Modularity of an odd icosahedral representation

Jehanne, Arnaud; Müller, Michael

doi:10.5802/jtnb.291

Cited by 6 publications

(6 citation statements)

References 1 publication

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“…Instead we can make use of one of the earlier examples of (odd) icosahedral representations, such as the one whose modularity is addressed by [4] and is described in [12]. This representation ρ is associated to the polynomial (which describes the splitting field) x 5 − 8x 3 − 2x 2 + 31x + 74, has conductor 3547, and odd quadratic determinant.…”

Section: Every Such Pair Of Irreducible Representations Gives An Irrementioning

confidence: 99%

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Further refinement of strong multiplicity one for GL(2)

Walji

2014

Trans. Amer. Math. Soc.

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Abstract. We obtain a sharp refinement of the strong multiplicity one theorem for the case of unitary non-dihedral cuspidal automorphic representations for GL(2). Given two unitary cuspidal automorphic representations for GL(2) that are not twist-equivalent, we also find sharp lower bounds for the number of places where the Hecke eigenvalues are not equal, for both the general and non-dihedral cases. We then construct examples to demonstrate that these results are sharp.

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Section: Every Such Pair Of Irreducible Representations Gives An Irrementioning

confidence: 99%

“…we apply the Chebotarev density theorem to determine the density of places at which pairs of the form (trρ(Frob v,K/F ), trρ(Frob v,K ′ /F )) occur: (12) (0,0) 9/18…”

Section: A Dihedral Example For Theoremmentioning

confidence: 99%

Further refinement of strong multiplicity one for GL(2)

Walji

2014

Trans. Amer. Math. Soc.

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“…We have applied the methods described in the previous sections to a few examples of splitting fields of polynomials with Galois group S 5 and A 5 , as listed in Table 6.1. For the A 5 cases, the Artin conjecture is true for all representations by known cases of functoriality [Kim94,JM01]. That speeds up the process, since we may apply Turing's method to the Artin L-functions directly.…”

Section: Numerical Resultsmentioning

confidence: 98%

“…For any particular ρ, one can search for the associated form; once found, comparing the representation constructed by Deligne-Serre to ρ via an effective version of the Cebotarev density theorem allows one to deduce the conjecture for ρ. This and other related techniques have been carried out in a number of cases; see [Buh78,Kim94,JM00,BS02].…”

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confidence: 99%

Artin's Conjecture, Turing's Method, and the Riemann Hypothesis

Booker

2006

Experimental Mathematics

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We present a group-theoretic criterion under which one may verify the Artin conjecture for some (non-monomial) Galois representations, up to finite height in the complex plane. In particular, the criterion applies to S 5 and A 5 representations. Under more general conditions, the technique allows for the possibility of verifying the Riemann hypothesis for Dedekind zeta functions of non-abelian extensions of Q.In addition, we discuss two methods for locating zeros of arbitrary L-functions. The first uses the explicit formula and techniques developed in [BS05] for computing with trace formulae. The second method generalizes that of Turing for verifying the Riemann hypothesis. In order to apply it we develop a rigorous algorithm for computing general L-functions on the critical line via the Fast Fourier Transform.Finally, we present some numerical results testing Artin's conjecture for S 5 representations, and the Riemann hypothesis for Dedekind zeta functions of S 5 and A 5 fields.1 Reading Turing's paper on the subject, which was his last, one marvels at what he accomplished with the limited computational resources of the day. His method was truly ahead of its time.

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“…First, there are a few technical difficulties in the even case that do not arise in the odd case. Second, there is already much in the way of evidence, both theoretical and numerical, in support of the strong Artin conjecture for odd representations (see [4], [3], [9], [14], [13], [5]); there are so far no known examples in the even case.…”

Section: Introductionmentioning

confidence: 99%