Holomorphy of adjoint L-functions for GL(n): $n\le 4$

Abstract

We show entireness of complete adjoint L-functions associated to any cuspidal representations of ${{\,\mathrm{GL}\,}}(3)$ or ${{\,\mathrm{GL}\,}}(4)$ over an arbitrary number field. Twisted cases are also investigated.

Tables of notation

F, ${\mathbb {A}}_F$	number field and its adele ring
$\Sigma _F,$ $\Sigma _{F,{{\,\mathrm{fin}\,}}},$ $\Sigma _{F,\infty }$	set of places of F;  set of finite places; set of archimedean places
E	field extension of F,  with norm denoted by $N_{E/F}$
G	the general linear group ${{\,\mathrm{GL}\,}}(n)$, with center $Z_G$
$\omega$	central character, i.e., a unitary character of $Z_G({\mathbb {A}}_F)$
$\varphi$	the test function in the trace formula
${\text {K}}(x,y)={\text {K}}^{\varphi }(x,y)$	the kernel function in the trace formula relative to $\varphi$
${\text {K}}_0(x,y),$ ${\text {K}}_{{\text {Res}}}(x,y),$ ${\text {K}}_{{\text {Eis}}}(x,y)$	part of ${\text {K}}(x,y)$ from cuspidal spectrum, residual spectrum, and continuous spectrum, respectively
${\text {K}}_{\infty }(x,y)={\text {K}}_{{\text {Eis}}}(x,y)+{\text {K}}_{{\text {Res}}}(x,y)$	non-cuspidal kernel
$\Xi _F$	set of unitary characters on $F^{\times }\backslash {\mathbb {A}}_F^{\times }$ which are trivial on ${\mathbb {R}}^{\times }_+$
$\mathcal {S}_0({\mathbb {A}}_F^n)$	space of generalized Gaussians defined in Sect. 1.3
$\eta =(0,0,\ldots ,0,1)$	a vector in $F^n$
$f(x,s)=f(x,\Phi ,\tau ;s)$	a Tate integral defined in Sect. 1.3, $\tau \in \Xi _F,$ $\Phi \in \mathcal {S}_0({\mathbb {A}}_F^n)$
P	in first two sections it means the standard parabolic subgroup of G of type $(n-1,1);$ from Sect. 3 it represents an arbitrary standard parabolic subgroup
B,  N	B is the standard Borel subgroup (i.e., nonsingular upper triangle matrices), with unipotent radical N
$P_0,$ $B_0$	$Z_G\backslash P$ (the mirabolic subgroup), $Z_G\backslash B,$ respectively
$E(x,s)=E_P(x,\Phi ,\tau ;s)$	Eisenstein series defined in Sect. 1.3
$\pi ,$ ${\tilde{\pi }}$	cuspidal representation and its contragredient
$\Lambda (s,\pi ,{\text {Ad}}\otimes \tau ),$ $\Lambda (s,\tau )$	complete adjoint L-function of $\pi$ twisted by $\tau$; complete Hecke L-function of $\tau$
$L(s,\pi ,{\text {Ad}}\otimes \tau ),$ $L(s,\tau )$	finite adjoint L-function of $\pi$ twisted by $\tau$; finite Hecke L-function of $\tau$ (i.e., with archimedean factors removed)
$\theta ,$ C(w),  $Q_k,$ $X_G,$ $R_k,$ $V_k,$ $V_k',$ $N_k,$ $\Psi _s,$ F(x, T)	notations defined in Sect. 2.1
${\mathbb {G}}_m$, ${\mathbb {G}}_a$	the multiplicative group and the additive group
S	parabolic subgroup of ${{\,\mathrm{GL}\,}}(4)$ of type (2, 1, 1);  and $S_0=Z_G\backslash S$
Q	parabolic subgroup of ${{\,\mathrm{GL}\,}}(4)$ of type (1, 2, 1)
${\text {K}}^{(k)}(x,y)={\text {K}}_{\infty }^{(k)}(x,y),$ $k\ge 2$	geometric expression for the spectral contribution ${\text {K}}_{\infty }^{(k)}(x,y)$ in the Fourier expansion (see (12))
$\mathcal {R}(1/2;\tau )^-$	the domain defined in (57)
$\mathcal {D}_{\chi }(\epsilon ),$ $\mathcal {R}(\beta ;\chi ,\varvec{\epsilon })$	zero free regions (see (63) and (64)); and $\mathcal {C}$ is certain boundary (see Sect. 3.1)
$\Psi (s,W_1,W_2;\lambda )$	Rankin–Selberg periods for non-cuspidal representations (see (55))
$R_{\varphi }(s,\lambda ;\phi _2)$	ratio of $\Psi (s,W_1,W_2;\lambda )$ and the L-function (see (59))
$\mathcal {F}(\varvec{\kappa };s)=\mathcal {F}(\varvec{\kappa };s,P,\chi )$	defined to be $R_{\varphi }(s,\varvec{\kappa };\phi )\Lambda (s,\pi _{\varvec{\kappa }}\otimes \tau \times {\widetilde{\pi }}_{-\varvec{\kappa }})$ (see Sect. 3.1)
$\mathcal {Z}_{m,P}(s,\tau )$	partial residues of $I_{\infty }^{(1)}(s,\tau )$ defined in (65)