Critical points of Eisenstein series. Issue 1 (29th March 2022) - Gun, Sanoli; Oesterlé, Joseph
Abstract: For any even integer k ⩾ 4 $k {\nobreakspace \geqslant \nobreakspace }4$, let E k be the normalized Eisenstein series of weight k for SL 2 ( Z ) ${\bf SL}_2({\bf Z})$ . Also let D be the closure of the standard fundamental domain of the Poincaré upper half plane modulo SL 2 ( Z ) ${\bf SL}_2({\bf Z})$ . Rankin and Swinnerton‐Dyer showed that all zeros of E k in D are of modulus one. In this article, we study the critical points of E k, that is to say the zeros of the derivative of E k . We show that they are simple. We count those belonging to D, prove that they are located on the two vertical edges of D and produce explicit intervals that separate them. We then count those belonging to γD, for any γ ∈ SL 2 ( Z ) $γ \in {\bf SL}_2({\bf Z})$ . This is the first instance where precise information about the location of all the zeros of an infinite family of (non‐modular) quasi‐modular forms is obtained.