BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Refinements of Strong Multiplicity One for $\rm{GL}(2)$ Wong, Peng-Jie


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 normalised Fourier coefficients of $f_1$ and $f_2$, respectively. If $a_{f_1}(p)=a_{f_2}(p)$ for almost all primes $p$, then it follows from the strong multiplicity one theorem that $f_1$ 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 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 $|a_{f_1}(p)| \neq |a_{f_2}(p)|$ when $f_1$ and $f_2$ are not twist-equivalent.

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