Petersson inner products of weight-one modular forms

n this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that these Petersson products posses remarkable arithmetic properties. Namely, such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. A similar result was obtained independently using a different method by W. Duke and Y. Li [5]. The main result of this paper is an explicit factorization formula for the algebraic number obtained by exponentiating a Petersson product

The sphere packing problem in dimension 24

Building on Viazovska's recent solution of the sphere packing problem in eight dimensions, we prove that the Leech lattice is the densest packing of congruent spheres in twenty-four dimensions and that it is the unique optimal periodic packing. In particular, we find an optimal auxiliary function for the linear programming bounds, which is an analogue of Viazovska's function for the eight-dimensional case.Comment: 17 page

Universal optimality of the E8E_8 and Leech lattices and interpolation formulas

We prove that the $E_8$ root lattice and the Leech lattice are universallyoptimal among point configurations in Euclidean spaces of dimensions $8$ and$24$, respectively. In other words, they minimize energy for every potentialfunction that is a completely monotonic function of squared distance (forexample, inverse power laws or Gaussians), which is a strong form of robustnessnot previously known for any configuration in more than one dimension. Thistheorem implies their recently shown optimality as sphere packings, and broadlygeneralizes it to allow for long-range interactions. The proof uses sharp linear programming bounds for energy. To construct theoptimal auxiliary functions used to attain these bounds, we prove a newinterpolation theorem, which is of independent interest. It reconstructs aradial Schwartz function $f$ from the values and radial derivatives of $f$ andits Fourier transform $\widehat{f}$ at the radii $\sqrt{2n}$ for integers$n\ge1$ in $\mathbb{R}^8$ and $n \ge 2$ in $\mathbb{R}^{24}$. To prove thistheorem, we construct an interpolation basis using integral transforms ofquasimodular forms, generalizing Viazovska's work on sphere packing and placingit in the context of a more conceptual theory.<br

Symmetry and disorder of the vitreous vortex lattice in an overdoped BaFe_{2-x}Co_xAs_2 superconductor: Indication for strong single-vortex pinning

The disordered flux line lattice in single crystals of the slightly overdoped aFe_{2-x}Co_xAs_2 (x = 0.19, Tc = 23 K) superconductor is studied by magnetization measurements, small-angle neutron scattering (SANS), and magnetic force microscopy (MFM). In the whole range of magnetic fields up to 9 T, vortex pinning precludes the formation of an ordered Abrikosov lattice. Instead, a vitreous vortex phase (vortex glass) with a short-range hexagonal order is observed. Statistical processing of MFM datasets lets us directly measure its radial and angular distribution functions and extract the radial correlation length \zeta. In contrast to predictions of the collective pinning model, no increase in the correlated volume with the applied field is observed. Instead, we find that \zeta decreases as 1.3*R1 ~ H^(-1/2) over four decades of the applied magnetic field, where R1 is the radius of the first coordination shell of the vortex lattice. Such universal scaling of \zeta implies that the vortex pinning in iron arsenides remains strong even in the absence of static magnetism. This result is consistent with all the real- and reciprocal-space vortex-lattice measurements in overdoped as-grown aFe_{2-x}Co_xAs_2 published to date and is thus sample-independent. The failure of the collective pinning model suggests that the vortices remain in the single-vortex pinning limit even in high magnetic fields up to 9 T.Comment: 11 pages, 6 figure

Symmetry and disorder of the vitreous vortex lattice in an overdoped BaFe_{2-x}Co_xAs_2 superconductor: Indication for strong single-vortex pinning

The disordered flux line lattice in single crystals of the slightly overdoped aFe_{2-x}Co_xAs_2 (x = 0.19, Tc = 23 K) superconductor is studied by magnetization measurements, small-angle neutron scattering (SANS), and magnetic force microscopy (MFM). In the whole range of magnetic fields up to 9 T, vortex pinning precludes the formation of an ordered Abrikosov lattice. Instead, a vitreous vortex phase (vortex glass) with a short-range hexagonal order is observed. Statistical processing of MFM datasets lets us directly measure its radial and angular distribution functions and extract the radial correlation length \zeta. In contrast to predictions of the collective pinning model, no increase in the correlated volume with the applied field is observed. Instead, we find that \zeta decreases as 1.3*R1 ~ H^(-1/2) over four decades of the applied magnetic field, where R1 is the radius of the first coordination shell of the vortex lattice. Such universal scaling of \zeta implies that the vortex pinning in iron arsenides remains strong even in the absence of static magnetism. This result is consistent with all the real- and reciprocal-space vortex-lattice measurements in overdoped as-grown aFe_{2-x}Co_xAs_2 published to date and is thus sample-independent. The failure of the collective pinning model suggests that the vortices remain in the single-vortex pinning limit even in high magnetic fields up to 9 T.Comment: 11 pages, 6 figure

Momentum dependence of the superconducting gap in Ba1−x_{1-x}Kx_{x}Fe2_2As2_2

The precise momentum dependence of the superconducting gap in the iron-arsenide superconductor with Tc = 32K (BKFA) was determined from angle-resolved photoemission spectroscopy (ARPES) via fitting the distribution of the quasiparticle density to a model. The model incorporates finite lifetime and experimental resolution effects, as well as accounts for peculiarities of BKFA electronic structure. We have found that the value of the superconducting gap is practically the same for the inner Gamma-barrel, X-pocket, and "blade"-pocket, and equals 9 meV, while the gap on the outer Gamma-barrel is estimated to be less than 4 meV, resulting in 2Delta/kT_c=6.8 for the large gap, and 2Delta/kT_c<3 for the small gap. A large (77 \pm 3%) non-superconducting component in the photoemission signal is observed below T_c. Details of gap extraction from ARPES data are discussed in Appendix.Comment: Images revised; details of gap extraction from ARPES spectra are added as an Appendi

Momentum-resolved superconducting gap in the bulk of Ba1−x_{1-x}Kx_{x}Fe2_2As2_2 from combined ARPES and μ\muSR measurements

Here we present a calculation of the temperature-dependent London penetration depth, $\lambda(T)$, in Ba$_{1-x}$K$_{x}$Fe$_2$As$_2$ (BKFA) on the basis of the electronic band structure [1,2] and momentum-dependent superconducting gap [3] extracted from angle-resolved photoemission spectroscopy (ARPES) data. The results are compared to the direct measurements of $\lambda(T)$ by muon spin rotation ($\mu$SR) [4]. The value of $\lambda(T=0)$, calculated with \emph{no} adjustable parameters, equals 270 nm, while the directly measured one is 320 nm; the temperature dependence $\lambda(T)$ is also easily reproduced. Such agreement between the two completely different approaches allows us to conclude that ARPES studies of BKFA are bulk-representative. Our review of the available experimental studies of the superconducting gap in the new iron-based superconductors in general allows us to state that all hole-doped of them bear two nearly isotropic gaps with coupling constants $2\Delta/k_{\rm B}T_{\rm c}=2.5\pm1.5$ and $7\pm2$