Critical points of Eisenstein series. Issue 1 (29th March 2022)
- Record Type:
- Journal Article
- Title:
- Critical points of Eisenstein series. Issue 1 (29th March 2022)
- Main Title:
- Critical points of Eisenstein series
- Authors:
- Gun, Sanoli
Oesterlé, Joseph - Abstract:
- 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 ) $\gamma \in {\bf SL}_2({\bf Z})$ . 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.
- Is Part Of:
- Mathematika. Volume 68:Issue 1(2022)
- Journal:
- Mathematika
- Issue:
- Volume 68:Issue 1(2022)
- Issue Display:
- Volume 68, Issue 1 (2022)
- Year:
- 2022
- Volume:
- 68
- Issue:
- 1
- Issue Sort Value:
- 2022-0068-0001-0000
- Page Start:
- 259
- Page End:
- 298
- Publication Date:
- 2022-03-29
- Subjects:
- Mathematics -- Periodicals
510.5 - Journal URLs:
- http://journals.cambridge.org/action/displayJournal?jid=MTK ↗
https://londmathsoc.onlinelibrary.wiley.com/journal/20417942 ↗
http://onlinelibrary.wiley.com/ ↗ - DOI:
- 10.1112/mtk.12124 ↗
- Languages:
- English
- ISSNs:
- 0025-5793
- Deposit Type:
- Legaldeposit
- View Content:
- Available online (eLD content is only available in our Reading Rooms) ↗
- Physical Locations:
- British Library DSC - BLDSS-3PM
British Library HMNTS - ELD Digital store - Ingest File:
- 27110.xml