Mathematische Annalen最新文献

Given a prime power q and $n\gg 1$ , we prove that every integer in a large subinterval of the Hasse-Weil interval $[{(\sqrt{q}-1)}^{2n},{(\sqrt{q}+1)}^{2n}]$ is $\#A\left(F_q\right)$ for some ordinary geometrically simple principally polarized abelian variety A of dimension n over $F_q$ . As a consequence, we generalize a result of Howe and Kedlaya for $F_2$ to show that for each prime power q, every sufficiently large positive integer is realizable, i.e., $\#A\left(F_q\right)$ for some abelian variety A over $F_q$ . Our result also improves upon the best known constructions of sequences of simple abelian varieties with point counts towards the extremes of the Hasse-Weil interval. A separate argument determines, for fixed n, the largest subinterval of the Hasse-Weil interval consisting of realizable integers, asymptotically as $q\to \infty$ ; this gives an asymptotically optimal improvement of a 1998 theorem of DiPippo and Howe. Our methods are effective: We prove that if $q\le 5$ , then every positive integer is realizable, and for arbitrary q, every positive integer $\ge q^{3\sqrt{q}logq}$ is realizable.

In the synthetic geometric setting introduced by Kunzinger and Sämann, we present an analogue of Toponogov's Globalisation Theorem which applies to Lorentzian length spaces with lower (timelike) curvature bounds. Our approach utilises a "cat's cradle" construction akin to that which appears in several proofs in the metric setting. On the road to our main result, we also provide a lemma regarding the subdivision of triangles in spaces with a local lower curvature bound and a synthetic Lorentzian version of the Lebesgue Number Lemma. Several properties of time functions and the null distance on globally hyperbolic Lorentzian length spaces are also highlighted. We conclude by presenting several applications of our results, including versions of the Bonnet-Myers Theorem and the Splitting Theorem for Lorentzian length spaces with local lower curvature bounds, as well as discussion of stability of curvature bounds under Gromov-Hausdorff convergence.

For an irreducible smooth representation of a connected reductive p-adic group, two important associated invariants are the wavefront set and the (partly conjectural) Langlands parameter. While a wavefront set consists of p-adic nilpotent orbits, one constituent of the Langlands parameter is a complex nilpotent orbit in the dual Lie algebra. For unipotent representations in the sense of Lusztig, the corresponding nilpotent orbits on the two sides are related via the Lusztig-Spaltenstein duality (Ciubotaru et al. in Am J Math arXiv:2112.14354v4, J Reine Angew Math (Crelles J) 823:191-253, 2025). In this paper, we formulate a general upper-bound conjecture and several variants relating the nilpotent orbits that appear in the wavefront set and in the Langlands parameter. We also verify these expectations in some cases, including the depth-zero supercuspidal representations of classical groups and all the irreducible representations of $G_2$ .

Random walk spaces are a general framework for the study of PDEs. They include as particular cases locally finite weighted connected graphs and nonlocal settings involving symmetric integrable kernels on $R^N$ . We are interested in the study of evolution problems involving two random walk structures so that the associated functionals have different growth on each structure. We also deal with the case of a functional with different growth on a partition of the random walk.

We introduce the notion of a regular integrable connection on a smooth log scheme over $C$ and construct an equivalence between the category of such connections and the category of integrable connections on its analytification, compatible with de Rham cohomology. This extends the work of Deligne (when the log structure is trivial), and combined with the work of Ogus yields a topological description of the category of regular connections in terms of certain constructible sheaves on the Kato-Nakayama space. The key ingredients are the notion of a canonical extension in this context and the existence of good compactifications of log schemes obtained recently by Włodarczyk.

In 1965, Erdős and Pósa proved that there is an (approximate) duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold for odd cycles, and Dejter and Neumann-Lara asked in 1988 to find all pairs  $(\ell ,z)$ of integers where such a duality holds for the family of cycles of length  $\ell$ modulo z. We characterise all such pairs, and we further generalise this characterisation to cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group. This unifies almost all known types of cycles that admit such a duality, and it also provides new results. Moreover, we characterise the obstructions to such a duality in this setting, and thereby obtain an analogous characterisation for cycles in graphs embeddable on a fixed compact orientable surface.

We study holomorphic maps F from a smooth Levi non-degenerate real hypersurface $M_{\ell }\subset C^n$ into a hyperquadric $H_{{\ell }^{\text{'}}}^N$ with signatures $\ell \le (n-1)/2$ and ${\ell }^{\text{'}}\le (N-1)/2,$ respectively. Assuming that $N-n<n-1,$ we prove that if $\ell ={\ell }^{\text{'}},$ then F is either CR transversal to $H_{\ell }^N$ at every point of $M_{\ell },$ or it maps a neighborhood of $M_{\ell }$ in $C^n$ into $H_{\ell }^N.$ Furthermore, in the case where ${\ell }^{\text{'}}>\ell ,$ we show that if F is not CR transversal at $0\in M_{\ell },$ then it must be transversally flat. The latter is best possible.

We present an adjunction formula for foliations on varieties and we consider applications of the adjunction formula to the cone theorem for rank one foliations and the study of foliation singularities.

Assuming a modular version of Schanuel's conjecture and the modular Zilber-Pink conjecture, we show that the existence of generic solutions of certain families of equations involving the modular j function can be reduced to the problem of finding a Zariski dense set of solutions. By imposing some conditions on the field of definition of the variety, we are also able to obtain versions of this result without relying on these conjectures, and even a result including the derivatives of j.

We develop a theory of quasisymmetries for finitely ramified fractals, with applications to finitely ramified Julia sets. We prove that certain finitely ramified fractals admit a naturally defined class of "undistorted metrics" that are all quasisymmetrically equivalent. As a result, piecewise-defined homeomorphisms of such a fractal that locally preserve the cell structure are quasisymmetries. This immediately gives a solution to the quasisymmetric uniformization problem for topologically rigid fractals such as the Sierpiński triangle. We show that our theory applies to many finitely ramified Julia sets, and we prove that any connected Julia set for a hyperbolic unicritical polynomial has infinitely many quasisymmetries, generalizing a result of Lyubich and Merenkov. We also prove that the quasisymmetry group of the Julia set for the rational function [Formula: see text] is infinite, and we show that the quasisymmetry groups for the Julia sets of a broad class of polynomials contain Thompson's group F.