The circle method and bounds for LL-functions - I

Let $f$ be a Hecke-Maass or holomorphic primitive cusp form of arbitrary level and nebentypus, and let $\chi$ be a primitive character of conductor $M$. For the twisted $L$-function $L(s,f\otimes \chi)$ we establish the hybrid subconvex bound $$L(1/2+it,f\otimes\chi)\ll (M(3+|t|))^{1/2-1/18+\varepsilon},$$ for $t\in \mathbb R$. The implied constant depends only on the form $f$ and $\varepsilon$.Comment: 8 page

Bounds for twisted symmetric square LL-functions

Let $f\in S_k(N,\psi)$ be a newform, and let $\chi$ be a primitive character of conductor $q^{\ell}$. Assume that $q$ is a prime and $\ell>1$. In this paper we describe a method to establish convexity breaking bounds of the form L(\tfrac{1}{2},\Sym f\otimes\chi)\ll_{f,\varepsilon} q^{3/4\ell-\delta_{\ell}+\varepsilon} for some $\delta_{\ell}>0$ and any $\varepsilon>0$. In particular, for $\ell=3$ we show that the bound holds with $\delta_{\ell}=1/4$.Comment: 19 pages, (Extensively revised version

Shifted convolution sums for GL(3)×GL(2)GL(3)\times GL(2)

For the shifted convolution sum $$D_h(X)=\sum_{m=1}^\infty\lambda_1(1,m)\lambda_2(m+h)V(\frac{m}{X})$$ where $\lambda_1(1,m)$ are the Fourier coefficients of a $SL(3,\mathbb Z)$ Maass form $\pi_1$, and $\lambda_2(m)$ are those of a $SL(2,\mathbb Z)$ Maass or holomorphic form $\pi_2$, and $1\leq |h| \ll X^{1+\varepsilon}$, we establish the bound $$D_h(X)\ll_{\pi_1,\pi_2,\varepsilon} X^{1-(1/20)+\varepsilon}.$$ The bound is uniform with respect to the shift $h$

Determination of GL(3)GL(3) Hecke-Maass forms from twisted central values

Suppose $\pi_1$ and $\pi_2$ are two Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ such that for all primitive character $\chi$ we have $$L(\tfrac{1}{2},\pi_1\otimes\chi)=L(\tfrac{1}{2},\pi_2\otimes\chi).$$ Then we show that $\pi_1=\pi_2$.Comment: First draf