Galois representations, automorphic forms, and the Sato-Tate Conjecture

Harris, Michael

doi:10.1007/s13226-014-0085-4

Cited by 10 publications

(2 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By the Hasse bound [16], conjectured by Artin [1] in his thesis, we have that | tr(E)| ≤ 2 √ p. Taking −1 ≤ a ≤ b ≤ 1 and a fixed elliptic curve E over Q, it was independently conjectured by Sato and Tate (see e.g. [14]) that if E does not have complex multiplication, then lim…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection

Kane¹,

Pujahari²

2020

Preprint

View full text Add to dashboard Cite

In this paper, we study moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. In particular, we fix the arithmetic progression and consider the ratio of the 2k-th moment to the zeroeth moment as one varies the size of the finite field F p r . We obtain asymptotic formulas for these ratios for cases for which the prime p goes to infinity for fixed r and cases where the power r goes to infinity with fixed p. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions which we investigate in this paper.

show abstract

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection

Kane¹,

Pujahari²

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…What we need for our purposes is a natural generalization of this result for the joint distribution of two newforms. This was done by Harris ([4], Theorem 5.4 1 ; see also [3], Theorem 2.4) for two nonisogenous elliptic curves. In [9] (Section 4), Murty and Pujahari have extended the argument for two Hecke eigenforms, provided that they are not twists of each other.…”

Section: Resultsmentioning

confidence: 99%

On the number of dominating Fourier coefficients of two newforms

Chiriac

2018

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

Arithmetic applications of the Langlands program

Harris

2010

Jpn. J. Math.

Self Cite

View full text Add to dashboard Cite

This expository article is an introduction to the Langlands functoriality conjectures and their applications to the arithmetic of representations of Galois groups of number fields. Thanks to the work of a great many people, the stable trace formula is now largely established in a version adequate for proving Langlands functoriality in the setting of endoscopy. These developments are discussed in the first two sections of the article. The final section describes the compatible families of`-adic Galois representations that can be attached to automorphic forms with the help of Shimura varieties. To illustrate the relevance of Langlands functoriality to number theory, the article concludes with a description of the Sato-Tate conjecture for elliptic modular forms, recently proved in joint work of Barnet-Lamb, Geraghty, Taylor, and the author.

show abstract

Galois representations, automorphic forms, and the Sato-Tate Conjecture

Cited by 10 publications

References 19 publications

Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection

Distribution of moments of Hurwitz class numbers in arithmetic progressions and holomorphic projection

On the number of dominating Fourier coefficients of two newforms

Arithmetic applications of the Langlands program

Contact Info

Product

Resources

About