Endoscopic lifts to the Siegel modular threefold related to Klein's cubic threefold

Abstract

Let $A_{11}^{lev}$ be the moduli space of (1,11)-polarized abelian surfaces with level structure of canonical type. Let $\chi$ be a finite character of order 5 with conductor 11. In this paper we construct five endoscopic lifts ${\Pi }_i,0\le i\le 4$ from two elliptic modular forms $f\otimes {\chi }^i$ of weight 2 and $g\otimes {\chi }^i$ of weight 4 with complex multiplication by $Q(\sqrt{-11})$ such that ${{\Pi }_i}_{\infty }$ gives a non-holomorphic differential form on $A_{11}^{lev}$ for each $i$. Then the spinor L-function is of form $L(f\otimes {\chi }^i,s-1)L(g\otimes {\chi }^i,s)$ such that $L(g\otimes {\chi }^i,s)$ does not appear in the L-function of $A_{11}^{lev}$ for any $i$. The existence of such lifts is motivated by the computation of the L-function of Klein's cubic hypersurface which is a birational smooth model of $A_{11}^{lev}$.

arXiv commentTo appear in American Journal of Math