178 papers
This paper establishes that rank-two Drinfeld modules with a specific -structure are isomorphic to their Taguchi duals, a result that enables the derivation of a dual Kodaira–Spencer isomorphism for the Hodge bundle on the corresponding Drinfeld modular curve.
This paper introduces the concept of "missing generators" for cyclic groups to establish a unique cyclic structure and additive properties for these groups, while demonstrating that factoring RSA numbers is computationally equivalent to computing a specific structural parameter under a conjecture regarding the density of primes in certain exponential sequences.
This paper establishes a quantitative form of the Green-Tao theorem for sparse sets by proving that any subset of primes with relative density lacking nontrivial arithmetic progressions of length must satisfy , an improvement achieved through a new quasipolynomial inverse theorem and a dense model theorem.
This paper constructs natural Frobenius structures on two families of rigid irregular -connections, utilizing them to analyze local monodromy, verify Reeder--Yu's predictions on epipelagic Langlands parameters, and confirm the cohomological and physical rigidity conjectures of Heinloth--Ngô--Yun.
This paper derives an asymptotic formula for the order of the 2-primary parts of even K-groups in the cyclotomic -extensions of real quadratic number fields by analyzing 2-adic divisibility of Dirichlet L-series, thereby determining their Iwasawa invariants and explicitly characterizing the structure of 2-primary tame kernels for specific families of fields.
This paper establishes an -duality converse to prior work by proving that the -dual of the action of on their tautological representations is the symplectic mirabolic space acting on and its tautological representations, while also formulating a corresponding global conjecture for the categorical theta-correspondence.
This paper reports the discovery of "murmurations," a new arithmetic phenomenon identified through AI-assisted analysis of large datasets that encodes subtle information about Frobenius traces and connects to the Birch and Swinnerton-Dyer conjecture and random matrix theory.
This paper refutes the conjecture that isolated periodic points of automorphisms on affine spaces have bounded height by providing a counterexample, while simultaneously proving that cohomologically hyperbolic dominant rational self-maps on projective varieties possess a Zariski open subset with height-bounded periodic points and offering evidence that such boundedness may fail for preperiodic points.
This paper investigates whether the divisor structure of influences Mersenne prime exponents by introducing a normalized parameter and demonstrating through statistical analysis that known Mersenne prime exponents exhibit moderately elevated values of compared to nearby primes, though a theoretical explanation for this phenomenon remains open.
This paper constructs exotic Hecke correspondences between the special fibers of PEL-type Shimura varieties with potentially bad reduction by utilizing Pappas-Rapoport splitting models, thereby establishing new geometric Jacquet-Langlands correspondences and verifying generic instances of the Tate conjecture.
The paper demonstrates that for any dimension , the existence of complex equiangular unit vectors implies the existence of such a set where all vector coefficients lie within a number field, a result motivated by the construction of SIC-POVMs in quantum physics.
This paper investigates the state complexity of automata recognizing relations and operations on -automatic sequences, establishing a link between the subword complexity of interior sequences and the state complexity of their linear subsequences while resolving a recent question by Zantema and Bosma regarding most-significant-digit-first inputs.
Inspired by Katz's work, this paper investigates the converse of the known property regarding rational torsion injection into reductions for -type abelian varieties and proposes a conjectural list of possible torsion orders for modular abelian varieties over with dimension up to 5.
Conditional on Zywina's effective Serre uniformity conjecture, this paper establishes a natural parameterization of non-CM rational points on all modular curves using finitely many such curves, thereby confirming Mazur and Ogg's philosophy that these points arise from the geometry of modular curves.
This paper provides explicit closed formulae for the even polynomials and , which represent the ratios and , by expressing them in terms of Eulerian numbers and analyzing their structural properties to aid future investigations into the arithmetic nature of these ratios.
This paper investigates the binomial sum by expressing it in terms of powers of for specific sequences like Fibonacci numbers and various polynomials, while also establishing its properties, probabilistic interpretations, and generating functions in relation to and Appell polynomials.
This paper surveys recent developments in the arithmetic of power monoids—structures formed by finite subsets of a monoid—and highlights their role in advancing factorization theory, particularly within non-cancellative and non-commutative settings.
This paper presents three families of modular Nahm sums for symmetrizable matrices of indices and with arbitrary rank , extending previous results for low ranks and utilizing these families to construct two vector-valued automorphic forms.
This paper claims to derive a strictly closed-form, non-iterative formula for the partition function by utilizing the Simplicial Spectral Decomposition of partition polytopes within the framework of root systems and Ehrhart theory.
This paper generalizes and improves upon existing results regarding the asymptotic behavior of the sum of the -th powers of the -th normalized Fourier coefficients of the -th symmetric power -functions for primitive holomorphic cusp forms, specifically for positive integers and satisfying .