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Modular symbolschap:modsymModular symbolsModular symbols permeate this thesis. In their simplest incarnation, modular symbols provide a finite presentation for the homology group H_{1}(X_{0}(N),) of the Riemann surface X_{0}(N). This presentation is equipped with such a rich structure that from it we can deduce the action of the Hecke operators; this is already sufficient information for us to compute a basis for the space S_{2}(Γ_{0}(N),) of cusp forms.

We recall the definition of spaces of modular symbols in Sections sec:defnofmodsyms–cuspidalsymbols. Then in Section sec:duality, we review the duality between modular symbols and modular forms. In Section sec:heckeops, we see that modular symbols are furnished with analogues of each of the standard operators that one finds on spaces of modular forms, and in Section sec:degeneracymaps we see that the same is true of the degeneracy maps. Section sec:maninsymbols describes Manin symbols, which supply a convenient finite presentation for the space of modular symbols. Finally, Section sec:tori introduces the complex torus attached to a newform, which appears in various guises throughout this thesis.

Before continuing, we offer an apology. We will only consider modular symbols that are already equipped with a fixed Dirichlet character. Though fixing a character complicates the formulas, the resulting increase in efficiency is of extreme value in computational applications. Fixing a character allows us to compute in just the part of the space of modular symbols for Γ_{1}(N) that interests us. We apologize for any inconvenience this may cause the less efficiency minded reader.

Acknowledgment. This chapter and the next were greatly influenced by the publications of Cremona cremona:gammaone, cremona:algsCremona and Merel merel:1585Merel, along with the foundational contributions of Manin manin:parabolic, Mazur mazur:arithmetic_{v}alues, mazur : symboles, andShokurov sokurov : modsym.Cremona′sbook cremona : algsprovidesamotivatedroadmapthatguidesthereaderwhowishestocomputewithmodularsymbolsinthefamiliarcontextofellipticcurves, andMerel′sMerelarticleprovidesanaccessibleoverviewoftheactionofHeckeoperatorsonhigherweightmodularsymbols, andtheconnectionbetweenmodularsymbolsandrelatedcohomologytheories.

The definition of modular symbols sec:defnofmodsyms Fix a positive integer N, an integer k ≥ 2, and a continuous homomorphism

: (/N)^{∗}^{∗}

such that (−1) = (−1)^{k}. We call N the levelLevel of modular symbols—textit, k the weightWeight of modular symbols—textit, and the Dirichlet character.Dirichlet character—textit

Consider the quotient of the abelian group generated by all formal symbols {, β}, with , β ∈ ¶^{1}() = {∞}, by the following relations:

{, β} + {β, γ} + {γ, } = 0,

for all , β, γ ∈ ¶^{1}(). Let be the torsion-free quotient of this group by its torsion subgroup. Because is torsion free, {, } = 0 and {, β} = −{β, }. Modular symbols!relations satisfied by

remark One is motivated to consider these relations by viewing {, β} as the homology class of an appropriate path from α to β in the upper half plane.