Refinements of Strong Multiplicity One for GL(2)

Abstract: Let $f_1$ and $f_2$ be holomorphic newforms of same weight and with same nebentypus, and let $a_{f_1}(n)$ and $a_{f_2}(n)$ denote the Fourier coefficients of $f_1$ and $f_2$, respectively. By the strong multiplicity one theorem, it is known that if $a_{f_1}(p)=a_{f_2}(p)$ for almost all primes $p$, then $f_1$ and $f_2$ are equivalent. Furthermore, a result of Ramakrishnan states that if $a_{f_1}(p)^2=a_{f_2}(p)^2$ outside a set of primes $p$ of density less than $\frac{1}{18}$, then $f_1$ and $f_2$ are twist-equivalent.

In this talk, we will discuss some refinements and variants of the strong multiplicity one theorem and Ramakrishnan's result for general $\rm{GL}(2)$-forms. In particular, we will analyse the set of primes $p$ for which $|a_{f_1}(p)| \neq |a_{f_2}(p)|$ for non-twist-equivalent $f_1$ and $f_2$.

number theory

Audience: researchers in the topic

Number Theory Online Conference 2020