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Journal articles on the topic 'Conjectures de Sato-Tate'

Author: Grafiati

Published: 4 June 2021

Last updated: 10 February 2022

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1

Arias-de-Reyna, Sara, Ilker Inam, and Gabor Wiese. "On conjectures of Sato–Tate and Bruinier–Kohnen." Ramanujan Journal 36, no. 3 (2014): 455–81. http://dx.doi.org/10.1007/s11139-013-9547-2.

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2

Hammonds, Trajan, Casimir Kothari, Noah Luntzlara, Steven J. Miller, Jesse Thorner, and Hunter Wieman. "The explicit Sato–Tate conjecture for primes in arithmetic progressions." International Journal of Number Theory 17, no. 08 (2021): 1905–23. http://dx.doi.org/10.1142/s179304212150069x.

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Let [Formula: see text] be Ramanujan’s tau function, defined by the discriminant modular form [Formula: see text] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer’s conjecture asserts that [Formula: see text] for all [Formula: see text]; since [Formula: see text] is multiplicative, it suffices to study primes [Formula: see text] for which [Formula: see text] might possibly be zero. Assuming standard conjectures for the twisted symmetric power [Formula: see text]-functions associated to [Formula: see text] (including GRH), we prove that if [Formula:

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3

du Sautoy, Marcus. "Natural boundaries for Euler products of Igusa zeta functions of elliptic curves." International Journal of Number Theory 14, no. 08 (2018): 2317–31. http://dx.doi.org/10.1142/s1793042118501415.

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We study the analytic behavior of adelic versions of Igusa integrals given by integer polynomials defining elliptic curves. By applying results on the meromorphic continuation of symmetric power L-functions and the Sato–Tate conjectures, we prove that these global Igusa zeta functions have some meromorphic continuation until a natural boundary beyond which no continuation is possible.

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4

Shparlinski, Igor. "On the Lang-Trotter and Sato-Tate conjectures on average for polynomial families of elliptic curves." Michigan Mathematical Journal 62, no. 3 (2013): 491–505. http://dx.doi.org/10.1307/mmj/1378757885.

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5

WANG, YINGNAN. "THE QUANTITATIVE DISTRIBUTION OF HECKE EIGENVALUES." Bulletin of the Australian Mathematical Society 90, no. 1 (2014): 28–36. http://dx.doi.org/10.1017/s0004972714000070.

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AbstractIn this paper, we prove that the Sato–Tate conjecture for primitive Maass forms holds on average. We also investigate the rate of convergence in the Sato–Tate conjecture and establish some estimates of the discrepancy with respect to the Sato–Tate measure on the average of primitive Maass forms.

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6

KUO, WENTANG. "A GENERALIZATION OF THE SATO–TATE CONJECTURE." International Journal of Number Theory 05, no. 01 (2009): 173–84. http://dx.doi.org/10.1142/s179304210900202x.

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The original Sato–Tate Conjecture concerns the angle distribution of the eigenvalues arisen from non-CM elliptic curves. In this paper, we formulate an analogue of the Sato–Tate Conjecture on automorphic forms of ( GL n) and, under a holomorphic assumption, prove that the distribution is either uniform or the generalized Sato–Tate measure.

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7

Banaszak, Grzegorz, and Kiran Kedlaya. "An algebraic Sato-Tate group and Sato-Tate conjecture." Indiana University Mathematics Journal 64, no. 1 (2015): 245–74. http://dx.doi.org/10.1512/iumj.2015.64.5438.

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8

Clozel, L. "The Sato-Tate conjecture." Current Developments in Mathematics 2006, no. 1 (2006): 1–34. http://dx.doi.org/10.4310/cdm.2006.v2006.n1.a1.

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9

Kuo, Wentang. "A Remark on a Modular Analogue of the Sato–Tate Conjecture." Canadian Mathematical Bulletin 50, no. 2 (2007): 234–42. http://dx.doi.org/10.4153/cmb-2007-025-7.

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AbstractThe original Sato–Tate Conjecture concerns the angle distribution of the eigenvalues arising from non-CM elliptic curves. In this paper, we formulate amodular analogue of the Sato–Tate Conjecture and prove that the angles arising from non-CM holomorphic Hecke eigenforms with non-trivial central characters are not distributed with respect to the Sate–Tatemeasure for non-CM elliptic curves. Furthermore, under a reasonable conjecture, we prove that the expected distribution is uniform.

