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1. Ordinary modules for vertex algebras of 𝔬𝔰𝔭 1|2𝑛

   https://doi.org/10.1515/crelle-2024-0060  

   Creutzig, Thomas ; Genra, Naoki ; Linshaw, Andrew ( August 2024 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   We show that the affine vertex superalgebra$V^k(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n})$V^{k}(\mathfrak{osp}_{1|2n})at generic level 𝑘 embeds in the equivariant 𝒲-algebra of$\mathfrak{s}{\mathfrak{p}}_{2n}$\mathfrak{sp}_{2n}times$4n$4nfree fermions.This has two corollaries:(1) it provides a new proof that, for generic 𝑘, the coset$\mathrm{Com}(V^k(\mathfrak{s}{\mathfrak{p}}_{2n}),V^k(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n}))$\operatorname{Com}(V^{k}(\mathfrak{sp}_{2n}),V^{k}(\mathfrak{osp}_{1|2n}))is isomorphic to${\mathcal{W}}^{\mathrm{\ell }}(\mathfrak{s}{\mathfrak{p}}_{2n})$\mathcal{W}^{\ell}(\mathfrak{sp}_{2n})for$\mathrm{\ell }=-(n+1)+(k+n+1)/(2k+2n+1)$\ell=-(n+1)+(k+n+1)/(2k+2n+1), and(2) we obtain the decomposition of ordinary$V^k(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n})$V^{k}(\mathfrak{osp}_{1|2n})-modules into$V^k(\mathfrak{s}{\mathfrak{p}}_{2n})\otimes {\mathcal{W}}^{\mathrm{\ell }}(\mathfrak{s}{\mathfrak{p}}_{2n})$V^{k}(\mathfrak{sp}_{2n})\otimes\mathcal{W}^{\ell}(\mathfrak{sp}_{2n})-modules.Next, if 𝑘 is an admissible level and ℓ is a non-degenerate admissible level for$\mathfrak{s}{\mathfrak{p}}_{2n}$\mathfrak{sp}_{2n}, we show that the simple algebra$L_k(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n})$L_{k}(\mathfrak{osp}_{1|2n})is an extension of the simple subalgebra$L_k(\mathfrak{s}{\mathfrak{p}}_{2n})\otimes {\mathcal{W}}_{\mathrm{\ell }}(\mathfrak{s}{\mathfrak{p}}_{2n})$L_{k}(\mathfrak{sp}_{2n})\otimes{\mathcal{W}}_{\ell}(\mathfrak{sp}_{2n}).Using the theory of vertex superalgebra extensions, we prove that the category of ordinary$L_k(\mathfrak{o}\mathfrak{s}{\mathfrak{p}}_{1|2n})$L_{k}(\mathfrak{osp}_{1|2n})-modules is a semisimple, rigid vertex tensor supercategory with only finitely many inequivalent simple objects.It is equivalent to a certain subcategory of${\mathcal{W}}_{\mathrm{\ell }}(\mathfrak{s}{\mathfrak{p}}_{2n})$\mathcal{W}_{\ell}(\mathfrak{sp}_{2n})-modules.A similar result also holds for the category of Ramond twisted modules.Due to a recent theorem of Robert McRae, we get as a corollary that categories of ordinary$L_k(\mathfrak{s}{\mathfrak{p}}_{2n})$L_{k}(\mathfrak{sp}_{2n})-modules are rigid.

    

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2. Jacobian determinants for nonlinear gradient of planar ∞-harmonic functions and applications

   https://doi.org/10.1515/crelle-2024-0016  

   Dong, Hongjie ; Peng, Fa ; Zhang, Yi Ru-Ya ; Zhou, Yuan ( April 2024 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   We introduce a distributional Jacobian determinant$detD{V}_{\beta }(Dv)$\det DV_{\beta}(Dv)in dimension two for the nonlinear complex gradient$V_{\beta }(Dv)={|Dv|}^{\beta }(v_{x_1},-v_{x_2})$V_{\beta}(Dv)=\lvert Dv\rvert^{\beta}(v_{x_{1}},-v_{x_{2}})for any$\beta >-1$\beta>-1, whenever$v\in W_{\mathrm{loc}}^{1,2}$v\in W^{1\smash{,}2}_{\mathrm{loc}}and$\beta {|Dv|}^{1+\beta }\in W_{\mathrm{loc}}^{1,2}$\beta\lvert Dv\rvert^{1+\beta}\in W^{1\smash{,}2}_{\mathrm{loc}}.This is new when$\beta \ne 0$\beta\neq 0.Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant$detD{V}_{\beta }(Du)$\det DV_{\beta}(Du)is a nonnegative Radon measure with some quantitative local lower and upper bounds.We also give the following two applications.

