STNB2017, (31st edition)
On Maeda's Conjecture
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Abstract
Let T_{n,k}(X) be the characteristic polynomial of the n-th Hecke operator acting on the space of cusp forms of weight k. A popular conjecture, called Maeda's Conjecture, asserts that, for every n, T_{n,k}(X) is irreducible in Q[X] and has full Galois group over Q. We will discuss some recent progress on the conjecture and will show the following: Let k be any positive integer congruent to 0 modulo 4. If one T_{n,k}(X) is irreducible and has full Galois group, then the same is true of T_{p,k}(X) for all primes p.
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