Adjoint L-value as period integrals

Abstract: For an elliptic cusp form $F$, an idea of Waldspurger of computing the period of a theta lift of $F$ for a quadratic space $V=W\oplus W^\perp$ over an orthogonal Shimura subvariety $S_W\times S_{W^\perp}$ is two-folds:

(S) Split $\theta(\Phi)(\tau,g,g^\perp)=\theta(\phi)(g)\cdot\theta(\phi^\perp)(g^\perp)$ ($\tau\in{\mathbb C}$ with $\mathrm{Im}(\tau)>0$ and $g^?\in{\mathrm O}_{W^?}({\mathbb A})$) for a decomposition $\Phi=\phi\otimes\phi^\perp$ ($\phi$ and $\phi^\perp$ Schwartz-Bruhat functions on $W_{\mathbb A}$ and $W_{\mathbb A}^\perp$);

(R) For the theta lift $\Theta(F)(g)=\int_{X_\tau}F(\tau)\theta(\phi)(\tau,g)d\mu_\tau$ with an $\SL(2)$-Shimura curve $X_\tau$, the period $P$ over the Shimura subvariety $S\times S^\perp$ ($S$ for ${\mathrm O}(W)$ and $S^\perp$ for ${\mathrm O}(W^\perp)$) is given by: $$ \int_{S\times S^\perp}\int_{X_\tau}F(\tau)\theta(\phi)(\tau;g)d\mu_\tau dg=\int_{X_\tau}F(\tau)\left(\int_{S^\perp}\theta(\phi^\perp)(\tau;g^\perp)dg^\perp\right)\cdot\left(\int_{S}\theta(\phi_0)(\tau;g_0)dg\right)d\mu_\tau. $$ Then invoke the Siegel-Weil formula to convert inner integrals into the Siegel-Weil Eisenstein series $E(\phi)$ and $E(\phi^\perp)$, reaching Rankin-Selberg integral $$P=\int_{X_\tau}F(\tau)E(\phi^\perp) E(\phi_0)d\mu_\tau=\text{$L$-value}.$$

Waldspurger applied in the early 1980s this scheme to $V=(M_2({\mathbb Q}),\mathrm{det})$ with a splitting $V:=M_2({\mathbb Q})\cong E\oplus E$ for a quadratic field $E={\mathbb Q}[\sqrt{D}]$ over ${\mathbb Q}$ and expressed the period by the central critical value of the Hecke L-function $L(s,F\otimes\left(\frac{D}{}\right))$. In this talk, we apply his idea to a 4-dimensional quadratic space over ${\mathbb Q}$ which produces the quaternionic Doi-Naganuma lift to the quadratic extension $E={\mathbb Q}[\sqrt{D}]$ of ${\mathbb Q}$ (including ${\mathbb Q}\times{\mathbb Q}$) as a theta left, and we compute the period in terms of the adjoint L-value $L(1,Ad(F)\otimes\left(\frac{D}{}\right))$.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)