arXiv subjects

Rafail Psyroukis

Publications and source records attributed to Rafail Psyroukis.

5 recordsLinked to original sources

arXiv2025

A Fourier-Jacobi Dirichlet series attached to modular forms of $SO(2,4)$

We consider a Dirichlet series $D(F,G;s)$ attached to two automorphic forms $F$ and $G$ of an orthogonal group of real signature $(2,4)$, involving their Fourier--Jacobi coefficients. When $F$ is a Hecke eigenform and $G$ a lift of a Jacobi-Poincar\'e series, our main result gives that $D(F,G;s)$ is equal to the standard $L$-function attached to $F$, up to an explicit constant. To establish this, we use a correspondence between binary Hermitian forms and ideals of quaternion algebras, as established by Latimer, together with the fact that the even Clifford algebra of a three-dimensional definite quadratic space can be identified with a quaternion division algebra. Our work should be seen as a generalisation of a work of Kohnen and Skoruppa, whose result corresponds to the case of the orthogonal group of real signature $(2,3)$.

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arXiv2025

A Fourier-Jacobi Dirichlet series for cusp forms on orthogonal groups

We investigate a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms $F,G$ for orthogonal groups of signature $(2,n+2)$. In the case when $F$ is a Hecke eigenform and $G$ is a Maass lift of a Poincaré series, we establish a connection with the standard $L$-function attached to $F$. What is more, we find explicit choices of orthogonal groups, for which we obtain a clear-cut Euler product expression for this Dirichlet series. Through our considerations, we recover a classical result for Siegel modular forms, first introduced by Kohnen and Skoruppa, but also provide a range of new examples, which can be related to other kinds of modular forms, such as paramodular, Hermitian, and quaternionic.

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arXiv2025

A note on the twisted degree $6$ $L$-function for Hermitian cusp forms of degree $2$

Let $F$ be a cuspidal Hermitian eigenform of degree two over $\mathbb{Q}(i)$, with first Fourier-Jacobi coefficient not identically zero. Building on a paper by Das and Jha, we prove the meromorphic continuation to $\mathbb{C}$ and the functional equation of a degree six $L$-function attached to $F$ by Gritsenko, twisted by a Dirichlet character.

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arXiv2024

Analytic Properties of an Orthogonal Fourier-Jacobi Dirichlet Series

We investigate the analytic properties of a Dirichlet series involving the Fourier-Jacobi coefficients of two cusp forms for orthogonal groups of signature $(2,n+2)$. Using an orthogonal Eisenstein series of Klingen type, we obtain an integral representation for this Dirichlet series. In the case when the corresponding lattice has only one $1$-dimensional cusp, we rewrite this Eisenstein series in the form of an Epstein zeta function. If additionally $4 \mid n$, we deduce a theta correspondence between this Eisenstein series and a Siegel Eisenstein series for the symplectic group of degree $2$. We obtain, in this way, the meromorphic continuation of the Dirichlet series to $\mathbb{C}$ as a corollary. In the case of the $E_8$ lattice, we are able to further deduce a precise functional equation for the Dirichlet series.

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arXiv2023

On a Rankin-Selberg integral of three Hermitian cusp forms

Let $K = \mathbb{Q}(i)$. We study the Petersson inner product of a Hermitian Eisenstein series of Siegel type on the unitary group $U_{5}(K)$, diagonally-restricted on $U_2(K)\times U_2(K)\times U_1(K)$, against two Hermitian cuspidal eigenforms $F, G$ of degree $2$ and an elliptic cuspidal eigenform $h$ (seen as a Hermitian modular form of degree 1), all having weight $k \equiv 0 \pmod 4$. We obtain, through this consideration, an integral representation of a certain Dirichlet series, together with an additional residue term. By taking $F$ to belong in the Maass space, we are able to show that the Dirichlet series possesses an Euler product. Moreover, its $p$-factor for an inert prime $p$ can be essentially identified with the twist by $h$ of a degree six Euler factor attached to $G$ by Gritsenko. The question of whether the same holds for the primes that split remains unanswered here, even though we make considerable steps in that direction too. Our paper is inspired by a work of Heim, who considered a similar question in the case of Siegel modular forms.

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