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1. -ADIC -FUNCTIONS FOR UNITARY GROUPS

   https://doi.org/10.1017/fmp.2020.4  

   EISCHEN, ELLEN; HARRIS, MICHAEL; LI, JIANSHU; SKINNER, CHRISTOPHER (January 2020, Forum of Mathematics, Pi)

   This paper completes the construction of $$p$$ -adic $$L$$ -functions for unitary groups. More precisely, in Harris, Li and Skinner [‘ $$p$$ -adic $$L$$ -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math. Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $$p$$ -adic $$L$$ -functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and $$p$$ -adic differential operators [Eischen, ‘A $$p$$ -adic Eisenstein measure for unitary groups’, J. Reine Angew. Math. 699 (2015), 111–142; ‘ $$p$$ -adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble) 62 (1) (2012), 177–243], Part II of the present paper provides the calculations of local $$\unicode[STIX]{x1D701}$$ -integrals occurring in the Euler product (including at $$p$$ ). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method. 

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2. A unipotent circle action on 𝑝-adic modular forms

   https://doi.org/10.1090/btran/52  

   Howe, Sean (November 2020, Transactions of the American Mathematical Society, Series B)

   null (Ed.)

   Following a suggestion of Peter Scholze, we construct an action of G m ^ \widehat {\mathbb {G}_m} on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p p -adic modular curve whose ring of functions is Serre’s space of p p -adic modular functions. This action is a local, p p -adic analog of a global, archimedean action of the circle group S 1 S^1 on the lattice-unstable locus of the modular curve over C \mathbb {C} . To construct the G m ^ \widehat {\mathbb {G}_m} -action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates q q ; along the way we also prove a natural generalization of Dwork’s equation τ = log ⁡ q \tau =\log q for extensions of Q p / Z p \mathbb {Q}_p/\mathbb {Z}_p by μ p ∞ \mu _{p^\infty } valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of G m ^ \widehat {\mathbb {G}_m} integrates the differential operator θ \theta coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p p -adic L L -functions. 

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3. Bounded Geometry and p-Harmonic Functions Under Uniformization and Hyperbolization

   https://doi.org/10.1007/s12220-020-00477-0  

   Björn, Anders; Björn, Jana; Shanmugalingam, Nageswari (May 2021, The Journal of Geometric Analysis)

   null (Ed.)

   Abstract The uniformization and hyperbolization transformations formulated by Bonk et al. in “Uniformizing Gromov Hyperbolic Spaces” , Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local p -Poincaré inequality, then the transformed measure is globally doubling and supports a global p -Poincaré inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly local p -Poincaré inequality, carries nonconstant globally defined p -harmonic functions with finite p -energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain. 

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4. Square Packings and Rectifiable Doubling Measures

   Badger, Matthew; Schul, Raanan (May 2025, Discrete analysis)

   We prove that for all integers 2≤m≤d−1, there exists doubling measures on ℝd with full support that are m-rectifiable and purely (m−1)-unrectifiable in the sense of Federer (i.e. without assuming μ≪m). The corresponding result for 1-rectifiable measures is originally due to Garnett, Killip, and Schul (2010). Our construction of higher-dimensional Lipschitz images is informed by a simple observation about square packing in the plane: N axis-parallel squares of side length s pack inside of a square of side length ⌈N1/2⌉s. The approach is robust and when combined with standard metric geometry techniques allows for constructions in complete Ahlfors regular metric spaces. One consequence of the main theorem is that for each m∈{2,3,4} and s0, f(E) has Hausdorff dimension s, and μ(f(E))>0. This is striking, because m(f(E))=0 for every Lipschitz map f:E⊂ℝm→ℍ1 by a theorem of Ambrosio and Kirchheim (2000). Another application of the square packing construction is that every compact metric space 𝕏 of Assouad dimension strictly less than m is a Lipschitz image of a compact set E⊂[0,1]m. Of independent interest, we record the existence of doubling measures on complete Ahlfors regular metric spaces with prescribed lower and upper Hausdorff and packing dimensions. 

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5. Universal differentiability sets in Laakso space

   https://doi.org/10.1016/j.na.2025.113752  

   Eriksson-Bique, Sylvester; Pinamonti, Andrea; Speight, Gareth (June 2025, Nonlinear Analysis)

   We show that there exists a family of mutually singular doubling measures on Laakso space with respect to which real-valued Lipschitz functions are almost everywhere differentiable. This implies that there exists a measure zero universal differentiability set in Laakso space. Additionally, we show that each of the measures constructed supports a Poincare inequality. 

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