On twisted forms and relative algebraic K-theory

代数族中的扭加倍方法

The Twisted Doubling Method in Algebraic Families

该研究在代数族框架下定义Cai-Friedberg-Ginzburg-Kaplan扭加倍zeta积分，证明其有理性与函数方程，进而定义了经典群乘一般线性群某类表示对应的非正规化$\gamma$-因子。

Sums of algebraic trace functions twisted by arithmetic functions

We obtain new bounds for short sums of isotypic trace functions associated to some sheaf modulo prime $p$ of bounded conductor, twisted by the Mobius function and also by the generalised divisor function. These trace functions include Kloosterman sums and several other classical number theoretic objects. Our bounds are nontrivial for intervals of length at least $p^{1/2+\varepsilon}$ with an arbitrary fixed $\varepsilon >0$, which is shorter than the length at least $p^{3/4+\varepsilon}$ in the case of the Mobius function and at least $p^{2/3+\varepsilon}$ in the case of the divisor function r…

A simple twisted relative trace formula

In this article we derive a simple twisted relative trace formula.

实二次域上扭三元积伽罗瓦表示的（代数）p-进阿廷形式化

(Algebraic) $ p $-adic Artin formalism of twisted triple product Galois representations over real quadratic fields

本文将Büyükboduk等人的代数分解框架推广到实二次域，研究扭三元积伽罗瓦表示的分解问题，并首次在秩(0,2)情形下取得分解结果，推动了全实域上三元积动机的研究。

Integrality of twisted L-values of elliptic curves

Under suitable, fairly weak hypotheses on an elliptic curve $E/\mathbb{Q}$ and a primitive non-trivial Dirichlet character $χ$, we show that the algebraic $L$-value $\mathscr{L}(E,χ)$ at $s=1$ is an algebraic integer. For instance, for semistable curves $\mathscr{L}(E,χ)$ is integral whenever $E$ admits no isogenies defined over $\mathbb{Q}$. Moreover we give examples illustrating that our hypotheses are necessary for integrality to hold.

Twisted linear periods and a new relative trace formula

We study the linear periods on $GL_{2n}$ twisted by a character using a new relative trace formula. We establish the relative fundamental lemma and the transfer of orbital integrals. Together with the spectral isolation technique of Beuzart-Plessis--Liu--Zhang--Zhu, we are able to compare the elliptic part of the relative trace formulae and to obtain new results generalizing Waldspurger's theorem in the $n=1$ case.