A Burgess-like subconvex bound for twisted L-functions

Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, X a primitive character of conductor q, and s a point on the critical line Rs = 1/2. It is proved that L(g circle times chi, s) 0 is arbitrary and theta = 7/64 is the current known approximation towards the RamannJan-Petersson conjecture (which would allow theta = 0); moreover, the dependence on s and all the parameters of g is polynomial. This result is an analog of Burgess' classical subconvex bound for Dirichlet L-functions. In Appendix 2 the above result is combined with a theorem of Waldspurger and the adelic calculations of Baruch-Mao to yield an improved uniform upper bound for the Fourier coefficients of holomorphic half-integral weight cusp forms

L-functions, automorphic forms, and arithmetic

We give a short, informal survey on the role of automorphic L-functions in number theory. We present the strongest currently known subconvexity bounds for twisted L-functions over number fields due to the authors and give various arithmetic applications. This is based on a talk of the first author

Bounds for Kloosterman sums on GL(n)

This paper establishes power-saving bounds for Kloosterman sums associated with the long Weyl element for GL(n), as well as for another type of Weyl element of order 2. These bounds are obtained by establishing an explicit representation as exponential sums. As an application we go beyond Sarnak's density conjecture for the principal congruence subgroup of prime level. We also bound all Kloosterman sums for GL(4).Comment: 20 page

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

We develop an explicit Kuznetsov formula on GL(3) for congruence subgroups. Applications include a Lindelöf 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. © 2017, Springer-Verlag GmbH Deutschland

Uniform Titchmarsh divisor problems

Asymptotic formulae for Titchmarsh-type divisor sums are obtained with strong error terms that are uniform in the shift parameter. This applies to more general arithmetic functions such as sums of two squares, improving the error term in the representation of the number as a sum of a prime and two squares, and to Fourier coefficients of cusp forms, generalizing a result of Pitt

Micro-CernVM: Slashing the Cost of Building and Deploying Virtual Machines

The traditional virtual machine building and and deployment process is centered around the virtual machine hard disk image. The packages comprising the VM operating system are carefully selected, hard disk images are built for a variety of different hypervisors, and images have to be distributed and decompressed in order to instantiate a virtual machine. Within the HEP community, the CernVM File System has been established in order to decouple the distribution from the experiment software from the building and distribution of the VM hard disk images. We show how to get rid of such pre-built hard disk images altogether. Due to the high requirements on POSIX compliance imposed by HEP application software, CernVM-FS can also be used to host and boot a Linux operating system. This allows the use of a tiny bootable CD image that comprises only a Linux kernel while the rest of the operating system is provided on demand by CernVM-FS. This approach speeds up the initial instantiation time and reduces virtual machine image sizes by an order of magnitude. Furthermore, security updates can be distributed instantaneously through CernVM-FS. By leveraging the fact that CernVM-FS is a versioning file system, a historic analysis environment can be easily re-spawned by selecting the corresponding CernVM-FS file system snapshot.Comment: Conference paper at the 2013 Computing in High Energy Physics (CHEP) Conference, Amsterda

A symplectic restriction problem

This is the final version. Available on open access from Springer via the DOI in this recordWe investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito–Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelöf hypothesis for the corresponding Koecher–Maaß series. The ingredients include a new relative trace formula for pairs of Heegner periods

Subconvexity for sup-norms of cusp forms on PGL(n)

Let F be an L2-normalized Hecke Maaß cusp form for Γ 0(N) ⊆ SL n(Z) with Laplace eigenvalue λF. If Ω is a compact subset of Γ 0(N) \ PGL n/ PO n, we show the bound ‖F|Ω‖∞≪ΩNελFn(n-1)/8-δ for some constant δ= δn> 0 depending only on n

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

Let $-d$ be a a negative discriminant and let $T$ vary over a set of representatives of the integral equivalence classes of integral binary quadratic forms of discriminant $-d$. We prove an asymptotic formula for $d \to \infty$ for the average over $T$ of the number of representations of $T$ by an integral positive definite quaternary quadratic form and obtain results on averages of Fourier coefficients of linear combinations of Siegel theta series. We also find an asymptotic bound from below on the number of binary forms of fixed discriminant $-d$ which are represented by a given quaternary form. In particular, we can show that for growing $d$ a positive proportion of the binary quadratic forms of discriminant $-d$ is represented by the given quaternary quadratic form.Comment: v5: Some typos correcte

The subconvexity problem for \GL_{2}

Generalizing and unifying prior results, we solve the subconvexity problem for the $L$-functions of \GL_{1} and \GL_{2} automorphic representations over a fixed number field, uniformly in all aspects. A novel feature of the present method is the softness of our arguments; this is largely due to a consistent use of canonically normalized period relations, such as those supplied by the work of Waldspurger and Ichino--Ikeda.Comment: Almost final version to appear in Publ. Math IHES. References updated