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: 19–64, doi:10.1070/IM1972v006n01ABEH001867, ISSN 0373-2436, MR 0314846 
• Manin, Yuri Ivanovich (2009), "Lectures on modular symbols", Arithmetic geometry, Clay Math. Proc. 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