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1. Weight 11 Compactly Supported Cohomology of Moduli Spaces of Curves

   https://doi.org/10.1093/imrn/rnad308  

   Payne, Sam; Willwacher, Thomas (January 2024, International Mathematics Research Notices)

   Abstract We study the weight 11 part of the compactly supported cohomology of the moduli space of curves $${\mathcal{M}}_{g,n}$$, using graph complex techniques, with particular attention to the case $n = 0$. As applications, we prove new nonvanishing results for the cohomology of $${\mathcal{M}}_{g}$$, and exponential growth with $$g$$, in a wide range of degrees. 

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2. Deligne Categories and the Limit of Categories Rep(GL(m|n))

   https://doi.org/10.1093/imrn/rny144  

   Entova-Aizenbud, Inna; Hinich, Vladimir; Serganova, Vera (June 2018, International Mathematics Research Notices)

   null (Ed.)

   Abstract For each integer $$t$$ a tensor category $$\mathcal{V}_t$$ is constructed, such that exact tensor functors $$\mathcal{V}_t\rightarrow \mathcal{C}$$ classify dualizable $$t$$-dimensional objects in $$\mathcal{C}$$ not annihilated by any Schur functor. This means that $$\mathcal{V}_t$$ is the “abelian envelope” of the Deligne category $$\mathcal{D}_t=\operatorname{Rep}(GL_t)$$. Any tensor functor $$\operatorname{Rep}(GL_t)\longrightarrow \mathcal{C}$$ is proved to factor either through $$\mathcal{V}_t$$ or through one of the classical categories $$\operatorname{Rep}(GL(m|n))$$ with $m-n=t$. The universal property of $$\mathcal{V}_t$$ implies that it is equivalent to the categories $$\operatorname{Rep}_{\mathcal{D}_{t_1}\otimes \mathcal{D}_{t_2}}(GL(X),\epsilon )$$, ($$t=t_1+t_2$$, $$t_1$$ not an integer) suggested by Deligne as candidates for the role of abelian envelope. 

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3. Overconvergent modular forms are highest-weight vectors in the Hodge-Tate weight zero part of completed cohomology

   https://doi.org/10.1017/fms.2021.16  

   Howe, Sean (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We construct a $$(\mathfrak {gl}_2, B(\mathbb {Q}_p))$$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $$0$$ of a sheaf on $$\mathbb {P}^1$$ , landing in the compactly supported completed $$\mathbb {C}_p$$ -cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest-weight vector for any overconvergent modular form of infinitesimal weight not equal to $$1$$ . For classical weight $$k\geq 2$$ , the Verma has an algebraic quotient $$H^1(\mathbb {P}^1, \mathcal {O}(-k))$$ , and on classical forms, the pairing factors through this quotient, giving a geometric description of ‘half’ of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $$\mathbb {P}^1$$ . Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan, but the perspective here is different, and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology. 

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4. Stability in the homology of Deligne–Mumford compactifications

   https://doi.org/10.1112/S0010437X21007582  

   Tosteson, Philip (December 2021, Compositio Mathematica)

   Abstract Using the theory of $${\mathbf {FS}} {^\mathrm {op}}$$ modules, we study the asymptotic behavior of the homology of $${\overline {\mathcal {M}}_{g,n}}$$ , the Deligne–Mumford compactification of the moduli space of curves, for $$n\gg 0$$ . An $${\mathbf {FS}} {^\mathrm {op}}$$ module is a contravariant functor from the category of finite sets and surjections to vector spaces. Via copies that glue on marked projective lines, we give the homology of $${\overline {\mathcal {M}}_{g,n}}$$ the structure of an $${\mathbf {FS}} {^\mathrm {op}}$$ module and bound its degree of generation. As a consequence, we prove that the generating function $$\sum _{n} \dim (H_i({\overline {\mathcal {M}}_{g,n}})) t^n$$ is rational, and its denominator has roots in the set $$\{1, 1/2, \ldots, 1/p(g,i)\},$$ where $p(g,i)$ is a polynomial of order $O(g^2 i^2)$ . We also obtain restrictions on the decomposition of the homology of $${\overline {\mathcal {M}}_{g,n}}$$ into irreducible $$\mathbf {S}_n$$ representations. 

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5. On the weight zero compactly supported cohomology of H_{g,n}

   https://doi.org/10.1017/fms.2024.53  

   Brandt, Madeline; Chan, Melody; Kannan, Siddarth (January 2024, Forum of Mathematics, Sigma)

   Abstract For$$g\ge 2$$and$$n\ge 0$$, let$$\mathcal {H}_{g,n}\subset \mathcal {M}_{g,n}$$denote the complex moduli stack ofn-marked smooth hyperelliptic curves of genusg. A normal crossings compactification of this space is provided by the theory of pointed admissible$$\mathbb {Z}/2\mathbb {Z}$$-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of$$\mathcal {H}_{g, n}$$. Using this graph complex, we give a sum-over-graphs formula for the$$S_n$$-equivariant weight zero compactly supported Euler characteristic of$$\mathcal {H}_{g, n}$$. This formula allows for the computer-aided calculation, for each$$g\le 7$$, of the generating function$$\mathsf {h}_g$$for these equivariant Euler characteristics for alln. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissibleG-covers of genus zero curves, whenGis abelian, as a symmetric$$\Delta $$-complex. We use these complexes to generalize our formula for$$\mathsf {h}_g$$to moduli spaces ofn-pointed smooth abelian covers of genus zero curves. 

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