arXiv subjects

Peter Xu

Publications and source records attributed to Peter Xu.

9 recordsLinked to original sources

arXiv2026

Localization and elliptic motivic relations

We observe that the motivic analogue of Suslin reciprocity (and similar degree-zero statements) is a formal consequence of localization (plus purity/some six functor formalism). In particular, the statement of Suslin reciprocity for smooth schemes over fields due to Kriz is a corollary of localization for higher Chow groups, over any base; we write down the framework yielding such relations with coefficients for schemes smooth over any base in $\A^1$-invariant motivic cohomology. As an application, we refine some relations between cup products of modular units to be integral in coefficients and in the base: first, we imitate the (rational-coefficients, complex-analytic) Busuioc--Park--Patashnick--Stevens argument for full-level-$N$ elliptic schemes, extending the result to integral bases and coefficients using the elementary reciprocity statement. We then refine the construction and resulting relations to the setting of motivic sheaves; in particular, this gives analogous relations at non-full level structure, as well as over any smooth global quotient stack.

math.AG↗

arXiv2026

Presenting restricted Steinberg modules of general linear groups

We give generators and presentations for various local restrictions of Steinberg modules over fields and relate them to \emph{partial} Borel--Serre compactifications of locally symmetric spaces in the case of number fields, extending the existing theory of partial modular symbols for $\mathrm{GL}_2(\Q)$. Along the way, we clarify the relationship between ``circuit'' and ``Bykovskii''-type presentations for such modules. In an appendix, we relate the existence of such presentations to Koszulity properties of the Steinberg $\mathrm{VB}$-algebra.

math.RT↗

arXiv2026

A note on cubical Bloch--Levine cycle complexes

We check that Levine's simplicial--cubical comparison argument for Bloch's cycle complexes also works over an arbitrary DVR. As a result, the sheaf of cubical Bloch cycle complexes computes motivic cohomology for smooth schemes over Dedekind bases.

math.AG↗

arXiv2026

A note on polyhedral cones and toric polylogarithms

We extend some methods of our previous work on special elements in Milnor K-theory of algebraic tori, exhibiting in particular a $\mathrm{GL}_n(\mathbb{Q})$-equivariant isomorphism between a chain complex of simplicial cones, computing the homology of $S^{n-1}$, and the trace-fixed part of the weight-n Gersten complex for the Milnor K- theory of $\mathbb{G}_m^n$ over $\mathbb{Q}$. Via a relationship between graded pieces of algebras of cones and Steinberg modules, this refines a result of Charlton-Radchenko-Rudenko.

math.KT↗

arXiv2025

Eisenstein class of a torus bundle and log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$

Starting from a topological treatment of the Eisenstein class of a torus bundle, we define log-rigid analytic classes for $\mathrm{SL}_n(\mathbb{Z})$. These are group cohomology classes for $\mathrm{SL}_n(\mathbb{Z})$ valued on log-rigid analytic functions on Drinfeld's $p$-adic symmetric domain. Such classes can be evaluated at points attached to totally real fields of degree $n$ where $p$ is inert. We conjecture that these values are $p$-adic logarithms of Gross--Stark units in the narrow Hilbert class field of totally real fields. We provide evidence for the conjecture by comparing our constructions to $p$-adic $L$-functions. In addition, we prove it in certain situations where the totally real field is Galois over $\mathbb{Q}$, as a consequence of the fact that in this case there is a conjugate of a Gross--Stark unit in $\mathbb{Q}_p$.

math.NT↗

arXiv2024

Explicit formula for the $(\text{GL}_2, \text{GL}_2)$ theta lift via Bruhat decomposition

Using combinations of weight-1 and weight-2 of Kronecker-Eisenstein series to construct currents in the distributional de Rham complex of a squared elliptic curve, we find a simple explicit formula for the type II $(\text{GL}_2, \text{GL}_2)$ theta lift without smoothing, analogous to the classical formula of Siegel for periods of Eisenstein series. For $K$ a CM field, the same technique applies without change to obtain an analogous formula for the $(\text{GL}_2(K),K^\times)$ theta correspondence.

math.NT↗

arXiv2024

Classical periods of Eisenstein series and Bernoulli polynomials in the equivariant cohomology of a torus

We find group cochains valued in currents giving explicit representatives for the $\text{GL}_2$-equivariant polylogarithm class of a torus. Based on the construction of weight-$2$ Eisenstein series for $\text{GL}_2$ from this polylogarithm class, we give a geometrically-flavored derivation of the classical formulas for the associated Dedekind-Rademacher homomorphisms, i.e. the periods of $E^2_{\alpha,\beta}$ for various nonzero torsion sections $(\alpha, \beta)$.

math.NT↗

arXiv2024

Assisting in Writing Wikipedia-like Articles From Scratch with Large Language Models

We study how to apply large language models to write grounded and organized long-form articles from scratch, with comparable breadth and depth to Wikipedia pages. This underexplored problem poses new challenges at the pre-writing stage, including how to research the topic and prepare an outline prior to writing. We propose STORM, a writing system for the Synthesis of Topic Outlines through Retrieval and Multi-perspective Question Asking. STORM models the pre-writing stage by (1) discovering diverse perspectives in researching the given topic, (2) simulating conversations where writers carrying different perspectives pose questions to a topic expert grounded on trusted Internet sources, (3) curating the collected information to create an outline. For evaluation, we curate FreshWiki, a dataset of recent high-quality Wikipedia articles, and formulate outline assessments to evaluate the pre-writing stage. We further gather feedback from experienced Wikipedia editors. Compared to articles generated by an outline-driven retrieval-augmented baseline, more of STORM's articles are deemed to be organized (by a 25% absolute increase) and broad in coverage (by 10%). The expert feedback also helps identify new challenges for generating grounded long articles, such as source bias transfer and over-association of unrelated facts.

cs.CL↗

arXiv2024

Symbols for toric Eisenstein cocycles and arithmetic applications

Using a complex parameterizing rational spherical chains, we construct explicit cocycles for $\mathrm{GL}_n(\Q)$ valued in the motivic cohomology of (open subsets of) the algebraic $n$-torus $\mathbb{G}_m^n$. The resulting cocycles directly generalize the work of Sharifi and Venkatesh from the case $n=2$ \cite{SV}. Even in this special case, our systematic use of pushforwards allows us to avoid the use of their ``connecting sequences,'' and allows us to refine the construction and Hecke properties of the Sharifi map $\varpi$ to the maximal expected statements, while inverting only the prime $2$. For general $n$, the $d\log$ regulator of our cocycle is related by convex conical duality to cocycles constructed from Shintani cones. This affords a systematic approach to $p$-adic $L$-functions for totally real fields without need for auxiliary data or logarithm sheaf coefficients, including a distribution-valued $\GL_n(\Z)$-cocycle specializing in a simple way to all such $p$-adic $L$-functions. It moreover provides a direct conceptual link between polylogarithmic constructions of Eisenstein classes (e.g., in \cite{BKL}), and those constructed using Shintani cones (e.g., in \cite{CDG}). We also show how our formalism gives an alternate proof of the exceptional divisibilities of the Deligne-Ribet $2$-adic $L$-function in almost all cases.

math.NT↗