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

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

New bounds on even cycle creating Hamiltonian paths using expander graphs

We say that two graphs on the same vertex set are $G$-creating if their union (the union of their edges) contains $G$ as a subgraph. Let $H_n(G)$ be the maximum number of pairwise $G$-creating Hamiltonian paths of $K_n$. Cohen, Fachini and K\"orner proved $$n^{\frac{1}{2}n-o(n)}\leq H_n(C_4) \leq n^{\frac{3}{4}n+o(n)}.$$ In this paper we close the superexponential gap between their lower and upper bounds by proving $$n^{\frac{1}{2}n-\frac{1}{2}\frac{n}{\log{n}}-O(1)}\leq H_n(C_4) \leq n^{\frac{1}{2}n+o\left(\frac{n}{\log{n}} \right)}.$$ We also improve the previously established upper bounds on $H_n(C_{2k})$ for $k>3$, 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

New bounds on even cycle creating Hamiltonian paths using expander graphs

We say that two graphs on the same vertex set are $G$-creating if their union (the union of their edges) contains $G$ as a subgraph. Let $H_n(G)$ be the maximum number of pairwise $G$-creating Hamiltonian paths of $K_n$. Cohen, Fachini and K\"orner proved $$n^{\frac{1}{2}n-o(n)}\leq H_n(C_4) \leq n^{\frac{3}{4}n+o(n)}.$$ In this paper we close the superexponential gap between their lower and upper bounds by proving $$n^{\frac{1}{2}n-\frac{1}{2}\frac{n}{\log{n}}-O(1)}\leq H_n(C_4) \leq n^{\frac{1}{2}n+o\left(\frac{n}{\log{n}} \right)}.$$ We also improve the previously established upper bounds on $H_n(C_{2k})$ for $k>3$, 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

On the sup-norm of Maass cusp forms of large level. III

Let $f$ be a Hecke--Maass cuspidal newform of square-free level $N$ and Laplacian eigenvalue $\lambda$. It is shown that \pnorm{f}_\infty \ll_{\lambda,\epsilon} N^{-1/6}+\epsilon} \pnorm{f}_2 for any $\epsilon>0$

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 formák és L-függvények = Automorphic forms and L-functions

Valentin Blomerrel Burgess-típusú szubkonvex becslést igazoltunk csavart Hilbert moduláris L-függvényekre, megjavítva Cogdell-PiatetskiShapiro-Sarnak és Venkatesh idevágó eredményeit. Közvetlen alkalmazásként az eddigieknél hatékonyabban tudjuk becsülni pozitív definit ternér kvadratikus formák előállításszámait egy teljesen valós számtest egészei felett. Valentin Blomerrel aszimptotikus formulát adtunk Rankin-Selberg L-függvények bizonyos archimédeszi családjaira. Az eredmény érdekessége, hogy amikor a Rankin-Selberg konvolúcióban a rögzített formát Eisenstein-sornak választjuk, az aszimptotikában a szokásos logaritmikus tagok mellett két forgó tag is megjelenik. Egy speciális esetben a holomorf csúcsformákhoz társított L-függvények negyedik momentumáról szól az eredmény. Nicolas Templier-val új becslést adtunk Hecke-Maass csúcsformák szuprémumára a szint aspektusban. Az eredmény analóg a Riemann zeta-függvényre vonatkozó szubkonvex Weyl-korláttal. A közelmúltban - más módszerrel - hasonló erejű tételt igazolt Blomer-Michel kompakt aritmetikus felületekre. Mi a kompaktság hiányát az Atkin-Lehner operátorok egy újszerű alkalmazásával kezeljük hatékonyan. | 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