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In mathematics, '''modular symbols''', introduced independently by [[Bryan John Birch]] and by {{harvs|txt|last=Manin|authorlink=Yuri Manin|year=1972}}, span a [[vector space]] closely related to a space of [[modular form]]s, on which the [[Group action (mathematics)|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 ''GL''<sub>2</sub>('''Q''') [[Group action (mathematics)|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 ''fd''τ, along the path corresponding to the symbol.

==References==

*{{Citation | last1=Manin | first1=Ju. I. | title=Parabolic points and zeta functions of modular curves | url=http://www.turpion.org/php/paper.phtml?journal_id=im&paper_id=1867 | doi=10.1070/IM1972v006n01ABEH001867 |mr=0314846 | year=1972 | journal=Math. USSR-Izv.  | issn=0373-2436 | volume=6 | pages=19–64}}
*{{Citation | last1=Manin | first1=Yuri Ivanovich | author1-link=Yuri Ivanovich Manin | title=Arithmetic geometry | url=http://www.claymath.org/publications/Arithmetic_Geometry/ | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Clay Math. Proc. | isbn=978-0-8218-4476-2 |mr=2498060 | year=2009 | volume=8 | chapter=Lectures on modular symbols | pages=137–152}}
*{{citation | zbl=0872.14041 | last=Cremona | first=J.E. | title=Algorithms for modular elliptic curves | edition=2nd | location=Cambridge | publisher=Cambridge University Press | year=1997 | isbn=0-521-59820-6 }}

[[Category:Modular forms]]