New Math Papers: Modular Forms & Number Theory Updates

Welcome to our latest roundup of cutting-edge research from the math.NT and math.RT categories on arXiv, dated December 5th, 2025. This week, we're diving into the fascinating world of modular forms, number theory, and representation theory, with some intriguing insights into algebraic geometry and combinatorics along the way. Let's explore the newest contributions to these vibrant fields!

Unpacking the Signs: Fourier Coefficients of Modular Forms

"Sign patterns of Fourier coefficients of modular forms" by Andrew R. Booker delves into a fundamental question concerning modular forms. The paper, just 5 pages long, presents conditions under which a self-dual holomorphic cusp form can be uniquely determined by the signs of its Fourier coefficients. This is a significant contribution to our understanding of the structure and properties of these highly symmetric functions, which play a crucial role in various areas of mathematics, from number theory to string theory. The Fourier coefficients are like the building blocks of these forms, and understanding their signs can reveal deep structural information. Booker's work provides a new lens through which to view this relationship, potentially opening doors to further classifications and conjectures in the study of modular forms. The elegance of a mathematical object being defined by the simple signs of its components is a recurring theme in number theory, and this paper offers a fresh perspective on this idea within the realm of self-dual modular forms.

Representation Theory Takes Center Stage

Several papers this week highlight the dynamism of representation theory. Il-Seung Jang's "Unipotent quantum coordinate ring and minuscule prefundamental representations: twisted case" (v2, 29 pages) explores the intricate connections between unipotent quantum coordinate rings and minuscule prefundamental representations within the context of twisted quantum loop algebras. This work builds upon previous research, providing a character formula that unifies twisted and untwisted cases and realizing specific representations for types $A_{2n-1}^{(2)}$ and $D_{n+1}^{(2)}$. This is advanced stuff, pushing the boundaries of our understanding of quantum groups and their representations. Meanwhile, GyeongHyeon Nam and Anna Puskás, in "The sparsity of character tables over finite reductive groups and its additive analogue" (26 pages), investigate the distribution of zeros in the character tables of finite reductive groups. They establish asymptotic lower bounds and show that for sequences of groups with increasing semisimple rank, the proportion of zeros approaches one. This offers a fascinating glimpse into the combinatorial structure of these fundamental objects in group theory. Continuing this thread, Buyan Li and Jie Xiao's "Notes on Chevalley Groups and Root Category II: Compact Lie Groups and Representations" (40 pages) revisits the classical theory of compact Lie groups and their representations, reconstructing it using the powerful framework of root categories. They show how classical results like the Peter-Weyl and Plancherel theorems can be recovered, bridging modern algebraic techniques with foundational theories. Lastly, in "Multiple rational normal forms in Lie theory" (28 pages), Dmitriy Voloshyn introduces a class of rational Weyl group elements that enable specific decompositions of elements within connected reductive complex algebraic groups, offering new insights into the structure of these groups. Gerhard Hiss and Caroline Lassueur's "On the source algebra equivalence class of blocks with cyclic defect groups, II" (35 pages) continues a multi-part series aiming to classify $p$-blocks of finite groups based on their source algebra equivalence classes, focusing here on quasisimple classical groups.

Number Theory and Dynamical Systems Intersect

Kasun Fernando and Tanja I. Schindler's paper, "Limit Theorems for a class of unbounded observables with an application to 'Sampling the Lindelöf hypothesis'" (journal version), connects the fields of dynamical systems and number theory. They prove Central Limit Theorems and related limit theorems for certain types of unbounded observables in expanding maps of an interval. The exciting part is the application to the Riemann zeta function, bringing probabilistic limit theorems to bear on the notoriously difficult Lindelöf hypothesis. This interdisciplinary approach is a hallmark of modern mathematical research, where tools from one area can unlock mysteries in another.

Exploring Non-Archimedean Realms and Exceptional Sets

Aihua Fan, Shilei Fan, and Hanfei Ye's "Non-Archimedean Koksma Theorems and Dimensions of Exceptional Sets" (math.NT) takes us into the world of non-Archimedean analysis. They establish a non-Archimedean analogue of Koksma's theorem concerning the uniform distribution of sequences. Their work reveals that while sequences like $[\alpha x^n]$ are uniformly distributed in the valuation ring for most $x$, the exceptional sets where this fails are surprisingly large and possess a rich fractal structure. This research offers deep insights into the distribution properties of sequences in a non-standard analytical setting.

Moments and Distributions in Number Theory

Further contributions to number theory include "On moments of the Erdős--Hooley Delta-function" by R. de la Bretèche and G. Tenenbaum. This paper focuses on deriving new upper bounds for the weighted real moments of the Erdős--Hooley $\Delta$-function, a measure of the multiplicative structure of integers. Additionally, Peng Gao and Xiaozhi Wu present "Upper Bounds for low moments of twisted Fourier coefficients of modular forms", providing new bounds for the moments of twisted Fourier coefficients of modular forms. Their results have implications for the distribution of these coefficients.

Algebraic Geometry and Number Theory Connections

In algebraic geometry, "More counterexamples to the Arithmetic Puncturing Problem" by Finn Bartsch introduces new examples of threefolds and surfaces with specific types of singularities that challenge existing conjectures about arithmetic puncturing. This work probes the delicate interplay between geometric properties and the distribution of rational points on algebraic varieties. Meanwhile, L. Alexander Betts and Ishai Dan-Cohen's extensive work, "A motivic Weil height machine for curves" (92 pages), develops a framework for studying rational points on curves through the lens of motivic algebra. They propose this as a more accessible alternative to Kim's conjecture and provide a finiteness result for specific curves.

Diverse Contributions Across Mathematical Fields

This week's arXiv dump also features several other noteworthy papers: "Inductive systems of the symmetric group, polynomial functors and tensor categories" by Kevin Coulembier explores connections between modular representations of symmetric groups and tensor categories. "Closed Colored Models and Demazure Crystals" by Yingzi Yang constructs solvable lattice models whose partition functions are Demazure characters. "Applying hypersurface bounds to a conjecture by Carlet" by Zoë Gemmell and Tim Trudgian refines estimates related to the sum-free property of the inverse function in finite fields. "On the Alexander polynomials of modular knots" by Soon-Yi Kang, Toshiki Matsusaka, and Kyungbae Park investigates topological invariants of modular knots. "Affine diagram categories, algebras and monoids" by David He and Daniel Tubbenhauer introduces affine versions of classical diagram algebras. "p-adic Hodge parameters in the crystalline representations of GSp4" by Xiaozheng Han generalizes work on crystalline representations in the context of $ ext{GSp}_4(\mathbb{Q}_p)$. Finally, "Restriction of the metaplectic representation over a $p$-adic field to an anisotropic torus" by Khemais Maktouf and Pierre Torasso examines the restriction of the metaplectic representation to specific tori.

For more details on these papers and to access the full articles, please visit the Github page. We encourage you to explore these exciting new developments in mathematics!

For further exploration into the fascinating world of number theory and related fields, we recommend visiting the Number Theory Web** and the American Mathematical Society (AMS)** website.**