A Note on the Irreducibility of Hecke Polynomials (original) (raw)

The irreducibility of some level 1 Hecke polynomials

2002

Abstract: Let $ T_ {p, k}(x) $ be the characteristic polynomial of the Hecke operator $ T_ {p} $ acting on the space of level 1 cusp forms $ S_ {k}(1) .Weshowthat. We show that .Weshowthat T_ {p, k}(x) $ is irreducible and has full Galois group over mathbfQ\ mathbf {Q} mathbfQ for $ k\ le 2000$ and $ p< 2000$, $ p $ prime.

Polynomial interpolation of modular forms for Hecke groups

2020

For m " 3, 4, ..., let λm " 2 cos π{m and let Jmpm " 3, 4, ...) be triangle functions for the Hecke groups Gpλmq with Fourier expansions Jmpτ q " ř 8 n"´1 anpmqq n m , where qmpτ q " exp 2πiτ {λm. (When normalized appropriately, J3 becomes Klein's j-invariant jpτ q " 1{e 2πiτ`7 44`....) For n "´1, 0, 1, 2 and 3, Raleigh gave polynomials Pnpxq such that a´1pmq n q 2n`2 m anpmq " Pnpmq for m " 3, 4, ..., and conjectured that similar relations hold for all positive integers n. This was proved by Akiyama. We apply work of Hecke to study experimentally similar polynomial interpolations of the Jm Fourier coefficents and the Fourier coefficients of other, positive weight, modular forms for Gpλmq. We connect these polynomials (again, only empirically) with variants of Dedekind's eta function, with the Fourier expansions of some standard Hauptmoduln, and, in the case of analogues of Eisenstein series for SLp2, Zq, with certain divisor sums. 43 The first few polynomials in table 10.5 agree with Raleigh's equation-group III in [35]. 44 Notebooks "conjecture 2.nb", "conjecture 2, clause 2.ipynb", [10]. 45 [10], Notebook "conjecture 1 clause 2 w code 14jun21.ipynb" 46 (proved in [1] 47 [27], equation (2).

Hecke nilpotency for modular forms mod 2 and an application to partition numbers

arXiv (Cornell University), 2022

A well-known observation of Serre and Tate is that the Hecke algebra acts locally nilpotently on modular forms mod 2 on SL2(Z). We give an algorithm for calculating the degree of Hecke nilpotency for cusp forms, and we obtain a formula for the total number of cusp forms mod 2 of any given degree of nilpotency. Using these results, we find that the degrees of Hecke nilpotency in spaces M k have no limiting distribution as k → ∞. As an application, we study the parity of the partition function using Hecke nilpotency.

Non vanishing of Central values of modular L-functions for Hecke eigenforms of level one

Arxiv preprint arXiv:1001.5181, 2010

Let F (z) = ∞ n=1 a(n)q n be a newform of weight 2k and level N with a trivial character, and assume that F (z) is a non-zero eigenform of all Hecke operators. For x > 0, let N F (x) := |{D fundamental | |D| < x, (D, N) = 1, L(F, D, k) = 0}|. A based on the Goldfeld's conjecture one expects to have N F (x) ≫ x (x → ∞). Kohnen [10] showed that if k ≥ 6 is a even integer, then for x ≫ 0 there is a normalized Hecke eigenform F of level 1 and weight 2k with the property that N F (x) ≫ k x (x → ∞). In this paper, we extend the result in [10] to the case when k is any integer, in particular when k is odd. So, we obtain that, when the level is 1, for each integer 2k such that the dimension of cusp forms of weight 2k is not zero, there is a normalized Hecke eigenform F of weight 2k satisfying N F (x) ≫ x (x → ∞).

Hecke operators on certain subspaces of integral weight modular forms

International Journal of Number Theory, 2014

Recent works of F. G. Garvan ([Congruences for Andrews' smallest parts partition function and new congruences for Dyson's rank, Int. J. Number Theory6(12) (2010) 281–309; MR2646759 (2011j:05032)]) and Y. Yang ([Congruences of the partition function, Int. Math. Res. Not.2011(14) (2011) 3261–3288; MR2817679 (2012e:11177)] and [Modular forms for half-integral weights on SL 2(ℤ), to appear in Nagoya Math. J.]) concern a certain family of half-integral weight Hecke-invariant subspaces which arise as multiples of fixed odd powers of the Dedekind eta-function multiplied by SL 2(ℤ)-forms of fixed weight. In this paper, we study the image of Hecke operators on subspaces which arise as multiples of fixed even powers of eta multiplied by SL 2(ℤ)-forms of fixed weight.

On a Generalization of a Theorem of Erich Hecke

Proceedings of The National Academy of Sciences, 1982

E. Hecike initiated the application of representation theory to the study of cusp forms. He showed that, for p a prime congruent to 3 mod 4, the difference ofmultiplicities ofcertain conjugate representations of SL4(F) on cusp forms of degree 1, level p, and weight 22 is given by the class number h(-p) of the field Q(Vj;). We apply the holomorphic Lefschetz theorem to actions on the Igusa compactification ofthe Siegel moduli space of degree 2 to compute the values of characters of the representations of SpN(FV) on certain spaces of cusp forms of degree 2 and level p at parabolic elements ofthis group. Our results imply that here too, the difference in multiplicities of conjugate representations of Sp4(F,) is a multiple of h(-p).