Endomorphism algebras of motives attached to elliptic modular forms.
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 \(k\geq 2\), level \(N\geq 1\), and nebentypus \(\varepsilon\). When \(k>2\), let \(M_f\) denote the motif attached to \(f\) constructed by \textit{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 \textit{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 \textit{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. 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.
scientific article; zbMATH DE number 2021759
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