First-Degree Prime Ideals of Biquadratic Fields dividing prescribed Principal Ideals

Primes in a prescribed arithmetic progression dividing the sequence a^k+b^k

Given positive integers a,b,c and d such that c and d are coprime we show that the primes p=c(mod d)dividing a^k+b^k for some k>=1 have a natural density and explicitly compute this density. We demonstrate our results by considering some claims of Fermat that he made in a 1641 letter to Mersenne.

计数具有指定分歧超尖点分量的新形式

Counting newforms with prescribed ramified supercuspidal components

本文给出了$S_k^{\rm new}(N)$中在素数集合$T$处具有指定分歧超尖点分量$π_p$的新形式数量公式，该公式基于Atkin--Lehner算子的迹，且在$T$为单个素数或$N$所有素因子时完全显式。

关于具有指定metacyclic Galois群的 cyclotomic $\mathbb{Z}_2$-扩展的最大无分歧pro-$2$-扩展的存在性

On the Existence of the Maximal Unramified Pro-$2$-Extension over the Cyclotomic $\mathbb{Z}_2$-Extension with Prescribed Metacyclic Galois Group

研究特定metacyclic非模群作为数域最大无分歧2-扩展的Galois群的可能性，并提出新方法探讨Greenberg猜想。

Singular values of multiple eta-quotients for ramified primes

We determine the conditions under which singular values of multiple $η$-quotients of square-free level, not necessarily prime to~6, yield class invariants, that is, algebraic numbers in ring class fields of imaginary-quadratic number fields. We show that the singular values lie in subfields of the ring class fields of index $2^{k' - 1}$ when $k' \geq 2$ primes dividing the level are ramified in the imaginary-quadratic field, which leads to faster computations of elliptic curves with prescribed complex multiplication. The result is generalised to singular values of modular functions on $X_0^+ (…

On number fields with given ramification

Let E/F be a CM field split above a finite place v of F, let l be a rational prime number which is prime to v, and let S be the set of places of E dividing lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and if u is a place of E dividing v, we show that any field embedding E_S \to \bar{E_u} has a dense image. The "unramified outside S" number fields we use are cut out from the l-adic cohomology of the "simple" Shimura varieties studied by Kottwitz and Harris-Taylor. The main ingredients of the proof are then the local Langlands correspondence for GL_n, the main glob…

复乘阶在普通同源类中的加权分布

Weighted Distributions of Complex Multiplication Orders in Ordinary Isogeny Classes

本文基于Deuring对应，研究有限域上普通椭圆同源类内端射影环的加权分布，通过类数公式得到全局分布，并揭示了同源图几何与类域理论的联系。