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Valentin Blomer

Publications and source records attributed to Valentin Blomer.

At least 55 records · Page 3Linked to original sources
arXiv2018

Triple correlation and long gaps in the spectrum of flat tori

We evaluate the triple correlation of eigenvalues of the Laplacian on generic flat tori in an averaged sense. As two consequence we show that (a) the limit inferior (resp. limit superior) of the triple correlation is almost surely at most (resp. at least) Poissonian, and that (b) almost all flat tori contain infinitely many gaps in their spectrum that are at least 2.006 times longer than the average gap. The significance of the constant 2.006 lies in the fact that there exist sequences with Poissonian pair correlation and with consecutive spacings bounded uniformly from above by 2, as we also prove in this paper. Thus our result goes beyond what can be deduced solely from the Poissonian behavior of the pair correlation.

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arXiv2018

Uniform subconvexity and symmetry breaking reciprocity

A non-symmetric reciprocity formula is established that expresses the fourth moment of automorphic L-functions of level q and primitive central character twisted by the l-th Hecke eigenvalue as a twisted mixed moment of automorphic L-functions of level l and trivial central character. As an application, uniform subconvexity bounds for L-functions in the level and the eigenvalue aspect are derived.

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arXiv2018

On the global sup-norm of GL(3) cusp forms

Let $ϕ$ be a spherical Hecke-Maass cusp form on the non-compact space $\mathrm{PGL}_3(\mathbb{Z})\backslash\mathrm{PGL}_3(\mathbb{R})$. We establish various pointwise upper bounds for $ϕ$ in terms of its Laplace eigenvalue $λ_ϕ$. These imply, for $ϕ$ arithmetically normalized and tempered at the archimedean place, the bound $\|ϕ\|_\infty\ll_ελ_ϕ^{39/40+ε}$ for the global sup-norm (without restriction to a compact subset). On the way, we derive a new uniform upper bound for the $\mathrm{GL}_3$ Jacquet-Whittaker function.

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arXiv2018

Subconvexity for L-functions of non-spherical cusp forms on GL(3)

Let f be a cusp form for SL(3, Z) associated with a generalized principal series representation of minimal weight d, spectral parameter r and associated L-function L(s, f). For $r \asymp d \asymp T$ the subconvexity bound $L(1/2, f) \ll T^{3/4 - 1/140000}$ is proved.

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arXiv2017

On the rank of universal quadratic forms over real quadratic fields

We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\mathbb Q(\sqrt D)$ and obtain lower and upper bounds for it in terms of certain sums of coefficients of the associated continued fraction. We also estimate such sums in terms of $D$ and establish a link between continued fraction expansions and special values of $L$-functions in the spirit of Kronecker's limit formula.

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arXiv2017

Applications of the Kuznetsov formula on GL(3): the level aspect

We develop an explicit Kuznetsov formula on GL(3) for congruence subgroups. Applications include a Lindelof on average type bound for the sixth moment of GL(3) L-functions in the level aspect, an automorphic large sieve inequality, density results for exceptional eigenvalues and density results for Maass forms violating the Ramanujan conjecture at finite places.

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arXiv2016

Small gaps in the spectrum of the rectangular billiard

We study the size of the minimal gap between the first N eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $α$, in comparison to the corresponding quantity for a Poissonian sequence. If $α$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size 1/N, which is essentially consistent with Poisson statistics. We also give related results for a set of $α$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $α$. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory.

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arXiv2016

Higher order divisor problems

An asymptotic formula is proved for the k-fold divisor function averaged over homogeneous polynomials of degree k in k-1 variables coming from incomplete norm forms.

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arXiv2016

Some applications of smooth bilinear forms with Kloosterman sums

We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet $L$-functions.

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arXiv2016

The sup-norm problem for the Siegel modular space of rank two

Let F be a square integrable Maass form on the Siegel upper half space of rank 2 for the Siegel modular group Sp(4, Z) with Laplace eigenvalue lambda. If, in addition, F is a joint eigenfunction of the Hecke algebra, we show a power-saving sup-norm bound in terms of lambda (relative to the generic bound) for F restricted to a compact set. As an auxiliary result of independent interest we prove new uniform bounds for spherical functions on semisimple Lie groups.

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arXiv2016

Epstein zeta-functions, subconvexity, and the purity conjecture

Subconvexity bounds are proved for general Epstein zeta functions of k-ary quadratic forms. This is related to sup-norm bounds for Eisenstein series on GL(k), and the exact sup-norm exponent is determined to be (k-2)/8 for k >= 2. In particular, if $k$ is odd, this exponent is not in Z/4, which shows that Sarnak's purity conjecture does not hold for Eisenstein series.

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arXiv2016

On moments of twisted $L$-functions

We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of $q$, and we use the theory of Deligne and Katz to estimate certain complete exponential sums in several variables and prove new bounds on bilinear forms in Kloosterman sums with power savings when both variables are near the square root of $q$. When at least one of the forms $f$ and $g$ is non-cuspidal, we obtain an asymptotic formula for the mixed second moment of twisted $L$-functions with a power saving error term. In particular, when both are non-cuspidal, this gives a significant improvement on M.~Young's asymptotic evaluation of the fourth moment of Dirichlet $L$-functions. In the general case, the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in Kloosterman sums.

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