arXiv subjects

Xuecai Ma

Publications and source records attributed to Xuecai Ma.

3 recordsLinked to original sources

arXiv2026

The principal series 2-representation for $\mathrm{GL}_n\times\mathrm{GL}_n$ over a 2-dimensional local field

Let $F=\mathbb{F}_q((u))((t))$ and $H=\mathrm{GL}_n(F)_L\times\mathrm{GL}_n(F)_R$. An ordered tuple $\alpha=(\alpha_1,\dots, \alpha_n) $ defines a cross-$K_2$ multiplier on the doubled Borel. Twisted equivariantization then yields an exact uniformly smooth categorical principal series $2$-representation of $H$. For unitary parameters, the $\mathbb R((X))$-measure gives a positive Hom-valued formal Hermitian inner product on the spherical part. These constructions provide two-dimensional categorical analogues of selected unramified structures.

math.NT↗

arXiv2026

Hecke algebra of $\mathrm{GL}_n$ over a 2-dimensional local field

Using the $\mathbb{R}((X))$-measure, we define and study certain $\mathbb{C}((X))$-valued functions on $\mathrm{GL}_n(F)$ for $F$ a two-dimensional local field. In particular, we define a convolution product on such suitable functions, which leads us to define the Hecke algebra of $\mathrm{GL}_n(F)$. We then define the measurable $\mathbb{C}((X))$-representations of $\mathrm{GL}_n(F)$, and prove that function space is a candidate for such representations.

math.NT↗

arXiv2025

Spectral moduli problems for level structures and an integral Jacquet-Langlands dual of Morava E-theory

Given an E-infinity ring spectrum R, with motivation from chromatic homotopy theory, we define relative effective Cartier divisors for a spectral Deligne-Mumford stack over Spet(R) and prove that, as a functor from connective R-algebras to topological spaces, it is representable. This enables us to solve various moduli problems of level structures on spectral abelian varieties, overcoming difficulty at primes dividing the level. In particular, we obtain higher-homotopical refinement for finite levels of a Lubin-Tate tower as E-infinity ring spectra, which generalizes Morava, Hopkins, Miller, Goerss, and Lurie's spectral realization of the deformation ring at the ground level. Moreover, passing to the infinite level and then descending along the equivariantly isomorphic Drinfeld tower, we obtain a Jacquet-Langlands dual to the Morava E-theory spectrum, along with homotopy fixed point spectral sequences dual to those studied by Devinatz and Hopkins. These serve as potential tools for computing higher-periodic homotopy types from pro-etale cohomology of p-adic general linear groups.

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