   Applying this result with$\beta =0$\beta=0, we develop an approach to build up a Liouville theorem, which improves that of Savin.Precisely, if 𝑢 is an ∞-harmonic function in the whole${\mathbb{R}}^2$\mathbb{R}^{2}with$\underset{R\to \mathrm{\infty }}{lim\; inf}\underset{c\in \mathbb{R}}{inf}\frac{1}{R}{⨍}_{B(0,R)}|u(x)-c|\mathit{d}x<\mathrm{\infty },$\liminf_{R\to\infty}\inf_{c\in\mathbb{R}}\frac{1}{R}\barint_{B(0,R)}\lvert u(x)-c\rvert\,dx<\infty,then$u=b+a\cdot x$u=b+a\cdot xfor some$b\in \mathbb{R}$b\in\mathbb{R}and$a\in {\mathbb{R}}^2$a\in\mathbb{R}^{2}.

   Denoting by$u_p$u_{p}the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that$detD{V}_{\beta }(D{u}_p)\to detD{V}_{\beta }(Du)$\det DV_{\beta}(Du_{p})\to\det DV_{\beta}(Du)as$p\to \mathrm{\infty }$p\to\inftyin the weak-⋆ sense in the space of Radon measure.Recall that$V_{\beta }(D{u}_p)$V_{\beta}(Du_{p})is always quasiregular mappings, but$V_{\beta }(Du)$V_{\beta}(Du)is not in general.

    

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3. Supersingular elliptic curves over ℤ 𝑝 -extensions

   https://doi.org/10.1515/crelle-2023-0029  

   Çiperiani, Mirela ( June 2023 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   Let$\mathrm{E}/\mathbb{Q}$\mathrm{E}/\mathbb{Q}be an elliptic curve and 𝑝 a prime of supersingular reduction for$\mathrm{E}$\mathrm{E}.Consider a quadratic extension$L/{\mathbb{Q}}_p$L/\mathbb{Q}_{p}and the corresponding anticyclotomic${\mathbb{Z}}_p$\mathbb{Z}_{p}-extension$L_{\mathrm{\infty }}/L$L_{\infty}/L.We analyze the structure of the points$\mathrm{E}(L_{\mathrm{\infty }})$\mathrm{E}(L_{\infty})and describe two global implications of our results.

    

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4. Irregular loci in the Emerton–Gee stack for GL 2

   https://doi.org/10.1515/crelle-2024-0045  

   Bellovin, Rebecca ; Borade, Neelima ; Hilado, Anton ; Kansal, Kalyani ; Lee, Heejong ; Levin, Brandon ; Savitt, David ; Wiersema, Hanneke ( July 2024 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   Let$K/{\mathbf{Q}}_p$K/\mathbf{Q}_{p}be unramified.Inside the Emerton–Gee stack${\mathcal{X}}_2$\mathcal{X}_{2}, one can consider the locus of two-dimensional mod 𝑝 representations of$\mathrm{Gal}(\bar{K}/K)$\mathrm{Gal}(\overline{K}/K)having a crystalline lift with specified Hodge–Tate weights.We study the case where the Hodge–Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms.We prove that if the gap between each pair of weights is bounded by 𝑝 (the irregular analogue of a Serre weight), then this locus is irreducible.We also establish various inclusion relations between these loci.

    

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5. Products of primes in arithmetic progressions

   https://doi.org/10.1515/crelle-2023-0096  

   Matomäki, Kaisa ; Teräväinen, Joni ( January 2024 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   A conjecture of Erdős states that, for any large primeq, every reduced residue class$(\mathrm{mod}q)${(\operatorname{mod}q)}can be represented as a product$p_1{p}_2${p_{1}p_{2}}of two primes$p_1,p_2\le q${p_{1},p_{2}\leq q}. We establish a ternary version of this conjecture, showing that, for any sufficiently large cube-free integerq, every reduced residue class$(\mathrm{mod}q)${(\operatorname{mod}q)}can be written as$p_1{p}_2{p}_3${p_{1}p_{2}p_{3}}with$p_1,p_2,p_3\le q${p_{1},p_{2},p_{3}\leq q}primes. We also show that, for any$\epsilon >0${\varepsilon>0}and any sufficiently large integerq, at least$(\frac{2}{3}-\epsilon )\phi (q)${(\frac{2}{3}-\varepsilon)\varphi(q)}reduced residue classes$(\mathrm{mod}q)${(\operatorname{mod}q)}can be represented as a product$p_1{p}_2${p_{1}p_{2}}of two primes$p_1,p_2\le q${p_{1},p_{2}\leq q}.The problems naturally reduce to studying character sums. The main innovation in the paper is the establishment of a multiplicative dense model theorem for character sums over primes in the spirit of the transference principle. In order to deal with possible local obstructions we establish bounds for the logarithmic density of primes in certain unions of cosets of subgroups of${\mathbb{Z}}_q^{\times }${\mathbb{Z}_{q}^{\times}}of small index and study in detail the exceptional case that there exists a quadratic character$\psi (\mathrm{mod}q)${\psi~{}(\operatorname{mod}\,q)}such that$\psi (p)=-1${\psi(p)=-1}for very many primes$p\le q${p\leq q}.

    

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