In mathematics, modular symbols, introduced independently by
Bryan John Birch
Bryan John Birch FRS (born 25 September 1931) is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture.
Biography
Bryan John Birch was born in Burton-on-Trent, the son of Arthur Jack and Mary Edith Birch. ...
and by , span a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
closely related to a space of
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...
s, on which the
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of the
Hecke algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators.
Properties
The algebra is a commutative ring.
In the classical elliptic modular form theory, the Hecke operators ''T'n'' with ''n'' coprime to the level acting ...
can be described explicitly. This makes them useful for computing with spaces of modular forms.
Definition
The
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
of (universal weight 2) modular symbols is spanned by symbols for α, β in the rational projective line Q ∪ subject to the relations
* + =
Informally, represents a homotopy class of paths from α to β in the
upper half-plane
In mathematics, the upper half-plane, \,\mathcal\,, is the set of points in the Cartesian plane with > 0.
Complex plane
Mathematicians sometimes identify the Cartesian plane with the complex plane, and then the upper half-plane corresponds to ...
.
The group ''GL''
2(Q)
acts
The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
on the rational
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
, 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 , 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
Modular forms