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10

Shieh, Yih-Dar. "Character theory approach to Sato–Tate groups." LMS Journal of Computation and Mathematics 19, A (2016): 301–14. http://dx.doi.org/10.1112/s1461157016000279.

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In this article, we propose to use the character theory of compact Lie groups and their orthogonality relations for the study of Frobenius distribution and Sato–Tate groups. The results show the advantages of this new approach in several aspects. With samples of Frobenius ranging in size much smaller than the moment statistic approach, we obtain very good approximation to the expected values of these orthogonality relations, which give useful information about the underlying Sato–Tate groups and strong evidence of the correctness of the generalized Sato–Tate conjecture. In fact, $2^{10}$ to $2

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11

Thorner, Jesse. "The error term in the Sato–Tate conjecture." Archiv der Mathematik 103, no. 2 (2014): 147–56. http://dx.doi.org/10.1007/s00013-014-0673-x.

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12

Barnet-Lamb, Thomas, Toby Gee, and David Geraghty. "The Sato-Tate conjecture for Hilbert modular forms." Journal of the American Mathematical Society 24, no. 2 (2011): 411. http://dx.doi.org/10.1090/s0894-0347-2010-00689-3.

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13

Lemke Oliver, Robert J., and Jesse Thorner. "Effective Log-Free Zero Density Estimates for Automorphic L-Functions and the Sato–Tate Conjecture." International Mathematics Research Notices 2019, no. 22 (2017): 6988–7036. http://dx.doi.org/10.1093/imrn/rnx309.

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Abstract Let $K/\mathbb{Q}$ be a number field. Let π and π′ be cuspidal automorphic representations of $\textrm{GL}_{d}(\mathbb{A}_{K})$ and $\textrm{GL}_{d^{\prime }}(\mathbb{A}_{K})$. We prove an unconditional and effective log-free zero density estimate for all automorphic L-functions L(s, π) and prove a similar estimate for Rankin–Selberg L-functions L(s, π × π′) when π or π′ satisfies the Ramanujan conjecture. As applications, we make effective Moreno’s analog of Hoheisel’s short interval prime number theorem and extend it to the context of the Sato–Tate conjecture; additionally, we bound

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14

Suh, Junecue. "Ordinary primes in Hilbert modular varieties." Compositio Mathematica 156, no. 4 (2020): 647–78. http://dx.doi.org/10.1112/s0010437x19007826.

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A well-known conjecture, often attributed to Serre, asserts that any motive over any number field has infinitely many ordinary reductions (in the sense that the Newton polygon coincides with the Hodge polygon). In the case of Hilbert modular cuspforms $f$ of parallel weight $(2,\ldots ,2)$, we show how to produce more ordinary primes by using the Sato–Tate equidistribution and combining it with the Galois theory of the Hecke field. Under the assumption of stronger forms of Sato–Tate equidistribution, we get stronger (but conditional) results. In the case of higher weights, we formulate the ord

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15

Baier, Stephan, and Liangyi Zhao. "The Sato-Tate conjecture on average for small angles." Transactions of the American Mathematical Society 361, no. 04 (2008): 1811–32. http://dx.doi.org/10.1090/s0002-9947-08-04498-x.

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16

Harris, Michael. "Galois representations, automorphic forms, and the Sato-Tate Conjecture." Indian Journal of Pure and Applied Mathematics 45, no. 5 (2014): 707–46. http://dx.doi.org/10.1007/s13226-014-0085-4.

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17

Johansson, Christian. "On the Sato-Tate conjecture for non-generic abelian surfaces." Transactions of the American Mathematical Society 369, no. 9 (2017): 6303–25. http://dx.doi.org/10.1090/tran/6847.

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18

Fité, Francesc, Josep González, and Joan-Carles Lario. "Frobenius Distribution for Quotients of Fermat Curves of Prime Exponent." Canadian Journal of Mathematics 68, no. 2 (2016): 361–94. http://dx.doi.org/10.4153/cjm-2015-028-x.

