Abstract
The conjecture of Serre referred to in the title is the one relating Galois representations to modular forms [12]. Let Q denote the field of algebraic numbers in C and Z the subring of algebraic integers. Fix a prime number ` and a prime ideal l of Z lying over `, and put F` = Z/l. Given a cuspidal Hecke eigenform f(z) = ∑ n>1 a(n)e 2πinz of level N , weight k, and character , we consider the continuous semisimple representation ρf : Gal(Q/Q) → GL(2,F`) associated to f by the work of Shimura (k = 2), Deligne (k > 2), and Deligne-Serre (k = 1). It is characterized up to isomorphism by the formulas
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