Related papers: A subconvex bound for twisted $L$-functions

Let $f$ be a $p$-primitive cusp form of level $p^{4r}$, where local representation of $f$ be supercuspidal at $p$, $p$ being an odd prime, $r\geq 1$ and $g$ be a Hecke-Maass or holomorphic primitive cusp form for…

In this paper, we prove hybrid subconvexity bounds for $\rm GL_2\times \rm GL_2$ Rankin--Selberg $L$-functions twisted by a primitive Dirichlet character $\chi$ modulo a prime power, in the $t$ and depth aspects.

We prove a Lindel\"{o}f-on-average upper bound for the second moment of the $L$-functions associated to a level 1 holomorphic cusp form, twisted along a coset of subgroup of the characters modulo $q^{2/3}$ (where $q = p^3$ for some odd…

For a $SL(2,\mathbb{Z})$ form $f$, we obtain the sub-Weyl bound \begin{equation*} L(1/2+it,f)\ll_{f,\varepsilon} t^{1/3-\delta+\varepsilon}, \end{equation*} where $\delta=1/174$, thereby crossing the Weyl barrier for the first time beyond…

We establish a sub-convexity estimate for Rankin-Selberg $L$-functions in the combined level aspect, using the circle method. If $p$ and $q$ are distinct prime numbers, $f$ and $g$ are non-exceptional newforms (modular or Maass) for the…

Let $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or the Eisenstien series $E(z,1/2)$ and $\pi$ be a $SL(3,\mathbb{Z})$ Hecke-Maass cusp form with its Langlands parameter $\mu$ in generic position i.e. away from Weyl chamber walls and…

Let $f$ be a fixed self-contragradient Hecke-Maass form for $SL(3,\mathbb Z)$, and $u$ an even Hecke-Maass form for $SL(2,\mathbb Z)$ with Laplace eigenvalue $1/4+k^2$, $k>0$. A subconvexity bound $O\big(k^{4/3+\varepsilon}\big)$ in the…

Let $f$ be a Maass cusp form for $\rm SL_2(\mathbb{Z})$ with Laplace eigenvalue $1/4+\mu_f^2$, $\mu_f>0$. Let $g$ be an arbitrary but fixed holomorphic or Maass cusp form for $\rm SL_2(\mathbb{Z})$. In this paper, we establish the following…

Let $\pi$ be a Hecke-Maass cusp form for $\mathrm{SL(3, \mathbb{Z})}$ and $f$ be a holomorphic cusp form for $\mathrm{SL(2,\mathbb{Z})}$ of weight $k$ or a Hecke-Maass cusp form corresponding to the Laplacian eigenvalue $1/4+k^2$, $k\geq…

In this note, we prove that given a Hecke-Maass cusp form $f$ for $SL_2(\mathbb{Z})$ and a sufficiently large integer $q=q_1q_2$ with $q_j\asymp \sqrt{q}$ being prime numbers for $j=1,2$, there exists a primitive Dirichlet character $\chi$…

Let f be a Hecke-Maass or holomorphic primitive cusp form for $SL(2,\mathbb{Z})$ with Fourier coefficients $\lambda_{f}(n)$. Let $\chi$ be a primitive Dirichlet character of modulus p, where p is a prime number. In this article we prove the…

Let $f $ be a holomorphic Hecke eigenforms or a Hecke-Maass cusp form for the full modular group $ SL(2, \mathbb{Z})$. In this paper we shall use circle method to prove the Weyl exponent for $GL(2)$ $L$-functions. We shall prove that \[ L…

We prove an upper bound for the twelfth moment of Hecke $L$-functions associated to holomorphic Hecke cusp forms of weight $k$ in a dyadic interval $T \leq k \leq 2T$ as $T$ tends to infinity. This bound recovers the Weyl-strength subconvex…

Let $f$ be a cuspidal eigenform (holomorphic or Maass) on the full modular group $SL(2, \mathbb{Z})$ . Let $\chi$ be a primitive character of modulus $P$. We shall prove the following results: 1. Suppose $P = p^r$, where $p$ is a prime and…

We revisit Munshi's proof of the $t$-aspect subconvex bound for $\rm GL(3)$ $L$-functions, and we are able to remove the `conductor lowering' trick. This simplification along with a more careful stationary phase analysis allows us to…

Let $M$ be a square-free integer and let $P$ be a prime not dividing $M$ such that $P \sim M^\eta$ with $0<\eta<2/21$. We prove subconvexity bounds for $L(\tfrac{1}{2}, f \otimes g)$ when $f$ and $g$ are two primitive holomorphic cusp forms…

Let f be a cusp form for the group SL(3, Z) with Langlands parameter mu and associated L-function L(s, f). If mu is in generic position, i.e. away from the Weyl chamber walls and away from the self-dual forms, we prove the subconvexity…

We estimate the $1$-level density of low-lying zeros of $L(s,\chi)$ with $\chi$ ranging over primitive Dirichlet characters of conductor $\in [Q/2,Q]$ and for test functions whose Fourier transform is supported in $[- 2 - 50/1093, 2 +…

For any large prime $q$, $x \leq 1$ and any real $k\geq 2$, we prove a lower bound for the following $2k$-th moment \begin{equation*} \sum_{\substack{\chi \in X_q^*}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k}, \end{equation*} where…

Let $\pi$ be a Hecke cusp form for $\mathrm{SL}_3(\mathbb{Z})$. We bound the second moment average of $L(s,\pi)$ over a short interval to obtain the subconvexity estimate $$ L(1/2+it, \pi) \ll_{\pi, \varepsilon}…