Related papers: Strong Hybrid Subconvexity for Twisted Selfdual $\…

In a recent important paper, Hoffstein and Hulse generalized the notion of Rankin-Selberg convolution $L$-functions by defining shifted convolution $L$-functions. We investigate symmetrized versions of their functions. Under certain mild…

We prove a subconvexity bound for the central value L(1/2, chi) of a Dirichlet L-function of a character chi to a prime power modulus q=p^n of the form L(1/2, chi)\ll p^r * q^(theta+epsilon) with a fixed r and theta\approx 0.1645 < 1/6,…

Let $F$ be a number field, $\pi$ either a unitary cuspidal automorphic representation of $\mathrm{GL}(2)/F$ or a unitary Eisenstein series, and $\chi$ a unitary Hecke character of analytic conductor $C(\chi).$ We develop a regularized…

We establish upper bounds for shifted moments of cubic and quartic Dirichlet $L$-functions under the generalized Riemann hypothesis. As an application, we prove bounds for moments of cubic and quartic Dirichlet character sums.

We compute an asymptotic formula for the twisted moment of GL(3)xGL(2) L-functions and their derivatives. As an application we prove that symmetric-square lifts of GL(2) Maass forms are uniquely determined by the central values of the…

We obtain a first moment formula for Rankin-Selberg convolution $L$-series of holomorphic modular forms or Maass forms of arbitrary level on $GL(2)$, with an orthonormal basis of Maass forms. One consequence is the best result to date,…

We consider the family of Rankin-Selberg convolution L-functions of a fixed SL(3, Z) Maass form with the family of Hecke-Maass cusp forms on SL(2, Z). We estimate the second moment of this family of L-functions with a "long" integration in…

We obtain the formula for the twisted harmonic second moment of the $L$-functions associated with primitive Hecke eigenforms of weight 2. A consequence of our mean value theorem is reminiscent of recent results of Conrey and Young on the…

We consider a variant of the strong multiplicity one theorem. Let $\pi_{1}$ and $\pi_{2}$ be two unitary cuspidal automorphic representations for $\mathrm{GL(2)}$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet…

We consider the Rankin-Selberg L-functions associated with a fixed modular form of full level and holomorphic cuspidal newforms of large even weight, fixed level and fixed primitive nebentypus. We compute the second moment of this family in…

Let $E/F$ be a CM extension of number fields, and let $H < G$ be a unitary Gan--Gross--Prasad pair defined with respect to $E/F$ that is compact at infinity. We consider a family $\mathcal{F}$ of automorphic representations of $G \times H$…

We describe a new method to obtain weak subconvexity bounds for $L$-functions with mild hypotheses on the size of the Dirichlet coefficients. We verify these hypotheses for all automorphic $L$-functions and (with mild restrictions) the…

We investigate the reciprocity law, studied by Conrey~\cite{Con07} and Young~\cite{You11a}, for the second moment of Dirichlet L-functions twisted by $\chi(a)$ modulo a prime $q$. We show that the error term in this reciprocity law can be…

We let $f$ be a half-integral weight modular form of weight $\kappa>4$ on $\Gamma_0(4)$ that is an eigenfunction of all Hecke operators $T_n$, so that $T_nf = \Lambda_f(n)n^{\frac{\kappa-1}{2}}f$. Let $\|f\|$ denote the Petersson norm of…

This is a sequel to our previous articles \cite{Kw23, Kw23a+}. In this work, we apply recent techniques that fall under the banner of `Period Reciprocity' to study moments of $GL(3)\times GL(2)$ $L$-functions in the non-archimedean aspects,…

Let $1\le N<M$ with $N$ and $M$ coprime and square-free. Through classical analytic methods we estimate the first moment of central $L$-values $ L(1/2,f\times g) $ where $f\in S^*_k(N)$ runs over primitive holomorphic forms of level $N$ and…

We establish sharp lower bounds for shifted (with two shifts) moments of Dirichlet $L$-function of fixed modulus under the generalized Riemann hypothesis.

In the prequel, a sharp bound in the level aspect on the fourth moment of Hecke--Maa{\ss} forms with an inexplicit (in fact exponential) dependency on the eigenvalue was given. In this paper, we develop further the framework of explicit…

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 give an asymptotic formula with power saving error term for the twisted first moment of symmetric square L-functions on GL(3) in the level aspect. As applications, we obtain non-vanishing results as well as lower bounds of the expected…