Let $f={\sum }_{n=1}^{\infty }a\left(n\right)q^n$ and $g={\sum }_{n=1}^{\infty }b\left(n\right)q^n$ be holomorphic common eigenforms of all Hecke operators for the congruence subgroup ${\Gamma }_0\left(N\right)$ of $S{L}_2\left(\mathbf{Z}\right)$ with “Nebentypus” character $\psi$ and $\xi$ and of weight $k$ and $\ell$, respectively. Define the Rankin product of $f$ and $g$ by$${𝒟}_N(s,f,g)=(\sum _{n=1}^{\infty }\psi \xi \left(n\right)n^{k+\ell -2s-2}\left)\right(\sum _{n=1}^{\infty }a\left(n\right)b\left(n\right)n^{-s}).$$Supposing $f$ and $g$ to be ordinary at a prime $p\ge 5$, we shall construct a $p$-adically analytic $L$-function of three variables which interpolate the values $\frac{{𝒟}_N(\ell +m,f,g)}{{\pi }^{\ell +2m+1}\<f,f\>}$ for integers $m$ with $0\le m\<k-1,$ by regarding all the ingredients $m$, $f$ and $g$ as variables. Here $⟨f,f⟩$ is the Petersson self-inner product of $f$.

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight $k$ for the full modular group $\mathrm{SL}(2,\mathbb{Z})$, and denote its $n$th Fourier coefficient by ${\lambda }_f\left(n\right)$. We consider the hybrid problem of quadratic forms with prime variables and Hecke eigenvalues of normalized primitive holomorphic cusp forms, which generalizes the result of D. Y. Zhang, Y. N. Wang (2017).

In the transformation formulas for the logarithms of the classical theta-functions, certain sums arise that are analogous to the Dedekind sums in the transformation of the logarithm of the eta-function. A new reciprocity law is established for one of these analogous sums and then applied to prove the law of quadratic reciprocity.

Lafforgue has proposed a new approach to the principle of functoriality in a test case, namely, the case of automorphic induction from an idele class character of a quadratic extension. For technical reasons, he considers only the case of function fields and assumes the data is unramified. In this paper, we show that his method applies without these restrictions. The ground field is a number field or a function field and the data may be ramified.

We prove the recursive integral formula of class one $M$-Whittaker functions on SL$(n,ℝ)$ conjectured and verified in case of $n=3,4$ by Stade.