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AbstractLet denote the Fermat curve over ℚ of prime exponent ℓ. The Jacobian Jac() of splits over ℚ as the product of Jacobians Jac(k), 1 ≤ k ≤ ℓ −2, where k are curves obtained as quotients of by certain subgroups of automorphisms of . It is well known that Jac(k) is the power of an absolutely simple abelian variety Bk with complex multiplication. We call degenerate those pairs (ℓ, k) for which Bk has degenerate CM type. For a non-degenerate pair (ℓ, k), we compute the Sato–Tate group of Jac(Ck), prove the generalized Sato–Tate Conjecture for it, and give an explicit method to compute the mom

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19

Hashimoto, Ki-ichiro, and Hiroshi Tsunogai. "On the Sato-Tate conjecture for QM-curves of genus two." Mathematics of Computation 68, no. 228 (1999): 1649–63. http://dx.doi.org/10.1090/s0025-5718-99-01061-3.

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20

Narayanan, Sridhar. "On the Non-Vanishing of a Certain Class of Dirichlet Series." Canadian Mathematical Bulletin 40, no. 3 (1997): 364–69. http://dx.doi.org/10.4153/cmb-1997-043-6.

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AbstractIn this paper, we consider Dirichlet series with Euler products of the form F(s) = Πp in > 1, and which are regular in ≥ 1 except for a pole of order m at s = 1. We establish criteria for such a Dirichlet series to be nonvanishing on the line of convergence. We also show that our results can be applied to yield non-vanishing results for a subclass of the Selberg class and the Sato-Tate conjecture.

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21

Rouse, Jeremy, and Jesse Thorner. "The explicit Sato-Tate Conjecture and densities pertaining to Lehmer-type questions." Transactions of the American Mathematical Society 369, no. 5 (2016): 3575–604. http://dx.doi.org/10.1090/tran/6793.

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22

Michel, P. "Autour de la conjecture de Sato-Tate pour les sommes de Kloosterman I." Inventiones Mathematicae 121, no. 1 (1995): 61–78. http://dx.doi.org/10.1007/bf01884290.

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23

Jameson, Marie, Jesse Thorner, and Lynnelle Ye. "Benford’s Law for coefficients of newforms." International Journal of Number Theory 12, no. 02 (2016): 483–94. http://dx.doi.org/10.1142/s1793042116500299.

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Let [Formula: see text] be a newform of even weight [Formula: see text] on [Formula: see text] without complex multiplication. Let [Formula: see text] denote the set of all primes. We prove that the sequence [Formula: see text] does not satisfy Benford’s Law in any integer base [Formula: see text]. However, given a base [Formula: see text] and a string of digits [Formula: see text] in base [Formula: see text], the set [Formula: see text] has logarithmic density equal to [Formula: see text]. Thus, [Formula: see text] follows Benford’s Law with respect to logarithmic density. Both results rely o

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24

Shparlinski, Igor E. "Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average." Michigan Mathematical Journal 56, no. 1 (2008): 99–111. http://dx.doi.org/10.1307/mmj/1213972400.

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25

Thorner, Jesse. "Effective forms of the Sato–Tate conjecture." Research in the Mathematical Sciences 8, no. 1 (2021). http://dx.doi.org/10.1007/s40687-020-00234-3.

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26

Gillman, Nate, Michael Kural, Alexandru Pascadi, Junyao Peng, and Ashwin Sah. "Patterns of primes in the Sato–Tate conjecture." Research in Number Theory 6, no. 1 (2019). http://dx.doi.org/10.1007/s40993-019-0184-8.

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27

Lau, Yuk-Kam, Ming Ho Ng, and Yingnan Wang. "Average Bound Toward the Generalized Ramanujan Conjecture and Its Applications on Sato–Tate Laws for GL(n)." International Mathematics Research Notices, October 14, 2020. http://dx.doi.org/10.1093/imrn/rnaa262.

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Abstract We give the 1st non-trivial estimate for the number of $GL(n)$ ($n\ge 3$) Hecke–Maass forms whose Satake parameters at any given prime $p$ fail the Generalized Ramanujan Conjecture and study some applications on the (vertical) Sato–Tate laws.

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28

Shparlinski, Igor E. "On the Sato–Tate conjecture on average for some families of elliptic curves." Forum Mathematicum, June 30, 2011, ———. http://dx.doi.org/10.1515/form.2011.141.

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