Kęstutis Česnavičius - 14h00 - The Manin constant and the modular degree

Abstract: By the modularity theorem, an elliptic curve E over Q of conductor N admits a surjection phi from the modular curve X_0(N). The Manin constant c of such a modular parametrization of E is the integer that scales the differential associated to the normalized newform on Gamma_0(N) determined by the isogeny class of E to the phi-pullback of a Néron differential of E. For optimal phi Manin conjectured his constant to be 1, and we show that in general it divides deg(phi) under mild assumptions at the primes 2 and 3. This gives new restrictions on the primes that could divide the Manin constant. The talk is based on joint work with Michael Neururer and Abhishek Saha.