Modular Curves and Abelian Varieties by Irene I. Bouw, Stefan Wewers (auth.), John E. Cremona,

By Irene I. Bouw, Stefan Wewers (auth.), John E. Cremona, Joan-Carles Lario, Jordi Quer, Kenneth A. Ribet (eds.)

This booklet offers lectures from a convention on "Modular Curves and Abelian Varieties'' on the Centre de Recerca Matem� tica (Bellaterra, Barcelona). The articles during this quantity current the most recent achievements during this super lively box and may be of curiosity either to experts and to scholars and researchers. Many contributions specialise in generalizations of the Shimura-Taniyama conjecture to forms equivalent to elliptic Q-curves and Abelian sorts of GL_2-type. The publication additionally contains a number of key articles within the topic that don't correspond to convention lectures.

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17 of [15], the fact that A is an algebraic integer, and the Jacquet-Langlands theorem) that Vp(A) ~ k - 1. On the other hand, if A < k - 1 then ()l-k f is in S~ (U; 1) and if it is non-zero then it is an eigenvector for Up with eigenvalue A/pk-l, which has negative valuation. On the other hand, Up is an operator with norm at most 1, and hence ()l-k f = O. Hence f is classical. 0 In the classical theory, one can find both classical and non-classical forms with A = k - 1, and it would be an interesting computational exercise to search for examples of this phenomenon in this situation.

Pages 179-197. Birkhiiuser, 1990. [12] M. Raynaud. Specialisation des revetements en caracteristique p > O. Ann. Sci. Ecole Norm. , 32(1):87-126, 1999. -P. Serre. Sur la topologie des varietes algebriques en caracteristique p. In Symp. Int. Top. , pages 24-53, 1958. -P. Serre. Topics in Galois Theory. Research notes in mathematics, 1. Jones and Bartlett Publishers, 1992. Lecture notes prepared by Henri Darmon. [15] K. Strambach and H. Volklein. On linear rigid triples. J. Reine Angew. , 510:57-62, 1999.

U, r) has a characteristic power series in R{ {T}} whose restriction to a weight K is the characteristic power series of Up on forms of weight K. The reader who has read the construction of the classical eigencurve in [7] will now see that we have all the ingredients to construct the analogous object in this case. (U,r) are orthonormizable, although in future work we shall show that this assumption is unnecessary. We point out here that we only define the "D" eigencurve in this setting, although no doubt one can also construct a "e" eigencurve by mimicking the construction in [7].

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