Related papers: An Analogue of the Dedekind Eta Function for Hecke…

We consider a special subsequence of the Fourier coefficients of powers of the Dedekind $\eta$-function, analogous to the sequence $\delta_\ell := 24^{-1} \pmod{\ell}$ on which exceptional congruences of the partition function are…

Groups of order $4$ are isomorphic to either $\mathbb{Z}/4\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. We give certain sufficient conditions permitting to specify the structure of class groups of order $4$ in the…

In this paper, we give an analogue of Wilton's product formula for Dirichlet series that satisfy Hecke's functional equation. We apply our results to obtain identities for Hecke series, L-functions associated to modular forms, Ramanujan's…

Recently, Allen et al. developed the Explicit Hypergeometric Modularity Method (EHMM) that establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. Motivated by this framework, we…

We investigate Eisenstein discriminants, which are squarefree integers $d \equiv 5 \pmod{8}$ such that the fundamental unit $\varepsilon_d$ of the real quadratic field $K=\mathbb{Q}(\sqrt{d})$ satisfies $\varepsilon_d \equiv 1…

Let $E(z,s)$ be the non-holomorphic Eisenstein series for the modular group $SL(2,{\mathbb Z})$. The classical Kronecker limit formula shows that the second term in the Laurent expansion at $s=1$ of $E(z,s)$ is essentially the logarithm of…

Let $k=\mathbb{Q}(\sqrt{-3})$, and let $c\in \mathfrak{O}_k$ be a square free algebraic integer such that $c\equiv 1~({\rm mod}~{\langle9\rangle})$. Let $\zeta_{k(c^{1/3})}(s)$ be the Dedekind zeta function of the cubic field $k(c^{1/3})$…

In this article we give an analogue of Hecke and Sturm bounds for Hilbert modular forms over real quadratic fields. Let $K$ be a real quadratic field and $\Om_K$ its ring of integers. Let $\Gamma$ be a congruence subgroup of $\SL_2(\Om_K)$…

The Dedekind eta function $\eta(\tau)$ is defined by \[\eta(\tau)=e^{\pi i\tau/12}\prod_{n=1}^{\infty}\left(1-e^{2\pi i n\tau}\right),\quad\text{when}\;\text{Im}\,\tau>0.\] It plays an important role in number theory, especially in the…

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…

In recent work, the author, in collaboration with Allen, Long, and Tu, developed the Explicit Hypergeometric Modularity Method (EHMM), which establishes the modularity of a large class of hypergeometric Galois representations in dimensions…

It is proved that the Hilbert class field of a real quadratic field ${\Bbb Q}(\sqrt{D})$ modulo a power $m$ of the conductor $f$ is generated by the Fourier coefficients of the Hecke eigenform for a congruence subgroup of level $fD$.

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 Poincar\'e series, we…

Let E be an elliptic curve having Complex Multiplication by the full ring O_K of integers of K=Q(\sqrt{-D}), let H=K(j(E)) be the Hilbert class field of K. Then the Mordell-Weil group E(H) is an O_K-module, and its structure denpends on its…

The theme of this work is the study of the Nekov\'a\v{r}-Selmer group H^1_f(K,T) attached to a twisted Hida family T of Galois representations and a quadratic number field K. The results that we obtain have the following shape: if a twisted…

In this paper we outline the Hecke theory for Hermitian modular forms in the sense of Hel Braun for arbitrary class number of the attached imaginary-quadratic number field. The Hecke algebra turns out to be commutative. Its inert part has a…

In this article, we establish an average behaviour of the normalised Fourier coefficients of the Hecke eigenforms supported at the integers represented by any primitive integral positive definite binary quadratic form of fixed discriminant…

We define an adelic version of a CM elliptic curve $E$ which is equipped with an action of the profinite completion of the endomorphism ring of $E$. The adelic elliptic curve so obtained is provided with a natural embedding into the adelic…

Let $\ell \geq 5$ be a prime, and let $\nu_\eta$ denote the Dedekind eta multiplier. For an odd integer $r$, and a real Dirichlet character $\psi$, recent work of Ahlgren, Andersen, and the author showed that quadratic congruences modulo…

An explicit formula for the quadratic mean value at $s=1$ of the Dirichlet $L$-functions associated with the odd Dirichlet characters modulo $f>2$ is known. Here we present a situation where we could prove an explicit formula for the…