Flach classes and generalised Hecke eigenvalues

Henri Darmon; Alice Pozzi

doi:10.1007/978-3-032-13123-2_3

Flach classes and generalised Hecke eigenvalues

Henri Darmon^*
, Alice Pozzi

^*Corresponding author for this work

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference Contribution (Conference Proceeding)

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Abstract

We describe the action of Hecke operators on generalised eigenspaces attached to certain mod p cuspidal eigenforms of weight two in terms of certain extension classes of Galois representations constructed by Matthias Flach. This description can be viewed as a partial generalisation to cusp forms of a formula of Barry Mazur for the Eisenstein series of weight two and prime level.

Original language	English
Title of host publication	Elliptic Curves and Modular Forms in Arithmetic Geometry
Subtitle of host publication	Celebrating Massimo Bertolini's 60th birthday. Milano, Italy, September 16-20, 2022
Editors	Matteo Longo, Marco Adamo Seveso, Rodolfo Venerucci, Stefano Vigni
Publisher	Springer Nature
Pages	51-74
Number of pages	24
ISBN (Electronic)	9783032131232
ISBN (Print)	9783032131225, 9783032131256
DOIs	https://doi.org/10.1007/978-3-032-13123-2_3
Publication status	Published - 1 May 2026
Event	Elliptic curves and modular forms in arithmetic geometry: Celebrating Massimo Bertolini's 60th birthday - , Italy Duration: 12 May 2022 → 16 May 2022 https://sites.google.com/view/bertolini60/home-page

Publication series

Name	Springer Proceedings in Mathematics & Statistics
Publisher	Springer Nature
Volume	527
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	Elliptic curves and modular forms in arithmetic geometry
Country/Territory	Italy
Period	12/05/22 → 16/05/22
Internet address	https://sites.google.com/view/bertolini60/home-page

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.

Access to Document

10.1007/978-3-032-13123-2_3

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Darmon, H., & Pozzi, A. (2026). Flach classes and generalised Hecke eigenvalues. In M. Longo, M. A. Seveso, R. Venerucci, & S. Vigni (Eds.), Elliptic Curves and Modular Forms in Arithmetic Geometry: Celebrating Massimo Bertolini's 60th birthday. Milano, Italy, September 16-20, 2022 (pp. 51-74). (Springer Proceedings in Mathematics & Statistics; Vol. 527). Springer Nature. https://doi.org/10.1007/978-3-032-13123-2_3

Darmon, Henri ; Pozzi, Alice. / Flach classes and generalised Hecke eigenvalues. Elliptic Curves and Modular Forms in Arithmetic Geometry: Celebrating Massimo Bertolini's 60th birthday. Milano, Italy, September 16-20, 2022. editor / Matteo Longo ; Marco Adamo Seveso ; Rodolfo Venerucci ; Stefano Vigni. Springer Nature, 2026. pp. 51-74 (Springer Proceedings in Mathematics & Statistics).

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abstract = "We describe the action of Hecke operators on generalised eigenspaces attached to certain mod p cuspidal eigenforms of weight two in terms of certain extension classes of Galois representations constructed by Matthias Flach. This description can be viewed as a partial generalisation to cusp forms of a formula of Barry Mazur for the Eisenstein series of weight two and prime level.",

author = "Henri Darmon and Alice Pozzi",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2026.; Elliptic curves and modular forms in arithmetic geometry : Celebrating Massimo Bertolini's 60th birthday ; Conference date: 12-05-2022 Through 16-05-2022",

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Darmon, H & Pozzi, A 2026, Flach classes and generalised Hecke eigenvalues. in M Longo, MA Seveso, R Venerucci & S Vigni (eds), Elliptic Curves and Modular Forms in Arithmetic Geometry: Celebrating Massimo Bertolini's 60th birthday. Milano, Italy, September 16-20, 2022. Springer Proceedings in Mathematics & Statistics, vol. 527, Springer Nature, pp. 51-74, Elliptic curves and modular forms in arithmetic geometry, Italy, 12/05/22. https://doi.org/10.1007/978-3-032-13123-2_3

Flach classes and generalised Hecke eigenvalues. / Darmon, Henri; Pozzi, Alice.
Elliptic Curves and Modular Forms in Arithmetic Geometry: Celebrating Massimo Bertolini's 60th birthday. Milano, Italy, September 16-20, 2022. ed. / Matteo Longo; Marco Adamo Seveso; Rodolfo Venerucci; Stefano Vigni. Springer Nature, 2026. p. 51-74 (Springer Proceedings in Mathematics & Statistics; Vol. 527).

Research output: Chapter in Book/Report/Conference proceeding › Conference Contribution (Conference Proceeding)

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AU - Pozzi, Alice

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N2 - We describe the action of Hecke operators on generalised eigenspaces attached to certain mod p cuspidal eigenforms of weight two in terms of certain extension classes of Galois representations constructed by Matthias Flach. This description can be viewed as a partial generalisation to cusp forms of a formula of Barry Mazur for the Eisenstein series of weight two and prime level.

AB - We describe the action of Hecke operators on generalised eigenspaces attached to certain mod p cuspidal eigenforms of weight two in terms of certain extension classes of Galois representations constructed by Matthias Flach. This description can be viewed as a partial generalisation to cusp forms of a formula of Barry Mazur for the Eisenstein series of weight two and prime level.

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DO - 10.1007/978-3-032-13123-2_3

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SN - 9783032131225

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EP - 74

BT - Elliptic Curves and Modular Forms in Arithmetic Geometry

A2 - Longo, Matteo

A2 - Seveso, Marco Adamo

A2 - Venerucci, Rodolfo

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PB - Springer Nature

T2 - Elliptic curves and modular forms in arithmetic geometry

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Darmon H, Pozzi A. Flach classes and generalised Hecke eigenvalues. In Longo M, Seveso MA, Venerucci R, Vigni S, editors, Elliptic Curves and Modular Forms in Arithmetic Geometry: Celebrating Massimo Bertolini's 60th birthday. Milano, Italy, September 16-20, 2022. Springer Nature. 2026. p. 51-74. (Springer Proceedings in Mathematics & Statistics). doi: 10.1007/978-3-032-13123-2_3

Flach classes and generalised Hecke eigenvalues

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