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

TL;DR

This paper constructs specific endoscopic lifts related to Klein's cubic threefold, revealing new connections between modular forms, L-functions, and the geometry of certain moduli spaces of abelian surfaces.

Contribution

It introduces five new endoscopic lifts from elliptic modular forms with complex multiplication, linking them to the geometry of Klein's cubic threefold and its associated L-functions.

Findings

Constructed five endoscopic lifts with specific properties.

Identified the form of the associated spinor L-functions.

Showed that certain L-functions do not appear in the cohomology of the moduli space.

Abstract

Let $A^{lev}_{11}$ 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 $A^{lev}_{11}$ 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 $A^{lev}_{11}$ 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 $A^{lev}_{11}$.