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
[edit]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
[edit]- Manin, Ju. I. (1972), "Parabolic points and zeta functions of modular curves", Math. USSR-Izv., 6: 19–64, 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