多项式样条插值素级数

CM模形式在惰性素数上的对称平方的Iwasawa理论

Iwasawa Theory for the Symmetric Square of a CM Modular Form at Inert Primes

设f是CM模形式，p是CM域中惰性的奇素数.我们构造了f的对称平方的两个p-adic L-函数，其中一个具有与Delbourgo-Dabrowski构造的函数相同的插值性质，而第二个具有类似的插值性质，但对应于Frobenius的不同特征值。这两个p-adic L-函数之间的对称性允许我们定义p-adic L-函数的正和负。我们还定义了类似于Kobayashi的Selmer群的正、负p-Selmer群。我们解释如何通过一个主要的猜想这两组对象。

On the distribution of products of two primes

For a real parameter $r$, the RSA integers are integers which can be written as the product of two primes $pq$ with $p<q\leq rp$, which are named after the importance of products of two primes in the RSA-cryptography. Several authors obtained the asymptotic formulas of the number of the RSA integers. However, the previous results on the number of the RSA integers were valid only in a rather restricted range of the parameter $r$. Dummit, Granville, and Kisilevsky found some bias in the distribution of products of two primes with congruence conditions. Moree and the first author studied some sim…

模素幂的离散非代数方程解的计数

Counting Solutions to Discrete Non-Algebraic Equations Modulo Prime Powers

随着社会越来越依赖计算机，密码安全变得越来越重要。当前的加密方案包括ElGamal签名方案，其依赖于离散对数问题的复杂性。人们认为，这些方案使用的函数具有计算上难以处理的逆。关于这一点，我们感兴趣的是计数的解决方案，以推广的离散对数问题模一个素数的权力。这是通过内插到p-adic函数，并使用亨塞尔引理，或其他方法的情况下奇异提升，和孙子剩余定理。

Iwasawa theory of automorphic representations of $\mathrm{GL}_{2n}$ at non-ordinary primes

Let $Π$ be a cuspidal automorphic representation of $\mathrm{GL}_{2n}(\mathbb{A_Q})$ and let $p$ be an odd prime at which $Π$ is unramified. In a recent work, Barrera, Dimitrov and Williams constructed possibly unbounded $p$-adic $L$-functions interpolating complex $L$-values of $Π$ in the non-ordinary case. Under certain assumptions, we construct two \textit{bounded} $p$-adic $L$-functions for $Π$, thus extending an earlier work of Rockwood by relaxing the Pollack condition. Using Langlands local-global compatibility, we define signed Selmer groups over the $p$-adic cyclotomic extension of $\…

Twisted triple product p-adic L-functions and Hirzebruch-Zagier cycles

Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a $p$-nearly ordinary family of unitary cuspidal automorphic representations of $\text{Res}_{L\times F/F}(\text{GL}_{2})$. Furthermore, when $L/\mathbb{Q}$ is a real quadratic number field and $p$ is a split prime, we prove a $p$-adic Gross-Zagier formula relating the value of the $p$-adic $L$-function outside the range of interpolation to the syntomic Abel-Jacobi imag…

$\mathfrak{p}$进$L$函数在 Eisenstein 素点处的整性

Integrality of $\mathfrak{p}$-Adic $L$-Functions at Eisenstein Primes

本文研究平凡新形式的$\mathfrak{p}$进$L$函数在 Eisenstein 同余素点处的整性，在温和分歧假设下证明其属于 Iwasawa 代数，并推出主猜想中的整除关系，权$2$情形亦成立。