Congruences between the ratios of Rankin-Selberg L-functions

Abstract

Let $h' \in S_{k'}(N',\chi')$ and $h \in S_{k}(N,\chi)$ be normalized newforms in the respective spaces with $k',k \geq 2$ and $k' - k \geq 2.$ Let $L(s, h\times h')$ denote the completed Rankin-Selberg $L$-function attached to $(h,h').$ It is well-known that for $m$ an integer and $\frac{k'+k}{2}-1<m < k'-1$ \begin{equation*} \frac{L(m, h\times h')}{L(m+1, h\times h')} \in \overline{\mathbb{Q}}. \end{equation*}

Let $h'' \in S_{k'}(N',\chi')$ be another newform and $\mathfrak{l} \subset \bar{\mathbb{Q}}$ be a prime ideal. For all $n \in \mathbb{N}$ assume $a(n,h') \equiv a(n,h'') \pmod{\mathfrak{l}}.$ This thesis is concerned with the question of whether the ratios of $L$-values are congruent modulo $\mathfrak{l}$, i.e., $$ a(n, h') \equiv a(n,h'') \pmod{\mathfrak{l}} \ \ \implies \ \ \frac{L(m, h \times h')}{L(m+1, h \times h')} \equiv \frac{L(m, h \times h'')}{L(m+1, h \times h'')} \pmod{\mathfrak{l}}? $$

First, we develop some algorithms to compute the special values of Rankin-Selberg $L$-functions from well-known results. Using them we verify in many instances that the ratios are congruent. Then, under some hypothesis on the prime $\mathfrak{l}$, the levels $N$ and $N'$ and the weights $k$ and $k'$ we show that the ratios are indeed congruent modulo $\mathfrak{l}$.