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Quadratic modular symbols on Shimura curves. (English. French summary) Zbl 1295.11052

Let \(f\in S_2(\Gamma_0(N))\) be a newform. Using the classical modular symbols, B. Mazur et al. [Invent. Math. 84, 1–48 (1986; Zbl 0699.14028)] constructed its cyclotomic \(p\)-adic \(L\)-function which interpolates the special values of the complex \(L\)-function \(L(f,s)\).
In a previous work [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 106, No. 1, 429–441 (2012; Zbl 1329.11065)], the authors proposed another construction. Instead of the classical modular symbols, they introduced the concept of quadratic modular symbols via integration along geodesics connecting two quadratic imaginary points of the complex upper half-plane, and defined quadratic \(p\)-adic \(L\)-functions. In the paper under review, the authors extend the above concepts of quadratic modular symbols and quadratic \(p\)-adic \(L\)-functions to cover cocompact Shimura curves, and observe the subtle differences from the classical case. In particular, they note that quadratic modular symbols span an infinite dimensional complex vector space, and quadratic \(p\)-adic \(L\)-functions take values in an infinite dimensional \(p\)-adic Banach space.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G18 Arithmetic aspects of modular and Shimura varieties
11G20 Curves over finite and local fields

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References:

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