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Overconvergent modular symbols and \(p\)-adic \(L\)-functions. (English) Zbl 1268.11075
Modular symbols (which are evidently closely related to modular forms) can be understood in a few different ways, and one of those ways is understanding them as a map from the \(0\)-divisor groups of rational cusps to \(k\)-symmetric product of \(\mathbb Q_p^2\) satisfying modular relations with respect to a given group. The authors consider a set of maps from the \(0\)-divisor groups of rational cusps to \(\mathcal D_k(\mathbb Z_p)\), the space of locally analytic \(p\)-adic distributions on \(\mathbb Z_p\) with the weight \(k\) action, and study the (Hecke-equivariant) specialization map between the two groups of modular symbols \(\text{Symb}_{\Gamma_0} (\mathcal D_k(\mathbb Z_p)) \to \text{Symb}_{\Gamma_0} (\text{Sym}^k (\mathbb Q_p^2))\). (For the purpose of this review, let us call the group on the left the set of overconvergent modular symbols.)
The strength of this idea is that one can write down an overconvergent modular symbol explicitly by finding elements of \(\mathcal D_k(\mathbb Z_p)\) satisfying the difference equation and the Manin relations.
Using this explicit construction, the authors show that the specialization map restricted to the subspace of slope less than \(k+1\) is a Hecke-equivariant isomorphism, which they refer to as a control theorem.
They also presents another application of their construction: They discuss an explicit method to compute \(p\)-adic \(L\)-functions of modular forms. (As mentioned above, modular symbols are closely related to modular forms, and one may say they are a different form of modular forms.) Also they discuss critical slope \(p\)-adic \(L\)-functions.
Many of the results here are deep and profound. Some of them are presented in the authors’ other papers, but they are not published yet, so this is the chief reference of their results at the moment.
(It must be noted that \(k\) in this paper is \(w-2\) where \(w\) is the usual weight of a modular form.)

11F85 \(p\)-adic theory, local fields
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11S40 Zeta functions and \(L\)-functions
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