On the distribution of cubic exponential sums

Louvel, Benoit

doi:10.1515/form.2011.167

Cited by 5 publications

(4 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [8] and [27], Bruggeman and Motohashi and Lokvenec-Guleska, respectively, construct a trace formula over an imaginary quadratic field with representations of all weights and expansions at different cusps, but the test functions involved-since they include all representations of various weights-are difficult to analyze. We mention Louvel in [29] does a similar analysis to what we need in this paper using the trace formula of [27]. However, his analysis of test functions on the spectral side of the trace formula is not strong enough for the results we need.…”

Section: 2mentioning

confidence: 87%

On Patterson's Conjecture: Sums of Quartic Exponential Sums

Herman

2013

Preprint

View full text Add to dashboard Cite

We give more evidence for Patterson's conjecture on sums of exponential sums by getting an asymptotic for a sum of quartic exponential sums over Q adjoined the eighth roots of unity. Previously, the strongest evidence of Patterson's conjecture over a number field is the paper of Livné and Patterson [28] on sums of cubic exponential sums overThe key ideas in getting such an asymptotic are a Kuznetsov-like trace formula for metaplectic forms over a quartic cover of GL 2 , and an identity on exponential sums relating Kloosterman sums and quartic exponential sums. To synthesize the spectral theory and the exponential sum identity, there is need for a good amount of analytic number theory.An unexpected aspect of the asymptotic of the sums of exponential sums is that there can be a secondary main term additional to the main term which is not predicted in Patterson's original paper [32].

show abstract

Section: 2mentioning

confidence: 87%

On Patterson's Conjecture: Sums of Quartic Exponential Sums

Herman

2013

Preprint

View full text Add to dashboard Cite

show abstract

“…It is thus natural to expect such a theorem should also exist for B(1, c). We would like to mention a similar result due to Louvel [Lo14] that such Bombieri-Vinogradov type equidistribution holds for cubic exponential sums modulo Eisenstein integers, for which he employed the spectral theory of cubic metaplectic forms, and cubic residue symbols can be well-introduced. However, as Louvel has pointed out, it is not yet known how to move from the cubic exponential sums modulo Eisenstein integers to those modulo rational integers in the horizontal aspect.…”

mentioning

confidence: 89%

When Kloosterman sums meet Hecke eigenvalues

2018

Preprint

View full text Add to dashboard Cite

By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from ℓ-adic cohomology, as well as a Bombieri-Vinogradov type mean value theorem for Kloosterman sums in arithmetic progressions, it is proved that for any given primitive Hecke-Maass cusp form of trivial nebentypus, the eigenvalue of the n-th Hecke operator does not coincide with the Kloosterman sum Kl(1, n) for infinitely many squarefree n with at most 100 prime factors. This provides a partial negative answer to a problem of Katz on modular structures of Kloosterman sums.

show abstract

“…It remains open to prove, for at least one choice of Ψ 1 , γ, the associated large sieve (or satisfying partial results towards it, if true). One approach might be to dualize and try to adapt [Lou14], at least over Z[ζ 3 ]. Alternatively, one could try elementary geometric and analytic arguments.…”

Section: 42mentioning

confidence: 99%

“…Remark C.3. It may or may not be natural to contrast Proposition C.1 with [Pat97, Theorem 3.1], a result showing that the "one-dimensional constituents of S c (q)" (before taking mixed 6th moments over a mod q) are "nice" over Q(ζ 3 ) (and in fact, amenable to statistical analysis via cubic metaplectic forms [Lou14]).…”

Section: B2 Counting On Other Varieties For Simplicity We Discuss Onl...mentioning

confidence: 99%

Diagonal cubic forms and the large sieve

Wang¹

2021

Preprint

View full text Add to dashboard Cite

Let F (x) be a diagonal integer-coefficient cubic form in m ∈ {4, 5, 6} variables. Excluding rational lines if m = 4, we bound the number of integral solutions x ∈ [−X, X] m to F (x) = 0 by O F, (X 3m/4−3/2+ ), conditionally on an "optimal large sieve inequality" (in a specific range of parameters) for "approximate Hasse-Weil L-functions" of smooth hyperplane sectionsWhen m is even, these results were previously established conditionally under Hooley's Hypothesis HW. Our 2 large sieve approach requires that certain bad factors be roughly 1 on average in 2 , while the ∞ Hypothesis HW approach only required the bound in 1 . Furthermore, the large sieve only accepts uniform vectors; yet our "initially given" vectors are only approximately uniform over c, due to variation in bad factors and in the archimedean component. Nonetheless, after some bookkeeping, partial summation, and Cauchy, the large sieve will still apply.

show abstract

On the distribution of cubic exponential sums

Cited by 5 publications

References 16 publications

On Patterson's Conjecture: Sums of Quartic Exponential Sums

On Patterson's Conjecture: Sums of Quartic Exponential Sums

When Kloosterman sums meet Hecke eigenvalues

Diagonal cubic forms and the large sieve

Contact Info

Product

Resources

About