Some applications of large sieve in Riemann surfaces

Chamizo, Fernando

doi:10.4064/aa-77-4-315-337

Cited by 21 publications

(27 citation statements)

References 5 publications

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“…After Lemmas and , the proof of Theorems and boils down to approximating sharp cut‐offs by smooth kernels and estimating S‐HC transforms. Following , with a hyperbolic version of a classical Euclidean device, it is possible to reduce the problem to estimate the S‐HC transform of the characteristic function of an interval (which is done in [, Lemma 2.4]). With this idea in mind, we write

χ_{V}

and

h_{V}

, respectively, for the characteristic function of

[0, V]

and its S‐HC transform:

χ_{V} false(x false) = \{\begin{matrix} left 1 & left if 0 ⩽ x < V, \\ left 0 & left otherwise \end{matrix} and h_{V} false(t false) = \int_{{u (z, i) ⩽ V}} y^{1 / 2 + i t} d μ false(z false) .

The slow decay of

h_{V}

would cause serious convergence problems when applying the spectral theorem.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

“…The key argument is a simple one and it is summarized in the following result. We merely sketch the details of the proof because it is essentially contained in that of [, Lemma 2.3] (see also [, § 5] and [, Lemma 2.2]). Lemma For

0 < v < V

consider the function

F : {double-struckH}^{2} \to R

given by

F false(z, w false) = \frac{1}{4 π v} \int_{double-struckH} χ_{V} (u false(z, ζ false)) χ_{v} (u false(ζ, w false)) d μ false(ζ false) .

Then there exists

f : [0, \infty) \to R

such that

F (z, w) = f (u (z, w))

and

χ_{V^{-}} ⩽ f ⩽ χ_{V^{+}} where V^{\pm} = {(() \sqrt{V (1 + v)} \pm \sqrt{v (1 + V)})}^{2} .

Moreover, the S‐HC transform of

f

h_{V} h_{v}

.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Sums of squares in real quadratic fields and Hilbert modular groups

Chamizo

Miatello

2020

Bull. London Math. Soc.

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χ_{V}

and

h_{V}

, respectively, for the characteristic function of

[0, V]

and its S‐HC transform:

χ_{V} false(x false) = \{\begin{matrix} left 1 & left if 0 ⩽ x < V, \\ left 0 & left otherwise \end{matrix} and h_{V} false(t false) = \int_{{u (z, i) ⩽ V}} y^{1 / 2 + i t} d μ false(z false) .

The slow decay of

h_{V}

would cause serious convergence problems when applying the spectral theorem.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

0 < v < V

consider the function

F : {double-struckH}^{2} \to R

given by

F false(z, w false) = \frac{1}{4 π v} \int_{double-struckH} χ_{V} (u false(z, ζ false)) χ_{v} (u false(ζ, w false)) d μ false(ζ false) .

Then there exists

f : [0, \infty) \to R

such that

F (z, w) = f (u (z, w))

and

χ_{V^{-}} ⩽ f ⩽ χ_{V^{+}} where V^{\pm} = {(() \sqrt{V (1 + v)} \pm \sqrt{v (1 + V)})}^{2} .

Moreover, the S‐HC transform of

f

h_{V} h_{v}

.…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Sums of squares in real quadratic fields and Hilbert modular groups

Chamizo

Miatello

2020

Bull. London Math. Soc.

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“…This exponent has not been improved but the natural conjecture (supported by average results [4]) is α = 1/2 + ε for every ε > 0.…”

Section: The Visible Lattice Point Problemmentioning

confidence: 95%

“…It is possible to write an explicit formula for the distance d (see [10] and [9] for it and its geometrical interpretation) that in the orbit of z = i acquires an especially simple form (see [4])…”

Section: Preliminaries and Symmetriesmentioning

confidence: 99%

Non-Euclidean visibility problems

Chamizo

2006

Proc. Indian Acad. Sci. (Math. Sci.)

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“…We do not investigate these but refer to [9,Theorem 12.1], [13,6,16,2]. To every conjugacy class {γ} of Γ corresponds a unique closed oriented geodesic of length l(γ).…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic lattice-point counting and modular symbols

Risager¹

2009

J. Théor. Nombres Bordeaux

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Résumé. Soit un sous-groupe Γ de SL 2 (R) cocompact et soit α une forme harmonique réelle (non nulle). Nousétudions le comportement asymptotique de la fonction comptant des points du réseau hyperbolique Γ sous hypothèses imposées par des symboles modulaires γ, α . Nous montrons que les valeurs normalisées des symboles modulaires, ordonnées selon ce comptage possèdent une répartition gaussienne.Abstract. For a cocompact group Γ of SL 2 (R) we fix a real nonzero harmonic 1-form α. We study the asymptotics of the hyperbolic lattice-counting problem for Γ under restrictions imposed by the modular symbols γ, α . We prove that the normalized values of the modular symbols, when ordered according to this counting, have a Gaussian distribution.

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Some applications of large sieve in Riemann surfaces

Cited by 21 publications

References 5 publications

Sums of squares in real quadratic fields and Hilbert modular groups

Sums of squares in real quadratic fields and Hilbert modular groups

Non-Euclidean visibility problems

Hyperbolic lattice-point counting and modular symbols

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