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In two recent papers by Barry (2010) [29] and (2011) [30], it is conjectured that Somos-4 admits a solution expressed in terms of Hankel determinant with its elements satisfying a convolution recursion relation. In this paper, Barrys conjecture on Somos-4 is firstly confirmed. Actually, we present a more generalized result. The proof is mainly based on new findings on properties for so-called BlockHankel determinants. The method can also be used to prove another conjecture proposed by Michael Somos, which has been solved by Guoce Xin.

Let X be a smooth, projective, geometrically connected curve over a finite field Fq, and let G be a split semisimple algebraic group over Fq. Its dual group Gˆ is a split reductive group over Z. Conjecturally, any l-adic Gˆ-local system on X (equivalently, any conjugacy class of continuous homomorphisms π1(X)→Gˆ(Q¯¯¯¯l)) should be associated with an everywhere unramified automorphic representation of the group G. We show that for any homomorphism π1(X)→Gˆ(Q¯¯¯¯l) of Zariski dense image, there exists a finite Galois cover Y→X over which the associated local system becomes automorphic.

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By applying the residue method for period integrals and Langlands-Shahidi’s theory for residues of Eisenstein series, we study the period integrals for six spherical varieties. For each spherical variety, we prove a relation between the period integrals and certain automorphic L-functions. In some cases, we also study the local multiplicity of the spherical varieties.