Modular symbol

In mathematics, modular symbols, introduced independently by Bryan John Birch and by Manin (1972), span a vector space closely related to a space of modular forms, on which the action of the Hecke algebra can be described explicitly. This makes them useful for computing with spaces of modular forms.

Definition

The abelian group of (universal weight 2) modular symbols is spanned by symbols {α,β} for α, β in the rational projective line Q ∪ {∞} subject to the relations

  • {α,β} + {β,γ} = {α,γ}

Informally, {α,β} represents a homotopy class of paths from α to β in the upper half-plane.

The group GL2(Q) acts on the rational projective line, and this induces an action on the modular symbols.

There is a pairing between cusp forms f of weight 2 and modular symbols given by integrating the cusp form, or rather fdτ, along the path corresponding to the symbol.

References

  • Manin, Ju. I. (1972), "Parabolic points and zeta functions of modular curves", Math. USSR Izv., 6 (1): 19–64, Bibcode:1972IzMat...6...19M, doi:10.1070/IM1972v006n01ABEH001867, ISSN 0373-2436, MR 0314846
  • Manin, Yuri Ivanovich (2009), "Lectures on modular symbols", Arithmetic geometry, Clay Math. Proc., vol. 8, Providence, R.I.: American Mathematical Society, pp. 137–152, ISBN 978-0-8218-4476-2, MR 2498060
  • Cremona, J.E. (1997), Algorithms for modular elliptic curves (2nd ed.), Cambridge: Cambridge University Press, ISBN 0-521-59820-6, Zbl 0872.14041

Content Disclaimer

Informasi ini disarikan dari Wikipedia dan disajikan kembali untuk tujuan edukasi. Konten tersedia di bawah lisensi CC BY-SA 3.0. Kami tidak bertanggung jawab atas ketidakakuratan data yang bersumber dari kontribusi publik tersebut.

  1. The information displayed on this website is sourced in part or in whole from Wikipedia and has been adapted for the purpose of restating it. We strive to provide accurate and relevant information, however:
  2. There is no guarantee of absolute accuracy. Wikipedia is an open, collaborative project that can be edited by anyone, so information is subject to change.
  3. It is not intended to constitute professional advice. The content displayed is for informational and educational purposes only. For important decisions (e.g., medical, legal, or financial), please consult a professional.
  4. Content copyright. Wikipedia is licensed under the Creative Commons Attribution-ShareAlike License (CC BY-SA). This means that content may be reused with appropriate attribution and shared under a similar license.
  5. Responsible use. Any risk arising from the use of information from this website is entirely the responsibility of the user.