Crossed product algebras attached to weight one forms

Abstract

Abstract. We completely determine the Brauer class of the crossed product algebraattached to a non-dihedral cusp form of weight one. 1. IntroductionLet f be a holomorphic cuspidal newform of weight at least one. In weights twoand higher, there is a Grothendieck motive M f (abelian variety A f in weight two)attached to f. It is a fundamental fact, due to Momose and Ribet [8] in weight twoand [1], [4] in higher weights, that the full algebra of endomorphisms of this motivehas the structure of an explicit crossed product algebra X, when the form f is ofnon-dihedral type. Moreover, much, but not all, is known about the Brauer class ofthis algebra [9], [6], [4].In this paper, we completely determine the Brauer class of X for non-dihedral cuspforms of weight one. The Brauer class in this case is closely related to the secondWitt invariant of the trace form of a number field determined by the projective Artinrepresentation associated to the form.We show that, much as in higher weights, the Brauer class of X is completelydetermined at a prime of good reduction by the parity of a certain slope, when thisslope is finite (Theorem 2). An interpretation of this slope in terms of the adjointrepresentation allows us to compute it, showing that, in contrast to what happens inhigher weights, the Brauer class is essentially unramified at all primes of good reduction(Theorem 4, Corollary 7). At primes of bad reduction we show that the Brauer classis determined purely by the nebentypus of the form (Theorem 5, Corollary 7). Finally,as examples, we determine the Brauer class of X for all non-dihedral weight one formsof prime level (Corollary 9).2. The crossed product algebra XThe crossed product algebra X we wish to study can be attached to non-dihedralforms in all weights. We recall the definition for forms of weight one.Let f =Pa