Abstract
Two-dimensional adelic objects were introduced by I. Fesenko in his study of the Hasse zeta function associated to a regular model \(\mathcal{E}\) of the elliptic curve E. The Hasse–Weil L-function L(E, s) of E appears in the denominator of the Hasse zeta function of \(\mathcal{E}\). The two-dimensional adelic analysis predicts that the integrand h of the boundary term of the two-dimensional zeta integral attached to \(\mathcal{E}\) is mean-periodic. The mean-periodicity of h implies the meromorphic continuation and the functional equation of L(E, s). On the other hand, if E is modular, several nice analytic properties of L(E, s), in particular the analytic continuation and the functional equation, are obtained by the theory of the cuspidal automorphic representation of GL(2) over the ordinary ring of adele (one dimensional adelic object). In this chapter, we try to relate in analytic way the theory of two-dimensional adelic object to the theory of cuspidal automorphic representation of GL(2) over the one-dimensional adelic object, under the assumption that E is modular. In the first approximation, they are dual each other.
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Acknowledgments
The author thanks Ivan Fesenko for his many helpful comments and questions to this research and also the School of Mathematical Sciences of the University of Nottingham for the hospitality on author’s stay in November 2007–February 2008, August 2008. The author had fruitful conversations and discussions with Guillaume Ricotta and Ivan Fesenko during this stay. This work was partially supported by Grant-in-Aid for JSPS Fellows.
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Suzuki, M. (2012). Two-Dimensional Adelic Analysis and Cuspidal Automorphic Representations of GL(2). In: Bump, D., Friedberg, S., Goldfeld, D. (eds) Multiple Dirichlet Series, L-functions and Automorphic Forms. Progress in Mathematics, vol 300. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8334-4_15
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DOI: https://doi.org/10.1007/978-0-8176-8334-4_15
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