We explicitly give an inner product formula and a Bessel period formula for theta series on ${\mathrm{GSp}}_4$, which was studied by Harris, Soudry, and Taylor. As a consequence, we prove a nonvanishing of the theta series of small weights and we give a criterion for the nonvanishing of the theta series modulo a prime.

In this paper, we consider a problem of counting multiplicities. We fix a counting function of multiplicity of rational points in a hypersurface of a projective space over a finite field, and we give an upper bound for the sum with respect to this counting function in terms of the degree of the hypersurface and the dimension and the cardinality of the finite field. This upper bound gives a description of the complexity of the singular locus of this hypersurface. In order to obtain this upper bound, we introduce a notion called an intersection tree by intersection theory. We construct a sequence of intersections, such that the multiplicity of a singular rational point is equal to that of one of the irreducible components in these intersections. The multiplicities of these irreducible components constructed above are bounded by their multiplicities in the intersection tree.

Our goal of this paper is to show that Ho’s vector-valued Morrey spaces can be realized as a special case of pointwise multiplier spaces. This extends Lemarié-Rieusset’s theorem. One cannot extend his theorem directly, because we are handling Banach lattices instead of Lebesgue spaces. It turns out that mixed Morrey spaces, Lorentz–Morrey spaces, and Orlicz–Morrey spaces fall under the scope of the framework in this paper. This work can be located as a continuation of the huge study of multiplier spaces collected in the book of Maz’ya.

It is shown that the direct limit of the semistandard decomposition tableau model for polynomial representations of the queer Lie superalgebra exists, which is believed to be the crystal for the upper half of the corresponding quantum group. An extension of this model to describe the direct limit combinatorially is given. Furthermore, it is shown that the polynomial representations may be recovered from the limit in most cases.

Let $f:{\mathbb{C}}^n⟶\mathbb{C}$ be a weighted homogeneous polynomial with a finite number of critical points. The aim of this paper is to give a formula for the Łojasiewicz exponent of f at infinity in terms of its weights $w_i$, when $w_i\ge 2$. We also give a characterization of properness for weighted homogeneous polynomial mapppings. As a corollary, we prove the Jacobian conjecture for weighted and preweighted homogeneous polynomial mappings with positive weights.

The aim of this paper is to give a characterization of the $L^p$ range $(p\ge 2)$ of a generalized Poisson transform of the hyperbolic space $B({\mathbb{H}}^n),(n\ge 2)$, over the classical field of the quaternions $\mathbb{H}$. Namely, if f is a hyperfunction in the boundary of $B({\mathbb{H}}^n)$, then we show that f is in $L^p(\partial B({\mathbb{H}}^n))$ if and only if its generalized Poisson transform satisfies a Hardy-type growth condition. An explicit expression of the generalized spherical functions is given.

For an arithmetic surface $X\to B=Spec{O}_K$, the Deligne pairing $⟨,⟩:Pic(X)\times Pic(X)\to Pic(B)$ gives the “schematic contribution” to the Arakelov intersection number. We present an idelic and adelic interpretation of the Deligne pairing; this is the first crucial step for a full idelic and adelic interpretation of the Arakelov intersection number. For the idelic approach, we show that the Deligne pairing can be lifted to a pairing ${⟨,⟩}_i:ker(d_{\times }^1)\times ker(d_{\times }^1)\to Pic(B)$, where $ker(d_{\times }^1)$ is an important subspace of the two-dimensional idelic group ${\mathbf{A}}_X^{\times }$. On the other hand, the argument for the adelic interpretation is entirely cohomological.