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Thanasis Bouganis

Publications and source records attributed to Thanasis Bouganis.

13 recordsLinked to original sources
arXiv2026

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-Poincaré 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 $(2,3)$.

math.NT↗
arXiv2026

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.

math.NT↗
arXiv2025

On an analogue of the doubling method in coding theory

It is well known that there is a deep relationship between codes and lattices. Concepts from coding theory are related to concepts of lattice theory as, for example, weight enumerators to theta series, MacWilliams identity to Jacobi identity, and Gleason's theorem to Hecke's theorem. In this framework, higher-genus (or multiple) weight enumerators are related to Siegel theta series, which opens up the possibility of introducing concepts from the theory of higher-rank modular forms to coding theory. There has been important work in this direction, for example Runge introduced a coding theory analogue of Siegel's $Φ$-operator and Nebe analogues of Hecke operators. In this paper, we show that the celebrated Doubling Method from the theory of higher-rank modular forms has a coding theory analogue. Given the impact that the Doubling Method has had in the study of higher-rank modular forms, one may expect that its analogue may prove useful to the study of higher-genus weight enumerators. In this paper we use it to solve an analogue of the "basis problem". That is, we express "cuspidal" polynomials which are invariant under a Clifford-Weil type group as an explicit linear combination of higher-genus weight enumerators of self-dual codes of that type.

math.NT↗
arXiv2023

The MacWilliams Identity for the Skew Rank Metric

The weight distribution of an error correcting code is a crucial statistic in determining it's performance. One key tool for relating the weight of a code to that of it's dual is the MacWilliams Identity, first developed for the Hamming metric. This identity has two forms: one is a functional transformation of the weight enumerators, while the other is a direct relation of the weight distributions via (generalised) Krawtchouk polynomials. The functional transformation form can in particular be used to derive important moment identities for the weight distribution of codes. In this paper, we focus on codes in the skew rank metric. In these codes, the codewords are skew-symmetric matrices, and the distance between two matrices is the skew rank metric, which is half the rank of their difference. This paper develops a $q$-analog MacWilliams Identity in the form of a functional transformation for codes based on skew-symmetric matrices under their associated skew rank metric. The method introduces a skew-$q$ algebra and uses generalised Krawtchouk polynomials. Based on this new MacWilliams Identity, we then derive several moments of the skew rank distribution for these codes.

cs.IT↗
arXiv2022

Algebraicity of L-values attached to Quaternionic Modular Forms

In this paper we prove the algebraicity of some L-values attached to quaternionic modular forms. We follow the rather well established path of the doubling method. Our main contribution is that we include the case where the corresponding symmetric space is of non-tube type. We make various aspects very explicit such as, the doubling embedding, coset decomposition, and the definition of algebraicity of modular forms via CM points.

math.NT↗
arXiv2020

Algebraicity of special $L$-values attached to Siegel-Jacobi modular forms

In this work we obtain algebraicity results on special $L$-values attached to Siegel-Jacobi modular forms. Our method relies on a generalization of the doubling method to the Jacobi group obtained in our previous work, and on introducing a notion of near holomorphy for Siegel-Jacobi modular forms. Some of our results involve also holomorphic projection, which we obtain by using Siegel-Jacobi Poincaré series of exponential type.

math.NT↗
arXiv2018

On the Rankin-Selberg method for vector valued Siegel modular forms

In this work we use the Rankin-Selberg method to obtain results on the analytic properties of the standard $L$-function attached to vector valued Siegel modular forms. In particular we provide a detailed description of its possible poles and obtain a non-vanishing result of the twisted $L$-function beyond the usual range of absolute convergence. We remark that these results were known in this generality only in the case of scalar weight Siegel modular forms. As an interesting by-product of our work we establish the cuspidality of some theta series.

math.NT↗
arXiv2011

Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups; the CM method

In this work we prove the so-called "torsion congruences" between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results in the special cases of $n=1$ and $n=2$. In both of these cases we also explain their implications for some particular "motives", as for example elliptic curves with complex multiplication.

math.NT↗
arXiv2010

On the non-commutative Main Conjecture for elliptic curves with complex multiplication

In arXiv:math/0404297 a non-commutative Iwasawa Main Conjecture for elliptic curves over $\mathbb{Q}$ has been formulated. In this note we show that it holds for all CM-elliptic curves $E$ defined over $\mathbb{Q}$. This was claimed in (loc.\ cit.) without proof, which we want to provide now assuming that the torsion conjecture holds in this case. Based on this we show firstly the existence of the (non-commutative) $p$-adic $L$-function of $E$ and secondly that the (non-commutative) Main Conjecture follows from the existence of the Katz-measure, the work of Yager and Rubin's proof of the 2-variable main conjecture. The main issues are the comparison of the involved periods and to show that the (non-commutative) $p$-adic $L$-function is defined over the conjectured in (loc.\ cit.) coefficient ring. Moreover we generalize our considerations to the case of CM-elliptic cusp forms.

math.NT↗
arXiv2010

Non-abelian congruences between special values of $L$-functions of elliptic curves; the CM case

In this work we prove congruences between special values of elliptic curves with CM that seem to play a central role in the analytic side of the non-commutative Iwasawa theory. These congruences are the analogue for elliptic curves with CM of those proved by Kato, Ritter and Weiss for the Tate motive. The proof is based on the fact that the critical values of elliptic curves with CM, or what amounts to the same, the critical values of Grössencharacters, can be expressed as values of Hilbert-Eisenstein series at CM points. We believe that our strategy can be generalized to provide congruences for a large class of $L$-values.

math.NT↗
arXiv2008

Special values of L-functions and false Tate curve extensions II

In this paper we show how one can combine the p-adic Rankin-Selberg product construction of Hida with freeness results of Hecke modules of Wiles to establish interesting congruences between special values of L-functions. These congruences is a part of some deep conjectural congruences that follow from the work of Kato on the non-commutative Iwasawa theory of the false Tate curve extension.

math.NT↗
arXiv2005

Algebraicity of L-values for elliptic curves in a false Tate curve tower

Let $E$ be an elliptic curve over $Q$, and $τ$ an Artin representation over $Q$ that factors through the non-abelian extension $Q(\sqrt[p^n]{m},μ_{p^n})/Q$, where $p$ is an odd prime and $n,m$ are positive integers. We show that $L(E,τ,1)$, the special value at $s=1$ of the $L$-function of the twist of $E$ by $τ$, divided by the classical transcendental period $Ω_{+}^{d^+}|Ω_{-}^{d^-}|ε(τ)$ is algebraic and Galois-equivariant, as predicted by Deligne's conjecture.

math.NT↗