Let $(G,X)$ be a Shimura datum of abelian type satisfying the hypotheses of the main theorem, and suppose that the associated Shimura varieties have hyperspecial good reduction at $p$. Their special fibres are stratified by the $\sigma$-conjugacy classes $b\in B(G_{\mathbb{Q}_p},\mu_h^{-1})$. For an individual Newton stratum, this paper constructs a stabilized formula for the alternating traces of Frobenius--Hecke correspondences on its compactly supported $\ell$-adic cohomology, with coefficients in an $\ell$-adic local system. The formula, obtained by modifying the Langlands--Kottwitz method, expresses these Lefschetz numbers as elliptic stable geometric distributions on endoscopic groups of $G$, placing the cohomological traces in a form suitable for comparison with automorphic spectral data. We also give a group-theoretic criterion determining which elliptic endoscopic groups can contribute to a given Newton stratum. For certain unitary Shimura varieties, we exhibit an intermediate Newton stratum with no non-trivial endoscopic contribution, although there are non-trivial endoscopic contributions in the cohomology of the ambient Shimura variety.
Let $X$ be a smooth projective complex surface with numerically trivial canonical class. With the help of GPT-5.6 Sol, we prove Qin's conjecture that the reduced generating series of intersection numbers of Chern characters of tautological bundles against the total Chern class of $X^{[n]}$ is a quasimodular form with t...
V. Alekseev, A. Chakravarty, Daebeom Choi et al.· 1 citation
We study the geometry of Shimura varieties associated with a Hermitian form of signature $(p,q)$ over an imaginary quadratic field $E$, where $2\leq p\leq q$. We prove that when $p\geq w_E$ and $(p,q)\neq(2,2),(2,3)$, where $w_E:=\#\mathscr{O}_E^\times$, there exists a toroidal compactification with at worst canonical...
Let $G=\mathrm{SL}_n(\mathbf C)$ and let $P\subset G$ be a standard parabolic subgroup with Levi factor $L$. For a $G$-dominant weight $\lambda$, consider the vector bundle on $T^*(G/P)$ obtained by pulling back the vector bundle on $G/P$ associated to the irreducible $L$-module $V_L(\lambda)^*$. We express its cohomol...
Let $X$ be a connected finite-type CW-complex with fundamental group $G$. We show that the profinite completion $\widehat{G}$ determines the cohomology jump loci $\mathcal{V}^q_s(X,\mathbb{C})$ under two hypotheses: that the loci are finite unions of torsion-translated subtori, as for smooth quasi-projective varieties,...
Let $X$ be a smooth projective surface over $\mathbb{C}$ with an effective action of a finite group $G$, and let $L$ be a $G$-linearized line bundle. We develop a residue-theoretic and birational approach to computing the equivariant Euler characteristic $\chi_G(X,L)$. Every such $L$ admits an equivariant divisor prese...
Let $\Gamma<\operatorname{PSL}_2(\mathbb R)$ be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let $\rho:\Gamma\to G$ be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing fo...
Dongryul M. Kim, Hee Oh, Wen-Yu Pan· 0 citations
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