数论

We construct integral models of Shimura varieties of abelian type with parahoric level structure over odd primes. These models are \'etale locally isomorphic to corresponding local models.

In this paper we prove rationality results of critical values for $L$-functions attached to representations in the residual spectrum of $\mathrm{GL}_4(\mathbb{A})$. We use the Jacquet-Langlands correspondence to describe their partial…

We study the distribution of large prime factors of a random element $u$ of arithmetic sequences satisfying simple regularity and equidistribution properties. We show that if such an arithmetic sequence has level of distribution $1$ the…

We compute the geometric Sen operator for arbitrary Shimura varieties in terms of equivariant vector bundles of flag varieties and the Hodge-Tate period map. As an application, we obtain the rational vanishing of completed cohomology in the…

The aim of this paper is twofold. We first present a construction of the overconvergent automorphic sheaves for Siegel modular forms by generalising the perfectoid method, originally introduced by Chojecki--Hansen--Johansson for automorphic…

Let $ G $ be a cyclic group, in this paper, we study the Herbrand quotient and $ 1-$th cohomology group on finitely generated $ G-$modules in some cases. When $ G $ is of order $ 2, $ the order of the cohomology group is explicitly related…

In this note, I develop step-by-step proofs of irrationality for $\,\zeta{(2)}\,$ and $\,\zeta{(3)}$. Though the proofs follow closely those based upon unit-square integrals proposed originally by Beukers, I introduce some modifications…

We introduce a gamma function $\Ga(x,z)$ in two complex variables which extends the classical gamma function $\Ga(z)$ in the sense that $\lim_{x\to 1}\Ga(x,z)=\Ga(z)$. We will show that many properties which $\Ga(z)$ enjoys extend in a…

Wan proved the rationality of partial toric $L$-functions using $\ell$-adic techniques. In this paper, we present a $p$-adic proof in the spirit of Dwork. We demonstrate that partial $L$-functions can be expressed as an alternating product…

Two elliptic curves defined over $\mathbb{Q}$ are called discriminant twins if they have the same minimal discriminant and the same conductor. Deines, in 2014, conjectured that there exist infinitely many semi-stable non-isogenous…

We show that the Galois cohomology of negative Tate twists can be organized by a single universal cyclotomic complex over the cyclotomic tower of $\mathbb{Q}$. Using cyclotomic descent and Teichm\"uller branch decomposition, we prove that a…

For a wide range of functions $W\colon\mathbb{N}\to\mathbb{N}$, we establish a general result for estimating weighted averages of the form \[ \mathbb{E}^{W}_{n \le N} f(\vartheta(n))= \frac{1}{W(N)}\sum_{n=1}^N (W(n)-W(n-1))f(\vartheta(n)),…

Gardiner, Lazarus, Metropolis, and Ulam introduced a variation of the sieve of Eratosthenes that (instead of producing the sequence of prime numbers) produces the sequence of "lucky numbers". The distribution of lucky numbers has a striking…

We establish the Hasse principle for $100\%$ of conic bundles over $\mathbb{P}^1_{\mathbb{Q}}$.

Let $p$ be a prime number, and let $\Delta_1,\Delta_2 < 0$ be two coprime fundamental discriminants. When $p$ splits in $\mathbb{Q}(\sqrt{\Delta_1})$ and $\mathbb{Q}(\sqrt{\Delta_2})$ the height pairings of the corresponding CM divisors on…

We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a…

Let $F_0$ be the function field of a curve over a $p$-adic field $K,$ and let $F$ be a quadratic extension over $F_0$. Let $A$ be a central simple algebra over $F$ of period $2,$ and let $\tau$ be a $F/F_0$-involution on $A$. We show the…

Andrews, Dixit, Schultz, and Yee conjecture the parity of a double Lambert series. In 2026, Amdeberhan, Andrews, and Ballantine offer some ideas that are pointing in the right direction for the proof. In this paper, we complete the rest of…

We prove that the ratio $B_n/D_n$ of the Ap\'ery-like sequence $B_n$ to the Domb numbers $D_n$ converges to $(7/24)\zeta(3)$, and that $\sum_{n=1}^{\infty} 64^n/(n^3 D_n D_{n-1}) = (56/3)\zeta(3)$. As a corollary we establish the value $Z_2…

The Hofstadter Q-sequence is a prominent example of nested recurrence. Despite decades of study, it is not even known whether Q(n) is defined for all n. Mantovanelli introduced a parity-perturbed variant $\widetilde{Q}$, obtained by adding…