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Zbl 1050.11062
Brown, Alexander F.; Ghate, Eknath P.
Endomorphism algebras of motives attached to elliptic modular forms. (English)
[J] Ann. Inst. Fourier 53, No. 6, 1615-1676 (2003). ISSN 0373-0956; ISSN 1777-5310/e

The authors study the endomorphism algebras of motives attached to cusp forms. Let $f=\sum a_n q^n$ be a normalized cuspidal eigen newform of weight $k\ge 2$, level $N\ge 1$, and nebentypus $\varepsilon$. When $k>2$, let $M_f$ denote the motif attached to $f$ constructed by {\it A. J. Scholl} [Invent. Math., 100, 419-430 (1990; Zbl 0760.14002)]. When $k=2$, let $M_f$ be the Abelian variety $A_f$ defined by Shimura and Eichler. Let $X_f$ denote the endomorphism algebra of $M_f$. Suppose that $f$ does not have CM, that is, it is not attached to a Hecke character of an imaginary quadratic field, and $f$ has an extra twist in the sense that $f$ is a twist of its conjugate. Following {\it K. A. Ribet} [Math. Ann. 253, 43--62 (1980; Zbl 0421.14008)], the authors consider a simple algebra $X$ associated to 2-cocycles in terms of Jacobi sums. As one of the main theorems, the authors show that $X$ is isomorphic to a subalgebra of $X_f$. This is an analogon of a {\it M. Momose} [J. Fac. Sci. Univ. Tokyo 28, 89-109 (1981; Zbl 0482.10023)] where $X_f$ is replaced by the endomorphism algebras of the Betti realization of $M_f$. The algebra $X$ is expected to be $X_f$, which is a consequence of the Tate conjecture. The equality is proved by Ribet when $k=2$, (loc. cit.) and it also follows from Faltings' theorem. \par The remaining part of this paper is to determine the simple algebra $X$. It is shown that $X$ is either a matrix algebra over a totally real field $F$ or a matrix algebra of a central quaternion division algebra over $F$. The authors relate the local invariants of $X$ and the valuation of the Fourier coefficients $a_p$ and the character values $\varepsilon(p)$. Among other results, $X$ is completely determined in the case where $\varepsilon$ is real and $p$ does not divide $2N$ (Theorem 4.1.11). The approach is to reduce the problems to information of the filtered $(\phi,N)$-modules associated to the local Galois representations by Fontaine's theory.
[Chia-Fu Yu (Taipei)]

MSC 2000:: *11G18 Arithmetic aspects of modular and Shimura varieties
11F11 Modular forms, one variable
11F32 Modular correspondences, etc.

Keywords: endomorphism algebras; modular motives; Tate conjecture; filtered $(\phi; N)$-modules; Newton polygons; symbols

Citations: Zbl 0760.14002; Zbl 0421.14008; Zbl 0482.10023

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