Results concerning automorphic forms and L-functions of higher rank
Talk by Paul Nelson
Motivating goal: Subconvexity for twists
\(F\) is a number field. \(\pi:\mathrm{GL}_n\), \(\chi:\mathrm{GL}_1\) \[\begin{align} \label{subconvex-twists} L(\pi\times\chi,&1/2)\ll_\pi C(\chi)^{n/4-\delta}\tag{$\star$} \\ &\Bigg\updownarrow \substack{\text{(Lapid-Mao) $n$ is even, $\pi$ comes from $\mathrm{SO}(n+1)$},\\ \text{$\chi$ quadratic, $C_\mathrm{fin}(\chi)\approx\square$-free}}\notag\\ \text{Fourier} & \text{ coefficients on $Mp_n$}\notag \end{align}\]
Theorem 1 (N., 2021). \(\eqref{subconvex-twists}\) for \(\chi=|\cdot|^{it}\), \(F=\mathbb{Q}\)
Theorem 2 (Yueke Hu, PN, today). variant of \(\eqref{subconvex-twists}\) for \(C_\mathrm{fin}(\chi)=\square\), e.g. \(p^2\rightarrow \infty\)
Subconvexity for \(L(\pi\times\check{\sigma},1/2)\), where
\(\pi:\mathrm{GL}_{n+1}\), \(\sigma:\mathrm{GL}_n\), \(|F|\ni \mathfrak{p}\)
Assuming:
bounded ramification outside \(\mathfrak{p}\)
\(\mathfrak{p}\) finite of norm \(\mathfrak{q}\).
principal series: \(\pi_\mathfrak{p}=\chi_1\boxplus\ldots\boxplus\chi_{n+1}\), \(\sigma_\mathfrak{p}=\eta_1\boxplus\ldots\boxplus\eta_n\)
uniform growth: \(C(\chi_i/\eta_j)=T\) (\(\Leftrightarrow\mathrm{Cond}(\pi\times\check{\sigma})=T^{n(n+1)}\)), where
\(T=\) max analytic conductor of \(\chi_i\), \(\eta_j\) \(=\) power of \(q\).
Even depth \(T=\mathfrak{q}^{2m}\)
comes from \(U(n+1)\times U(n)\) (something about anisotropic)
Theorem 1 (N., 2020-2023). \(\mathfrak{p}\) archimedean, remove (3), (5)
Theorem 2 (N., 2021). remove (6), \(\sigma\) Eisenstein.
Marshall: \(\mathfrak{p}\) fixed, \(C(\chi_i/\chi_j)=C(\eta_i/\eta)=T\), for \(i\neq j\) (wall-avoidance)
Proof scheme
\(\pi:G=\mathrm{GL}(n+1)\), \(\sigma:H=\mathrm{GL}(n)\)
Choose \(u\in\sigma\) Estimate both sides: \(\omega\in C_c^\infty(G(\mathbb{A}))\)
\[\sum_\pi|L^{(\mathfrak{p})}(\pi\times\check{\sigma},1/2)|^2\sum_{v\in\mathcal{B}(\pi_\mathfrak{p})} \int_{h\in H(F_\mathfrak{p})}\langle h\pi(\omega)v,v\rangle\langle u,hu\rangle\mathrm{d}h=(\ldots)\] This is called the spectral side
\[(\ldots)=\int_{x,y\in H(F)\backslash H(\mathbb{A})}u(x)\overline{u(y)}\sum_{\gamma\in G(F)} \omega(x^{-1}\gamma y)\mathrm{d}x\mathrm{d}y\]
Spectral side
\(\mathfrak{p}\subseteq\mathfrak{o}\) (local) \[1\neq \psi: \mathfrak{p}/\mathfrak{p}^2\rightarrow U(1)\] \[\chi:\mathrm{GL}_1(\mathfrak{o}/\mathfrak{p}^2)\rightarrow\mathbb{C}^{\ast}\leadsto\xi_\chi\in\mathfrak{o}/\mathfrak{p}\] \[\chi(1+x)=\psi(\xi_\chi x),\,\forall x\in\mathfrak{p}\]
\[G,H,M,M_H=\mathrm{GL}_{n+1},\mathrm{GL}_n,\operatorname{Mat}_{n+1}, \operatorname{Mat}_n\] \[\tau\in M(\mathfrak{o}/\mathfrak{p}),\quad \pi=\text{representation of } G(\mathfrak{o}/\mathfrak{p}^2)\]
Call \(v\in \pi\) localized at \(\tau\) if \(\forall x\in M(\mathfrak{o}/\mathfrak{p}^2)\), \(x\equiv 0 \pmod{\mathfrak{p}}\)
\[\pi(1+x)v=\psi(\operatorname{trace}(x\tau))v\]
Fact 1. “Often” \(\dim{|{v|}}=1\). \(P_\pi:=P_\tau=\) characteristic polynomial depends only on \(\pi\).
