Modular symbols for \(F_ q(T)\)
Let \(A\) be the polynomial ring \(\mathbb{F}_ q[T]\) with quotient field \(k=\mathbb{F}_ q(T)\) over a finite field \(\mathbb{F}_ q\). H. Hasse, E. Witt, A. Weil, and others established deep and far-reaching analogies between the respective arithmetics of \(A\) and of \(\mathbb{Q}\). In 1973, V. G. Drinfeld constructed two kinds of analogues for the classical modular setting (modular groups, modular curves, modular forms): The first one, dealing with functions on the ``Drinfeld upper half- plane'' leads to (rigid analytic) Drinfeld modular forms and curves. The second one uses the Bruhat-Tits tree \({\mathfrak T}\) of \(\text{PGL}_ 2(K_ \infty)\) as a substitute for the complex upper half-plane, and leads to automorphic forms on a certain adelic group. Both of these are related (and are simultaneously related to the classification of elliptic curves over \(k\) à la Shimura-Taniyama-Weil) by work of M. Reversat and the reviewer. Let \(\text{GL}_ 2(A)\) be the ``modular group'', which acts on \({\mathfrak T}\), and \(\Gamma=\Gamma_ 0(N)\) a ``Hecke congruence subgroup'' \((0\neq N\in A)\). As has become apparent in the last years, the ``right'' analogue of the \(\mathbb{Z}\)-module of cusp forms of weight two with integral Fourier coefficients for a Hecke congruence subgroup of \(\text{SL}_ 2(\mathbb{Z})\) is the \(\mathbb{Z}\)-module \(C_{\text{har}}(\Gamma_ 0(N),\mathbb{Z})\) of harmonic \(\Gamma_ 0(N)\)-invariant cuspidal functions on the set of edges of \({\mathfrak T}\). Some questions (mainly about Hecke correspondence on \(\Gamma\setminus{\mathfrak T}\) and the associated modular curve \(X_ \Gamma\)) are technically easier to handle after dualizing \(C_{\text{har}}\), which leads to the definition of modular symbols given in the paper. It follows the one given by Manin in the classical case. Essentially, a modular symbol is some functional on \(C_{\text{har}}\) induced by ``path integration''. The contents of the paper are as follows: --- definition and elementary properties of (the space \(M_ \Gamma\) of) modular symbols; --- let \(\text{Jac}(X_ \Gamma)\) be the Jacobian with group \(\Phi(\Gamma)\) of connected components of its Néron model at \(\infty\). A description of \(\Phi(\Gamma)\) through \(M_ \Gamma\) is given (Theorem 14); --- generators and relations for the \(\mathbb{Z}\)-module \(M_ \Gamma\) (Theorem 21); --- applications: The author shows how to use \(M_ \Gamma\) for explicit calculations in specific instances. Notably he works out an example of a certain elliptic curve over \(\mathbb{F}_ 7(T)\) and shows how the calculated data fit with the relevant conjectures. The author finally states: ``We \dots feel that we have raised many more questions than we have answered.'' This is true (not in the pejorative sense), in view of the many lines of future research indicated in the last section. There are two minor inaccuracies: \(w(e)\) should be the order of the stabilizer in \(\Gamma\) modulo its center (Definition 10), and the elliptic curve treated in the applications should be replaced by its unramified quadratic twist.
\(\mathbb{Z}\)-module of harmonic cuspidal functions
scientific article; zbMATH DE number 179215
scientific article