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

Let A 11 be the moduli space of (1, 11)-polarized abelian surfaces with a canonical level structure. Let χ be a primitive character of order 5 with conductor 11. In this paper we construct five endoscopic lifts Πi, 0 6 i 6 4 from two elliptic modular forms f ⊗ χ of weight 2 and g ⊗ χ of weight 4 with complex multiplication by Q( √ −11) such that Πi∞ gives a non-holomorphic differential form on A 11 for each i, 0 6 i 6 4. Then their spinor L-functions are of form L(s − 1, f ⊗ χ)L(s, g ⊗ χ) such that L(s, g ⊗ χ) does not appear in the L-function of A 11 for any i, 0 6 i 6 4. The existence of such lifts is motivated by the computation of the L-function of Klein’s cubic threefold which is a birational smooth model of A 11 .