60 research outputs found
Uniform approximate functional equation for principal L-functions
We prove an approximate functional equation for the central value of the
L-series attached to an irreducible cuspidal automorphic representation of
GL(m) over a number field with unitary central character. We investigate the
decay rate of the terms involved using the analytic conductor of Iwaniec and
Sarnak as a guideline. Straightforward extensions of the results exist for
products of central values. We hope that these formulae will help further
understanding of the central values of principal L-functions, such as finding
good bounds on their various power means, or establishing subconvexity or
nonvanishing results in certain families. A crucial role in the proofs is
played by recent progress on the Ramanujan--Selberg conjectures achieved by
Luo, Rudnick and Sarnak. The bounds at the non-Archimedean places enter through
the work of Molteni.Comment: 8 pages, LaTeX2e; v2: shorter abstract in paper, recent amsart
package used; v3: small alterations in the text (most importantly correcting
the definition of the analytic conductor (4)), references updated; to appear
soon in Internat. Math. Res. Notice
An additive problem in the Fourier coefficients of cusp forms
We establish an estimate on sums of shifted products of Fourier coefficients
coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus.
These sums are analogous to the binary additive divisor sum which has been
studied extensively. As an application we derive, extending work of Duke,
Friedlander and Iwaniec, a subconvex estimate on the critical line for
L-functions associated to character twists of these cusp forms.Comment: 16 pages, LaTeX2e; v2: lots of changes, Theorem 2 is new, notation
changed to standard one, abstract and further references added; v3: minor
changes, some restriction imposed in Theorem 2, additional references; v4:
introduction revised, references added, typos corrected; v5: final, revised
version incorporating suggestions by the referee (e.g. Section 5 was added
New bounds on even cycle creating Hamiltonian paths using expander graphs
We say that two graphs on the same vertex set are -creating if their union
(the union of their edges) contains as a subgraph. Let be the
maximum number of pairwise -creating Hamiltonian paths of . Cohen,
Fachini and K\"orner proved In this paper we close the superexponential gap
between their lower and upper bounds by proving
We also improve the
previously established upper bounds on for , and we present
a small improvement on the lower bound of F\"uredi, Kantor, Monti and Sinaimeri
on the maximum number of so-called pairwise reversing permutations. One of our
main tools is a theorem of Krivelevich, which roughly states that (certain
kinds of) good expanders contain many Hamiltonian paths.Comment: 14 pages, LaTeX2e; v2: updated Footnote 1 on Page 5; v3: revised
version incorporating suggestions by the referees (the changes are mainly in
Section 5); v4: final version to appear in Combinatoric
A hybrid asymptotic formula for the second moment of Rankin-Selberg L-functions
We consider the Rankin-Selberg L-functions associated with a fixed modular
form of full level and holomorphic cuspidal newforms of large even weight,
fixed level and fixed primitive nebentypus. We compute the second moment of
this family in fairly general ranges, and obtain an asymptotic formula with a
power saving error term. A special case treats the fourth moment of L-functions
associated with holomorphic cusp forms.Comment: 30 pages, LaTeX2e, submitted; v2: revised version incorporating
suggestions by the refere
On the sup-norm of Maass cusp forms of large level. III
Let be a Hecke--Maass cuspidal newform of square-free level and
Laplacian eigenvalue . It is shown that \pnorm{f}_\infty
\ll_{\lambda,\epsilon} N^{-1/6}+\epsilon} \pnorm{f}_2 for any
New bounds on even cycle creating Hamiltonian paths using expander graphs
We say that two graphs on the same vertex set are -creating if their union
(the union of their edges) contains as a subgraph. Let be the
maximum number of pairwise -creating Hamiltonian paths of . Cohen,
Fachini and K\"orner proved In this paper we close the superexponential gap
between their lower and upper bounds by proving
We also improve the
previously established upper bounds on for , and we present
a small improvement on the lower bound of F\"uredi, Kantor, Monti and Sinaimeri
on the maximum number of so-called pairwise reversing permutations. One of our
main tools is a theorem of Krivelevich, which roughly states that (certain
kinds of) good expanders contain many Hamiltonian paths.Comment: 14 pages, LaTeX2e; v2: updated Footnote 1 on Page 5; v3: revised
