1996/11/12 by Robert F. Coleman, Coleman, Robert F., Bas Edixhoven +1
Computer Science · Mathematics · #11F11 (Primary) 14G35 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.alg-geom/9611013
openalex publication_date 1996/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let p be a prime number and N an integer prime to p. We show that the operator Up on the space of cuspidal modular forms of level pN and weight two is semi-simple. It follows from this that the Hecke algebra acting on the space of weight two forms of level M is reduced if M is cube free. Assuming Tate's conjecture for cycles on smooth projective varieties over finite fields, we generalize these results to higher weights. The main point in the proof is that the crystalline Frobenius of the reduction mod p of the motive associated to a newform of level prime to p and weight at least two cannot be a scalar. Assuming Tate's conjecture, it follows that Ramanujan's inequality is strict. For N prime, we relate the discriminant of the weight two Hecke algebra to the height of the modular curve X0(N), for which we get an upper bound.