On the semi-simplicity of the $U_p$ -operator on modular forms

Abstract

For and positive integers, let denote the -vector space of cuspidal modular forms of level and weight . This vector space is equipped with the usual Hecke operators , . If we need to consider several levels or weights at the same time, we will denote this by , or ! . If " is a prime number dividing , our $# is also known under the name %&# . One of our main results can be stated very easily: if ('*) and ",+ does not divide , then the operator $# is semi-simple. We can prove the same result for weight /. , under the assumption that certain crystalline Frobenius elements are semi-simple. Milne has shown in [11, 0 2] that this semi-simplicity is implied by Tate’s conjecture claiming that for 1 projective and smooth over a finite field of characteristic " , and 23 54 , 687:9 =@? AB C1D FEHGJILKNMO equals the order of PQ C1R SB at 2 . Ulmer proved in [17] that $# is semi-simple, for 5'T. and ",U not dividing , under the assumption of the Birch-Swinnerton-Dyer conjecture for elliptic curves over function fields in characteristic " . His method is quite different from ours: assuming that $# is not semi-simple, he really shows that the Birch-Swinnerton-Dyer conjecture does not hold for an explicitly given elliptic curve. The structure of our proof is as follows. Using the theory of newforms, the problem is shown to be equivalent to the problem of showing that, for a normalized newform V of weight , prime-to-" level and character W , the polynomial X UZY\[ #]X<^\W8 _" `" !ba,c has no double root. This polynomial happens to be the characteristic polynomial of the Frobenius element at " in the two-dimensional Galois representations associated to V ; it is also the characteristic polynomial of the crystalline Frobenius asociated to V . We show that this crystalline Frobenius cannot be a scalar. In Sections 2 and 3 we prove the results concerning these Frobenius elements for 'd) and e f) , respectively. Section 2 is quite elementary, whereas in Section 3 we use a lot of the machinary for comparing " -adic etale and crystalline cohomology. In Section 4 we give some applications: the Ramanujan inequality is a strict inequality in certain cases, certain Hecke algebras are reduced, hence have non-zero discriminant. Section 5 gives some results, due to Abbes and Ullmo, concerning the discriminants of certain Hecke algebras. g partially supported by the Institut Universitaire de France