On PAC Extensions and Scaled Trace Forms

Trace forms of Galois extensions in the presence of a fourth root of unity

We study quadratic forms that can occur as trace forms of Galois field extensions L/K, under the assumption that K contains a primitive 4th root of unity. M. Epkenhans conjectured that any such form is a scaled Pfister form. We prove this conjecture and classify the finite groups G which admit a G-Galois extension L/K with a non-hyperbolic trace form. We also give several applications of these results.

二级K3包的积分磁性：CM西塔提升和2-同构迹收缩

Integral magneticity of the level-two K3 packet: CM theta lifts and a 2-isogeny trace contraction

研究二级K3包中三个亚纯模形式\(C_4\)、\(C_{6a}\)、\(C_{6b}\)的积分磁性。通过超几何变换、形式等同及相关公式，证明其分母为1的性质及奇素数可除性，处理2的情况，得出二级K3包具有全局分母为1的磁性这一主要贡献。

A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the archimedean case

In this paper, we prove, following earlier work of Waldspurger ([Wa1], [Wa4]), a sort of local relative trace formula which is related to the local Gan-Gross-Prasad conjecture for unitary groups over a local field $F$ of characteristic zero. As a consequence, we obtain a geometric formula for certain multiplicities $m(π)$ appearing in this conjecture and deduce from it a weak form of the local Gan-Gross-Prasad conjecture (multiplicity one in tempered L-packets). These results were already known over $p$-adic fields and thus are only new when $F=\mathbb{R}$.