Simple zeros of GL(2) L-functions

Abstract: I will discuss my recent work on simple zeros of automorphic L-functions of degree 2. For a primitive holomorphic form $f$ of arbitrary weight and level, I show that its completed L-function has $\Omega(T^\delta)$ simple zeros with imaginary part in $[-T, T]$, for any $\delta < \frac{2}{27}$. This provides the first power bound in this problem for $f$ of non-trivial level, where the previous best bound was $\Omega(\log \log \log T)$. The proof uses a method of Conrey-Ghosh combined with ideas of Booker and Booker-Milinovich-Ng, in addition to a new ingredient coming from zero-density estimates for twists of $f$. I will explain the basic method, the obstructions that arise when $f$ has non-trivial level, and how to unconditionally get around such obstructions to obtain a power bound. This argument gives a curious connection between the quality of zero-density estimates for a certain family and the number of simple zeros for a single element of that family.

number theory

Audience: researchers in the topic

Heilbronn number theory seminar

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