Latest Math Papers July 26, 2025 - Number Theory And Representation Theory

Hey math enthusiasts! 👋 Check out the latest batch of fascinating papers from ArXiv's math.NT and math.RT categories, as of July 26, 2025. This week's collection covers a diverse range of topics, from number theory and representation theory to algebraic geometry and combinatorics. Whether you're a seasoned researcher or just curious about the cutting edge of mathematical research, there's something here for everyone. Don't forget to explore the Github page for even more papers and details! Let's dive in!

Latest Papers from math.NT, math.RT

[2507.18601] Global Fluctuations for Standard Young Tableaux

Date: 2025-07-24 | Authors: Gabriel Raposo | Category: math.PR | Comment: 82 pages

Global fluctuations for standard Young tableaux is a central topic in modern combinatorics and representation theory, and this paper by Gabriel Raposo delves deep into the asymptotic behavior of these objects. The paper, a substantial 82 pages, suggests a comprehensive exploration of the topic. For those unfamiliar, Young tableaux are fundamental combinatorial objects used in various areas of mathematics, including representation theory, algebraic combinatorics, and the study of symmetric functions. The standard Young tableaux, in particular, are those where the entries are strictly increasing along rows and columns, filled with the numbers 1 through n. Understanding their global fluctuations involves studying how their shapes and properties vary as their size grows. This is not just a niche topic; it has connections to random matrix theory, the longest increasing subsequence problem, and other areas of probability and mathematical physics. The study of global fluctuations often requires sophisticated techniques from probability theory, analysis, and combinatorics. Expect to see tools like the central limit theorem, large deviation principles, and maybe even connections to stochastic processes. For example, the Tracy-Widom distribution, which arises in random matrix theory, often pops up in the asymptotic analysis of combinatorial structures. Raposo's paper likely builds on previous work in this area, possibly extending existing results or providing new perspectives on the fluctuations of standard Young tableaux. If you're a graduate student or researcher working in combinatorics, probability, or representation theory, this is one paper you'll definitely want to check out. The length indicates a thorough treatment of the subject, possibly including detailed proofs and extensive examples. Given the breadth of applications for Young tableaux, this research could have far-reaching implications. It would be interesting to see if this work makes connections to other areas, such as the study of crystal structures or even the analysis of algorithms.

[2507.18589] Evaluation of a Determinant Involving Legendre Symbols

Date: 2025-07-24 | Authors: Chen-Kai Ren, Zhi-Wei Sun | Category: math.NT | Comment: 14 pages

In this evaluation of a determinant involving Legendre symbols, Chen-Kai Ren and Zhi-Wei Sun tackle a fascinating problem in number theory. The Legendre symbol, denoted as (a/p), is a crucial tool in understanding quadratic residues modulo a prime number p. It tells us whether an integer 'a' is a square modulo p. Now, imagine building a matrix where the entries involve these Legendre symbols. The question then becomes, what is the determinant of this matrix? Calculating determinants can often reveal deep arithmetic properties and patterns, and the involvement of Legendre symbols suggests a connection to quadratic reciprocity and related theorems. The determinant itself is a fundamental concept in linear algebra, representing the scaling factor of a linear transformation. In this context, evaluating it could uncover hidden structures within the distribution of quadratic residues. The authors' 14-page paper likely dives into intricate calculations and employs a mix of algebraic and number-theoretic techniques. We might expect to see applications of quadratic reciprocity, character sums, or even more advanced tools like the theory of modular forms if the structure is rich enough. Such determinants often arise in various contexts, including the study of elliptic curves, coding theory, and cryptography. It is possible that the results in this paper could have implications in these areas, providing new insights or computational tools. For number theory enthusiasts, this paper promises a deep dive into the interplay between linear algebra and arithmetic. The evaluation of such determinants is not just an exercise in computation; it's a way to probe the underlying structure of the integers and their relationships. It would be interesting to see if the authors also explore generalizations to other characters or different types of matrices, potentially opening up further avenues of research.

[2507.18574] Infinitely Many Pairs of Non-Isomorphic Elliptic Curves Sharing the Same BSD Invariants

Date: 2025-07-24 | Authors: Asuka Shiga | Category: math.NT | Comment: 10 pages

Asuka Shiga's paper on infinitely many pairs of non-isomorphic elliptic curves sharing the same BSD invariants is a significant contribution to the arithmetic theory of elliptic curves. Elliptic curves, defined by cubic equations, are central objects in number theory, with deep connections to cryptography, algebraic geometry, and even physics. The Birch and Swinnerton-Dyer (BSD) conjecture is one of the most important unsolved problems in mathematics, linking the arithmetic properties of an elliptic curve to its analytic behavior. The BSD conjecture predicts a relationship between the rank of the elliptic curve (a measure of the number of independent rational solutions) and the vanishing order of its L-function at s=1. The invariants mentioned in the title likely refer to quantities that appear in the BSD conjecture, such as the L-function, the regulator, the Tamagawa numbers, and the order of the torsion subgroup. If two elliptic curves share the same BSD invariants, it means these crucial quantities are identical, at least up to some expected relationships dictated by the conjecture. However, the curves themselves might be fundamentally different, i.e., non-isomorphic. The fact that there are infinitely many such pairs is quite striking. It suggests a rich and complex landscape of elliptic curves, where curves with similar arithmetic properties can still have distinct algebraic structures. This 10-page paper likely uses advanced techniques from number theory and algebraic geometry. We might expect to see constructions involving families of elliptic curves, careful analysis of their invariants, and potentially the use of modular forms or Galois representations. This result could have implications for our understanding of the BSD conjecture itself. By identifying families of curves with shared invariants, we might gain insights into the conjecture's structure and potential approaches to proving it. For researchers in number theory and arithmetic geometry, this paper is a must-read. It addresses a fundamental question about the distribution of elliptic curves and their invariants, pushing the boundaries of our knowledge in this area. It would be interesting to see what specific families of curves Shiga constructs and whether these constructions shed light on other open problems in the field.

[2407.01152] Invariants of Finite Orthogonal Groups of Plus Type in Odd Characteristic

Date: 2025-07-24 | Authors: H. E. A. Campbell, R. James Shank, David L. Wehlau | Category: math.AC | Comment: The second version includes number of minor corrections

This paper, titled Invariants of Finite Orthogonal Groups of Plus Type in Odd Characteristic, delves into the realm of invariant theory, a classical area of mathematics with connections to algebra, geometry, and representation theory. The focus here is on finite orthogonal groups, which are groups of matrices that preserve a certain quadratic form. The