a unified framework that yields numerous new discrete regularity results. Moreover, as a consequence of the continuous-time theory, we establish several important properties of discrete stochastic maximal regularity such as extrapolation in the exponent p and with respect to a power weight. Furthermore, employing the $H^{\infty }$ -functional calculus, we derive a powerful discrete maximal estimate in the trace space norm $D_A(1-\frac{1}{p},p)$ for $p\in [2,\infty )$ .

We study the one-phase Alt-Phillips free boundary problem, focusing on the case of negative exponents $\gamma \in (-2,0)$ . The goal of this paper is twofold. On the one hand, we prove smoothness of $C^{1,\alpha }$ -regular free boundaries by reducing the problem to a class of degenerate quasilinear PDEs, for which we establish Schauder estimates. Such a method provides a unified proof of the smoothness for general exponents. On the other hand, by exploiting the higher regularity of solutions, we derive a new stability condition for the Alt-Phillips problem in the negative exponent regime, ruling out the existence of nontrivial axially symmetric stable cones in low dimensions. Finally, we provide a variational criterion for the stability of cones in the Alt-Phillips problem, which recovers the one for minimal surfaces in the singular limit as $\gamma \to -2$ .

We prove that Boolean matrices with bounded ${\gamma }_2$ -norm or bounded normalized trace norm must contain a linear-sized all-ones or all-zeros submatrix, verifying a conjecture of Hambardzumyan, Hatami, and Hatami. We also present further structural results about Boolean matrices of bounded ${\gamma }_2$ -norm and discuss applications in communication complexity, operator theory, spectral graph theory, and extremal combinatorics. As a key application, we establish an inverse theorem for MaxCut. A celebrated result of Edwards states that every graph G with m edges has a cut of size at least $\frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}$ , with equality achieved by complete graphs with an odd number of vertices. To contrast this, we prove that if the MaxCut of G is at most $\frac{m}{2}+O\left(\sqrt{m}\right)$ , then G must contain a clique of size $\Omega \left(\sqrt{m}\right)$ .

We prove that single G-weighted $b$ -Hurwitz numbers with internal faces are computed by refined topological recursion on a rational spectral curve, for certain rational weights G. Consequently, the $b$ -Hurwitz generating function analytically continues to a rational curve. In particular, our results cover the cases of $b$ -monotone Hurwitz numbers, and the enumeration of maps and bipartite maps (with internal faces) on non-oriented surfaces. As an application, we prove that the correlators of the Gaussian, Jacobi and Laguerre $\beta$ -ensembles are computed by refined topological recursion.

We give a general formula for generators of the NL cone on an orthogonal modular variety. This is the cone of effective divisors linearly equivalent to an effective linear combination of irreducible components of Noether-Lefschetz divisors. We apply this to describe, in terms of minimal generators, the NL cone of various moduli spaces of geometric origin such as those of polarized K3 surfaces, cubic fourfolds, and hyperkähler manifolds. Additionally, we establish uniruledness for many moduli spaces of primitively polarized hyperkähler manifolds of $\text{OG6}$ and ${\text{Kum}}_n$ -type. Finally, in analogy with the case of K3 surfaces of degree 2, we show that any family of polarized ${\text{Kum}}_2$ -type hyperkähler manifolds with divisibility 2 and polarization degree 2 over a projective base is isotrivial.

We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest k eigenvalues of the Ricci tensor. If $(M^n,g)$ is a Riemannian manifold satisfying such curvature bounds for $k=2$ , then we show that M is contained in a neighbourhood of controlled width of an isometrically embedded 1-dimensional sub-manifold. From this, we deduce several metric and topological consequences: M has at most linear volume growth and at most two ends, it has bounded 1-Urysohn width, the first Betti number of M is bounded above by 1, and there is precise information on elements of infinite order in ${\pi }_1\left(M\right)$ . If $(M^n,g)$ is a Riemannian manifold satisfying such bounds for $k\ge 2$ , then we show that M has at most $(k-1)$ -dimensional behavior at large scales. If $k=n=\text{dim}\left(M\right)$ , so that the integral lower bound is on the scalar curvature, assuming in addition that the $(n-2)$ -Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to $n-2$ . From the above results we deduce topological restrictions, such as upper bounds on the first Betti number.

We study the shifted convolution problem for the divisor function in function fields in the large degree limit, that is, the average value of $d\left(f\right)d(f+h)$ where f runs over monic polynomials in $F_q\left[T\right]$ of a given degree, and h is a given monic polynomial. We prove an asymptotic formula in the range $deg\left(h\right)<(2-ϵ)deg\left(f\right)$ . We also consider mixed correlations and self-correlations of $r_{\chi }=1\star \chi$ , the convolution of 1 with a Dirichlet character mod $\ell$ , where $\ell$ is a monic irreducible polynomial, proving asymptotic formulae in various ranges. This includes the case of quadratic characters, which yields results about correlations of norm-counting functions of quadratic extensions of $F_q\left[T\right]$ . A novel feature of our work is a Voronoi summation formula (equivalently, a functional equation for the Estermann function) in $F_q\left[T\right]$ which was not previously available.