Example 1. \(P_{\chi}=t-\xi_t,\quad P_{\boxplus \pi_i}=\prod_iP_{\pi_i}\)
Method of Nelson-Venkatesh:
\(\forall (\pi,\sigma)\) of \((\mathrm{GL}_{n+1}(\mathfrak{o}/\mathfrak{p}^2), \mathrm{GL}_n(\mathfrak{o}/\mathfrak{p}^2))\) with \((P_\pi,P_\sigma)=(1)\), there exists \[\begin{pmatrix}\tau_H&\ast\\\ast&\ast\end{pmatrix}=\tau\in M(\mathfrak{o}/\mathfrak{p}),\quad \tau_H\in M_H(\mathfrak{o}/\mathfrak{p})\] such that \(P_\tau=P_\pi\) and \(P_{\tau_H}=P_\sigma\)
There exists unit vectors \(v\in\pi,\,u\in \sigma\) localized at \(\tau,\tau_H\), respectively, such that for \(h\in H(\mathfrak{o}/\mathfrak{p}^2)\)
\[\langle hv,v\rangle\langle u,hu\rangle= \begin{cases} 1\text{ if }h\equiv 1\pmod{\mathfrak{p}},\\ 0\text{ else}. \end{cases}\] (+ something similar over local fields)
Geometric side after Cauchy-schwarz
\(F=\mathfrak{o}/\mathfrak{p}=\mathbb{F}_q\)
\[G,H,M,M_H=\text{ points over }F\]
\[\tau\in M,e=\begin{pmatrix}0\\ \vdots\\0\\1\end{pmatrix}, e^{\ast}=\begin{pmatrix}0\!&\!\ldots\!&\!0\!&\!1\end{pmatrix}\]
Lemma 1. \[(P_\tau,P_{\tau_H})=(1)\Leftrightarrow e,e^\ast:\tau-\text{cyclic}\]
Definition 1. If any (hence both) of the above holds, we say that \(\tau\) is stable
For \(\tau\in M_{\text{stable}}, a\in G_\tau\), then the set \[X_{\tau,a}=\{y=H_{\tau_H}:ay\in HG_\tau\}\] is a closed subscheme.
Theorem 3 (HN). Assume \(q\) is odd, \(n+1\neq 3\). Then \[\frac{X_{\tau,a}}{H_{\tau_H}}\ll \frac{1}{q} \Leftarrow X_{\tau,a}\neq H_{\tau_H}\]
\(\mu(y)\nu(y^{-1})=1\), where \(\mu,\nu:M_{H,\tau_H}\rightarrow M_{\tau}\Big|_{\substack{e^\ast\mu(x)=e^\ast axa^{-1}\\\nu(x)e=axa^{-1}e}}\)
Lemma 2. If \(X_{\tau,a}=H_{\tau_H}\) then, \[\begin{align} \mu&=\nu \label{lin}\tag{linear} \\ \mu(xy)&=\mu(x)\mu(y) \label{quad}\tag{quadratic} \end{align}\]
Theorem 4. \(\eqref{lin} \Leftrightarrow a^2\in Z\)
Theorem 5. \(\eqref{lin}+\eqref{quad}\) \(\Leftrightarrow a\in Z\)
Proof. (of Theorem 4) Let \(A_j=e^{\ast}\tau^jae\), \(B_j=e^{\ast}\tau^ja^{-1}e\) \[\mu(\mathbf{1}_H)-v(\mathbf{1}_H)=B_0a-A_0a^{-1}\] If either \(B_0\neq 0\) or \(A_0\neq 0\) then \(a^2\in Z\). Else continue with \(\tau_H,\tau_H^2,\tau_H^3,\ldots\) ◻
For Theorem 5, use that \(\mu(\mathbf{1}_H)^2=\mu(\mathbf{1}_H)\), \(\tau^2+c_1\tau+c_0=0\).
Ramon Nunes
Assistant Professor of Mathematics
My reasearch interests lie in Number theory. More specifically, in Analytic Number Theory. I am also interested in Automorphic forms, specially in its applications to Number Theory.