Related papers: $L^4$-norms of Hecke newforms of large level

Let $K/F$ be an unramified quadratic extension of non-Archimedian local fields with residue character not equals to 2. We prove the linear Arithmetic Fundamental Lemma for GL$_4$ with the unit element in the spherical Hecke Algebra. In this…

Given a congruence of Hecke eigenvalues between newforms of weight $2$, we prove, under certain conditions, a congruence between corresponding weight-$3/2$ forms.

Let $F$ be a $G L(3)$ Hecke-Maass cusp form of prime level $P_1$ and let $f$ be a $G L(2)$ Hecke-Maass cuspform of prime level $P_2$. In this article, we will prove a subconvex bound for the $G L(3) \times G L(2)$ Rankin-Selberg…

Following the trend of coefficient bound problems in Geometric Function Theory, in the present paper, we obtain the sharp bound of $|H_3(1)|$ for the class $\mathcal{S}^*$, of starlike functions and $\mathcal{SL}_q^*$, of $q$- starlike…

Let $f$ be a newform of prime level $p$ with any central character $\chi\, (\bmod\, p)$, and let $g$ be a fixed cusp form or Eisenstein series for $\hbox{SL}_{2}(\mathbb{Z})$. We prove the subconvexity bound: for any $\varepsilon>0$,…

It is believed that Dirichlet series with a functional equation and Euler product of a particular form are associated to holomorphic newforms on a Hecke congruence group. We perform computer algebra experiments which find that in certain…

Let T_k denote the Hecke algebra acting on newforms of weight k and level N. We prove that the power of p dividing the index of T_k inside its normalisation grows at least linearly with k (for fixed N), answering a question of Serre. We…

Let $\phi$ be a spherical Hecke-Maass cusp form on the non-compact space $\mathrm{PGL}_3(\mathbb{Z})\backslash\mathrm{PGL}_3(\mathbb{R})$. We establish various pointwise upper bounds for $\phi$ in terms of its Laplace eigenvalue…

We investigate some key analytic properties of Fourier coefficients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms $F$ of degree 2, weight $k$ and level $N$. First, assuming that $F$ is a Hecke eigenform that is not of…

Let q be a prime and -D < -4 be an odd fundamental discriminant such that q splits in Q(\sqrt{-D}). For f a weight zero Hecke-Maass newform of level q and h the weight one theta series of level D corresponding to an ideal class group…

We prove a Weyl-type subconvexity bound for the central value of the $L$-function of a Hecke-Maass form or a holomorphic Hecke eigenform twisted by a quadratic Dirichlet character, uniform in the archimedean parameter as well as the…

We study a mean value of the shifted convolution problem over the Hecke eigenvalues of a fixed non-holomorphic cusp form. We attain a result also for a weighted case. Furthermore, we point out that the proof yields analogous upper bounds…

In this paper a theory of Hecke operators for higher order modular forms is established. The definition of cusp forms and attached L-functions is extended beyond the realm of parabolic invariants. The role of representation theoretic…

In this paper, we study quadratic forms in spaces of holomorphic cusp forms. We show, conditionally, that when two quadratic forms in Hecke eigenforms share no common diagonal terms, their inner product is expected to converge to the sum of…

Let f be an L^2-normalized Hecke--Maass cuspidal newform of level N and Laplace eigenvalue \lambda. It is shown that |f|_\infty <<_{\lambda, \epsilon} N^{-1/12 + \epsilon} for any \epsilon>0. The exponent is further improved in the case…

Let $S_2^*(q)$ be the set of primitive Hecke eigenforms of weight 2 and prime level $q$. For $p$ prime and $t\in \mathbb{R}$, we prove asymptotic formulas for the sums $$ \mathcal {A}(p^j,q,t)=\sum_{f\in S_2^*(q)}…

A reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q twisted by the ell-th Hecke eigenvalue as the fourth moment of automorphic L-functions of level ell twisted by the q-th Hecke…

In this work, we establish several results on distinguishing Siegel cusp forms of degree two. In particular, a Hecke eigenform of level one can be determined by its second Hecke eigenvalue under a certain assumption. Moreover, we can…

We prove the optimal upper bound sum || f ||_4^4 << q^{eps} where f runs over an orthonormal basis of Maass cusp forms of prime level q and bounded spectral parameter.

In this note, following results from Henri Cohen and Winfried Kohnen, we show that for all integer k greater than 12 and divisible by 4, there exists a cuspidal eigenform of weight k for the full modular group SL2(Z) such that its Hecke…