Ome new theorems on generating functions and their applications on odd and even certain numbers attached to p and q parameters

非普通素数处模形式p进L函数的Artin形式化

Artin formalism for $p$-adic $L$-functions of modular forms at non-ordinary primes

本文研究非普通素数处模形式的双变量带符号p进L函数在虚二次域上的Artin形式化，并为Castella–Ciperiani–Skinner–Sprung的预测提供证据。

On the special values of certain Rankin-Selberg L-functions and applications to odd symmetric power L-functions of modular forms

We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and Kasten-Schmidt. As an application of this result, we prove algebraicity results for certain critical values of the fifth and the seventh symmetric power L-functions attached to a holomorphic cusp form. Assuming Langlands functoriality one can prove similar algebraicity results for the special values of any odd symmetric power L-function. We also prove a conjecture of Blas…

格罗斯字符族中的一级密度研究

One-level densities in families of Grössencharakters associated to CM elliptic curves

研究了与CM椭圆曲线相关的L函数族中的一级密度，发现其分解为symplictic和orthogonal子族，并计算了子族的一级项。

Evaluation of Certain Hypergeometric Functions over Finite Fields

For an odd prime $p$, let $ϕ$ denote the quadratic character of the multiplicative group ${\mathbb F}_p^\times$, where ${\mathbb F}_p$ is the finite field of $p$ elements. In this paper, we will obtain evaluations of the hypergeometric functions $ {}_2F_1\left(\begin{matrix} ϕψ& ψ\\ & ϕ\end{matrix};x\right)$, $x\in {\mathbb F}_p$, $x\neq 0, 1$, over ${\mathbb F}_p$ in terms of Hecke character attached to CM elliptic curves for characters $ψ$ of ${\mathbb F}_p^\times$ of order $3$, $4$, $6$, $8$, and $12$.

半判别式为$\boldsymbol{N^2}$的某些偶三元格对应的Hecke模

Hecke modules associated to certain even ternary lattices of half-discriminant $\mathbf{N^2}$

该研究针对半判别式为$N^2$的定偶三元格对应的Hecke模，借助Watson型变换序列，建立其与经典模形式空间的Jacquet-Langlands型显式对应，为相关数论问题提供了新的研究路径。

奇数秩基本 $\mathrm{GL}_n$ 局部志村塔的 Stein 性

Steinness of the Basic $\mathrm{GL}_n$ Local Shimura Tower in Odd Rank

本文证明奇数秩基本局部志村塔在有限层和任意素数下具有 Stein 性，通过构造全局解析函数并证明其有限性，涵盖 $p=2$。

Counting Restricted Partitions of Integers into Fractions: Symmetry and Modes of the Generating Function and a Connection to $ω(t)$

Motivated by the study of integer partitions, we consider partitions of integers into fractions of a particular form, namely with constant denominators and distinct odd or even numerators. When numerators are odd, the numbers of partitions for integers smaller than the denominator form symmetric patterns. If the number of terms is restricted to $h$, then the nonzero terms of the generating function are unimodal, with the integer $h$ having the most partitions. Such properties can be applied to a particular class of nonlinear Diophantine equations. We also examine partitions with even numerator…

算术级数中模素数幂的三重除数型函数

Triple divisor type functions in arithmetic progressions to prime power moduli

本文研究非尖点GL_3自守形式Hecke特征值在素数幂模算术级数中的等分布，利用Kowalski-Lin-Michel和Milićević技术证明指数1/2+7/458-ε可取。

关于平坦偶变形环

On flat even deformation rings

研究显示在非平凡双Selmer群存在时，某些全球偶变形环为有限且在Z_p上平坦。通过全球类域论技术，计算了偶表示的显式例子，并观察到若Leopoldt猜想对相关数域成立，则全球偶变形环在最小水平上平坦。

On the $p$-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

Let $E/K$ be a finite Galois extension of totally real number fields with Galois group $G$. Let $p$ be an odd prime and let $r>1$ be an odd integer. The $p$-adic Beilinson conjecture relates the values at $s=r$ of $p$-adic Artin $L$-functions attached to the irreducible characters of $G$ to those of corresponding complex Artin $L$-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the `$p$-part' of the equivariant Tamagawa number conjecture for the pair $(h^0(\mathrm{Spec}(E))(r), \mathbb Z[G])$. If $r>1$ is even we obtain a sim…

Partitioning the Primes and Efficient Non-trivial Factor Generation using the Least Odd Partition Identity in Conjunction with Dirichlet Linear Progressions and the Reduced Residue System Modulo 18

In this work we introduce the numerical constant, LOPI, N LOPI is congruent to LOPI mod 18, equal to the lowest odd partition identity in conjunction with the reduced residue system Modulo 18, a complete disjoint covering residue system when considered in its whole set of residues from 0 to 17. By convolution of specific LOPI Dirichlet linear progressions for each LOPI superset, the non-prime elements of the 6 reduced residue congruence classes mod 18 are generated sequentially and uniquely. Through set generation and complete composite number subset generation based on the pattern of sequenti…

Circular sets of prime numbers and p-extension of the rationals

Let p be an odd prime number and let S be a finite set of prime numbers congruent to 1 modulo p. We prove that the group G_S(Q)(p) has cohomological dimension 2 if the linking diagram attached to S and p satisfies a certain technical condition, and we show that G_S(Q)(p) is a duality group in these cases. Furthermore, we investigate the decomposition behaviour of primes in the extension Q_S(p)/Q and we relate the cohomology of G_S(Q)(p) to the etale cohomology of the scheme Spec(Z)-S. Finally, we calculate the dualizing module.

Shtukas and the Taylor expansion of $L$-functions (II)

For arithmetic applications, we extend and refine our results in \cite{YZ} to allow ramifications in a minimal way. Starting with a possibly ramified quadratic extension $F'/F$ of function fields over a finite field in odd characteristic, and a finite set of places $Σ$ of $F$ that are unramified in $F'$, we define a collection of Heegner--Drinfeld cycles on the moduli stack of $\mathrm{PGL}_{2}$-Shtukas with $r$-modifications and Iwahori level structures at places of $Σ$. For a cuspidal automorphic representation $π$ of $\mathrm{PGL}_{2}(\mathbb{A}_{F})$ with square-free level $Σ$, and $r\in\m…