version incorporating suggestions by the referees (the changes are mainly in
Section 5); v4: final version to appear in Combinatoric
The spectral decomposition of shifted convolution sums
We obtain a spectral decomposition of shifted convolution sums in Hecke
eigenvalues of holomorphic or Maass cusp forms.Comment: 15 pages, LaTeX2e; v2: corrected and slightly expanded versio
Automorf formĂĄk Ă©s L-fĂŒggvĂ©nyek = Automorphic forms and L-functions
Valentin Blomerrel Burgess-tĂpusĂș szubkonvex becslĂ©st igazoltunk csavart Hilbert modulĂĄris L-fĂŒggvĂ©nyekre, megjavĂtva Cogdell-PiatetskiShapiro-Sarnak Ă©s Venkatesh idevĂĄgĂł eredmĂ©nyeit. Közvetlen alkalmazĂĄskĂ©nt az eddigieknĂ©l hatĂ©konyabban tudjuk becsĂŒlni pozitĂv definit ternĂ©r kvadratikus formĂĄk elĆĂĄllĂtĂĄsszĂĄmait egy teljesen valĂłs szĂĄmtest egĂ©szei felett. Valentin Blomerrel aszimptotikus formulĂĄt adtunk Rankin-Selberg L-fĂŒggvĂ©nyek bizonyos archimĂ©deszi csalĂĄdjaira. Az eredmĂ©ny Ă©rdekessĂ©ge, hogy amikor a Rankin-Selberg konvolĂșciĂłban a rögzĂtett formĂĄt Eisenstein-sornak vĂĄlasztjuk, az aszimptotikĂĄban a szokĂĄsos logaritmikus tagok mellett kĂ©t forgĂł tag is megjelenik. Egy speciĂĄlis esetben a holomorf csĂșcsformĂĄkhoz tĂĄrsĂtott L-fĂŒggvĂ©nyek negyedik momentumĂĄrĂłl szĂłl az eredmĂ©ny. Nicolas Templier-val Ășj becslĂ©st adtunk Hecke-Maass csĂșcsformĂĄk szuprĂ©mumĂĄra a szint aspektusban. Az eredmĂ©ny analĂłg a Riemann zeta-fĂŒggvĂ©nyre vonatkozĂł szubkonvex Weyl-korlĂĄttal. A közelmĂșltban - mĂĄs mĂłdszerrel - hasonlĂł erejƱ tĂ©telt igazolt Blomer-Michel kompakt aritmetikus felĂŒletekre. Mi a kompaktsĂĄg hiĂĄnyĂĄt az Atkin-Lehner operĂĄtorok egy ĂșjszerƱ alkalmazĂĄsĂĄval kezeljĂŒk hatĂ©konyan. | In joint work with Valentin Blomer we proved a Burgess-like subconvex bound for twisted Hilbert modular L-functions, improving on the relevant results of Cogdell-PiatetskiShapiro-Sarnak and Venkatesh. As a direct application, we can estimate more efficiently the number of representations by a positive definite ternary quadratic form over the integers of a totally real number field. In joint work with Valentin Blomer we established an asymptotic formula for certain archimedean families of Rankin-Selberg L-functions. As an interesting feature of the result, when the fixed form in the Rankin-Selberg convolution is chosen to be an Eisenstein series, two winding terms appear in addition to the usual logarithmic terms. A special case treats the fourth moment of L-functions associated with holomorphic cusp forms. In joint work with Nicolas Templier we established a new bound for the sup-norm of Hecke-Maass cusp forms in the level aspect. The result is analogous to the subconvex Weyl bound for the Riemann zeta function. Very recently, Blomer-Michel proved, with a different method, a theorem of similar strength for compact arithmetic surfaces. We handle the lack of compactness efficiently by a novel application of Atkin-Lehner operators
New bounds for automorphic L-functions
This thesis contributes to the analytic theory of automorphic L-functions.
We prove an approximate functional equation for the central value of the
L-series attached to an irreducible cuspidal automorphic representation of
GL(m) over a number field. We investigate the decay rate of the cutoff function
and its derivatives in terms of the analytic conductor introduced by Iwaniec
and Sarnak. We also see that the truncation can be made uniformly explicit at
the cost of an error term. The results extend to products of central values.
We establish, via the Hardy-Littlewood circle method, a nontrivial bound on
shifted convolution sums of Fourier coefficients coming from classical
holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums
are analogous to the binary additive divisor sum which has been studied
extensively. We achieve polynomial uniformity in all the parameters of the cusp
forms by carefully estimating the Bessel functions that enter the analysis. As
an application we derive, extending work of Duke, Friedlander and Iwaniec, a
subconvex estimate on the critical line for L-functions associated to character
twists of these cusp forms.
We also study the shifted convolution sums via the Sarnak-Selberg spectral
method. For holomorphic cusp forms this approach detects optimal cancellation
over any totally real number field. For Maass cusp forms the method is burdened
with complicated integral transforms. We succeed in inverting the simplest of
these transforms whose kernel is built up of Gauss hypergeometric functions.Comment: Ph. D. thesis, Princeton University, 2003, vii+82 page
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