On the search of genuine -adic modular -functions for . With a correction to: On -adic -functions of over totally real fields
Haruzo Hida (1996)
Mémoires de la Société Mathématique de France
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Haruzo Hida (1996)
Mémoires de la Société Mathématique de France
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Ami Fischman (2002)
Annales de l’institut Fourier
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We explore the question of how big the image of a Galois representation attached to a -adic modular form with no complex multiplication is and show that for a “generic” set of -adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all have a large image.
John Coates (1988-1989)
Séminaire Bourbaki
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Eknath Ghate, Vinayak Vatsal (2004)
Annales de l'Institut Fourier
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Let be a primitive cusp form of weight at least 2, and let be the -adic Galois representation attached to . If is -ordinary, then it is known that the restriction of to a decomposition group at is “upper triangular”. If in addition has CM, then this representation is even “diagonal”. In this paper we provide evidence for the converse. More precisely, we show that the local Galois representation is not diagonal, for all except possibly finitely many of the arithmetic members...
Haruzo Hida (1986)
Annales scientifiques de l'École Normale Supérieure
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Haruzo Hida (1991)
Annales de l'institut Fourier
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Let be the Rankin product -function for two Hilbert cusp forms and . This -function is in fact the standard -function of an automorphic representation of the algebraic group defined over a totally real field. Under the ordinarity assumption at a given prime for and , we shall construct a -adic analytic function of several variables which interpolates the algebraic part of for critical integers , regarding all the ingredients , and as variables.