P-irreducibility of binding polynomials

Irreducibility of Hecke Polynomials

In this note, we show that if the characteristic polynomial of some Hecke operator Tn acting on the space of weight k cusp forms for the group SL2(Z ) is irreducible, then the same holds for Tp, where p runs through a density one set of primes. This proves that if Maeda’s conjecture is true for some Tn, then it is true for Tp for almost all primes p.

IRREDUCIBILITY OF POLYNOMIALS MODULO p VIA NEWTON POLYTOPES

Ostrowski established in 1919 that an absolutely irreducible integral polynomial remains absolutely irreducible modulo all sufficiently large prime numbers. We obtain a new lower bound for the size of such primes in terms of the number of integral points in the Newton polytope of the polynomial, significantly improving previous estimates for sparse polynomials.

On the irreducibility of Hecke polynomials

Let Tn,k(X) be the characteristic polynomial of the nth Hecke operator acting on the space of cusp forms of weight k for the full modular group. We record a simple criterion which can be used to check the irreducibility of the polynomials Tn,k(X). Using this criterion with some machine computation, we show that if there exists n ≥ 2 such that Tn,k(X) is irreducible and has the full symmetric gr...

Irreducibility testing of lacunary 0, 1-polynomials

A reciprocal polynomial g(x) ∈ Z[x] is such that g(0) 6= 0 and if g(α) = 0 then g(1/α) = 0. The non-reciprocal part of a monic polynomial f(x) ∈ Z[x] is f(x) divided by the product of its irreducible monic reciprocal factors (to their multiplicity). This paper presents an algorithm for testing the irreducibility of the nonreciprocal part of a 0, 1-polynomial (a polynomial having each coefficien...

Irreducibility of Polynomials with Low Absolute Values