One Level Density for Cubic Galois Number Fields

Meisner, Patrick

doi:10.4153/cmb-2018-002-4

Cited by 7 publications

(4 citation statements)

References 8 publications

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“…Obtaining an asymptotic for the second moment for cubic Dirichlet L-functions is still an open question, over functions fields or number fields. Moreover, for the case of cubic Dirichlet L-functions, computing the one-level density can only be done for limited support of the Fourier transform of the test function, and that is not enough to lead to a positive proportion of nonvanishing for the full family, even under the GRH [5,29]. Recently David and Güloğlu [14] obtained a positive proportion of nonvanishing for Luo's thin family [28] by computing the one-level density.…”

Section: Introductionmentioning

confidence: 99%

Nonvanishing for cubic L-functions

David

Florea

Laĺın

2021

Forum of Mathematics, Sigma

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We prove that there is a positive proportion of L-functions associated to cubic characters over $\mathbb F_q[T]$ that do not vanish at the critical point $s=1/2$ . This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester and Radziwiłł, which in turn develops further ideas from the work of Soundararajan, Harper and Radziwiłł. We work in the non-Kummer setting when $q\equiv 2 \,(\mathrm {mod}\,3)$ , but our results could be translated into the Kummer setting when $q\equiv 1\,(\mathrm {mod}\,3)$ as well as into the number-field case (assuming the generalised Riemann hypothesis). Our positive proportion of nonvanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.

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Section: Introductionmentioning

confidence: 99%

Nonvanishing for cubic L-functions

David

Florea

Laĺın

2021

Forum of Mathematics, Sigma

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“…In recent years, there has been an increased interest in a variety of different aspects of higher order characters and twists; see, e.g., [1,7,8,9,10,11,12,13,27,28]. Motivated by this development, we investigate expected values of traces of high powers of the Frobenius class and the one-level density of families of cubic twists of elliptic curves of the form y 2 = x 3 + B defined over F q (t).…”

Section: Introductionmentioning

confidence: 99%

Low-lying zeros in families of elliptic curve $L$-functions over function fields

Meisner¹,

Södergren²

2021

Preprint

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We investigate the low-lying zeros in families of L-functions attached to quadratic and cubic twists of elliptic curves defined over Fq(T ). In particular, we present precise expressions for the expected values of traces of high powers of the Frobenius class in these families with a focus on the lower order behavior. As an application we obtain results on one-level densities and we verify that these elliptic curve families have orthogonal symmetry type. In the quadratic twist families our results refine previous work of Comeau-Lapointe. Moreover, in this case we find a lower order term in the one-level density reminiscent of the deviation term found by Rudnick in the hyperelliptic ensemble. On the other hand, our investigation is the first to treat these questions in families of cubic twists of elliptic curves and in this case it turns out to be more complicated to isolate lower order terms due to a larger degree of cancellation among lower order contributions.

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Section: Introductionmentioning

confidence: 99%

Non-vanishing for cubic $L$--functions

David¹,

Florea²,

Laĺın³

2020

Preprint

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We prove that there is a positive proportion of L-functions associated to cubic characters over F q [T ] that do not vanish at the critical point s = 1/2. This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L-functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Lester-Radziwi l l, which in turn develops further ideas from the work of Soundararajan, Harper, and Radziwi l l-Soundararajan. We work in the non-Kummer setting when q ≡ 2 (mod 3), but our results could be translated into the Kummer setting when q ≡ 1 (mod 3) as well as into the number field case (assuming the Generalized Riemann Hypothesis). Our positive proportion of non-vanishing is explicit, but extremely small, due to the fact that the implied constant in the upper bound for the mollified second moment is very large.

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One Level Density for Cubic Galois Number Fields

Cited by 7 publications

References 8 publications

Nonvanishing for cubic L-functions

Nonvanishing for cubic L-functions

Low-lying zeros in families of elliptic curve $L$-functions over function fields

Non-vanishing for cubic $L$--functions

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