On the summation of root numbers for holomorphic cusp forms of $\mathrm{PGL}_2$ with simple ramifications.

Seminar: 
Algebra and Number Theory Seminar
Event time: 
Tuesday, February 9, 2021 - 8:00pm
Location: 
Zoom
Speaker: 
Zhilin Luo
Speaker affiliation: 
University of Minnesota
Event description: 

Over a totally real field $F$ with $F\otimes_{\mathbb{Q}}\mathbb{R} \simeq \mathbb{R}^{n}$, an explicit formula for
$$
\sum_{\pi\in \mathcal{A}_{\mathrm{cusp}}(\mathfrak{N}^3,\vec{k})}\varepsilon(\frac{1}{2},\pi)
$$
is established. Here $\mathcal{A}_{\mathrm{cusp}}(\mathfrak{N}^3,\vec{k})$ is the set of cuspidal automorphic representations of $\mathrm{PGL}_2(\mathbb{A})$ whose Archimedean component is a discrete series representation of $\mathrm{PGL}_2(F\otimes_{\mathbb{Q}}\mathbb{R})$ of weight $4\vec{k}$ where $\vec{k} = (k_1,...,k_n)\in \mathbb{N}^n$, and $\mathfrak{N}$ is a square free integral ideal of $F$ corresponding to the ramified places of automorphic representations in $\mathcal{A}_{\mathrm{cusp}}(\mathfrak{N}^3,\vec{k})$. In particular, for any $\mathfrak{p}\mid \mathfrak{N}$ and $\pi\in \mathcal{A}_{\mathrm{cusp}}(\mathfrak{N}^3,\vec{k})$, $\pi_\mathfrak{p}$ is always assumed to be simple supercuspidal. This is a joint work in progress with Q. Pi and H. Wu.