Interpolated sequences and critical LL-values of modular forms

Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for $\zeta(3)$ in terms of a critical $L$-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier's six sporadic sequences are essentially critical $L$-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown's cellular integrals and critical $L$-values of modular forms of odd weight.Comment: 23 pages, to appear in Proceedings for the KMPB conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theor

On certain finiteness questions in the arithmetic of modular forms

We investigate certain finiteness questions that arise naturally when studying approximations modulo prime powers of p-adic Galois representations coming from modular forms. We link these finiteness statements with a question by K. Buzzard concerning p-adic coefficient fields of Hecke eigenforms. Specifically, we conjecture that for fixed N, m, and prime p with p not dividing N, there is only a finite number of reductions modulo p^m of normalized eigenforms on \Gamma_1(N). We consider various variants of our basic finiteness conjecture, prove a weak version of it, and give some numerical evidence.Comment: 25 pages; v2: one of the conjectures from v1 now proved; v3: restructered parts of the article; v4: minor corrections and change

Interacting quantum rotors in oxygen-doped germanium

We investigate the interaction effect between oxygen impurities in crystalline germanium on the basis of a quantum rotor model. The dipolar interaction of nearby oxygen impurities engenders non-trivial low-lying excitations, giving rise to anomalous behaviors for oxygen-doped germanium (Ge:O) below a few degrees Kelvin. In particular, it is theoretically predicted that Ge:O samples with oxygen-concentration of 10$^{17-18}$cm$^{-3}$ show (i) power-law specific heats below 0.1 K, and (ii) a peculiar hump in dielectric susceptibilities around 1 K. We present an interpretation for the power-law specific heats, which is based on the picture of local double-well potentials randomly distributed in Ge:O samples.Comment: 13 pages, 11 figures; to be published in Phys. Rev.

Magnetic and structural quantum phase transitions in CeCu6-xAux are independent

The heavy-fermion compound CeCu$_{6-x}$Au$_x$ has become a model system for unconventional magnetic quantum criticality. For small Au concentrations $0 \leq x < 0.16$, the compound undergoes a structural transition from orthorhombic to monoclinic crystal symmetry at a temperature $T_{s}$ with $T_{s} \rightarrow 0$ for $x \approx 0.15$. Antiferromagnetic order sets in close to $x \approx 0.1$. To shed light on the interplay between quantum critical magnetic and structural fluctuations we performed neutron-scattering and thermodynamic measurements on samples with $0 \leq x\leq 0.3$. The resulting phase diagram shows that the antiferromagnetic and monoclinic phase coexist in a tiny Au concentration range between $x\approx 0.1$ and $0.15$. The application of hydrostatic and chemical pressure allows to clearly separate the transitions from each other and to explore a possible effect of the structural transition on the magnetic quantum critical behavior. Our measurements demonstrate that at low temperatures the unconventional quantum criticality exclusively arises from magnetic fluctuations and is not affected by the monoclinic distortion.Comment: 5 pages, 3 figure

Arithmetic Spacetime Geometry from String Theory

An arithmetic framework to string compactification is described. The approach is exemplified by formulating a strategy that allows to construct geometric compactifications from exactly solvable theories at $c=3$. It is shown that the conformal field theoretic characters can be derived from the geometry of spacetime, and that the geometry is uniquely determined by the two-dimensional field theory on the world sheet. The modular forms that appear in these constructions admit complex multiplication, and allow an interpretation as generalized McKay-Thompson series associated to the Mathieu and Conway groups. This leads to a string motivated notion of arithmetic moonshine.Comment: 36 page

On higher congruences between cusp forms and Eisenstein series

In this paper we present several finite families of congruences between cusp forms and Eisenstein series of higher weights at powers of prime ideals. We formulate a conjecture which describes properties of the prime ideals and their relation to the weights. We check the validity of the conjecture on several numerical examples.Comment: 20 page

GeSn/Ge heterostructure short-wave infrared photodetectors on silicon

status: publishe

Thermal disc emission from a rotating black hole: X-ray polarization signatures

Thermal emission from the accretion disc around a black hole can be polarized, due to Thomson scattering in a disc atmosphere. In Newtonian space, the polarization angle must be either parallel or perpendicular to the projection of the disc axis on the sky. As first pointed out by Stark and Connors in 1977, General Relativity effects strongly modify the polarization properties of the thermal radiation as observed at infinity. Among these effects, the rotation of the polarization angle with energy is particularly useful as a diagnostic tool. In this paper, we extend the Stark and Connors calculations by including the spectral hardening factor, several values of the optical depth of the scattering atmosphere and rendering the results to the expected performances of planned X-ray polarimeters. In particular, to assess the perspectives for the next generation of X-ray polarimeters, we consider the expected sensitivity of the detectors onboard the planned POLARIX and IXO missions. We assume the two cases of a Schwarzschild and an extreme Kerr black hole with a standard thin disc and a scattering atmosphere. We compute the expected polarization degree and the angle as functions of the energy as they could be measured for different inclinations of the observer, optical thickness of the atmosphere and different values of the black hole spin. We assume the thermal emission dominates the X-ray band. Using the flux level of the microquasar GRS 1915+105 in the thermal state, we calculate the observed polarization.Comment: 8 pages, 7 figures, accepted by MNRA

On the multiplicativity of quantum cat maps

The quantum mechanical propagators of the linear automorphisms of the two-torus (cat maps) determine a projective unitary representation of the theta group, known as Weil's representation. We prove that there exists an appropriate choice of phases in the propagators that defines a proper representation of the theta group. We also give explicit formulae for the propagators in this representation.Comment: Revised version: proof of the main theorem simplified. 21 page

Shimura varieties in the Torelli locus via Galois coverings of elliptic curves

We study Shimura subvarieties of $\mathsf{A}_g$ obtained from families of Galois coverings $f: C \rightarrow C'$ where $C'$ is a smooth complex projective curve of genus $g' \geq 1$ and $g= g(C)$. We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of $\mathsf{A}_g$ for $g' =1,2$ and for all $g \geq 2,4$ and for $g' > 2$ and $g \leq 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the case $g'=0$. Here we find 6 families of Galois coverings, all with $g' = 1$ and $g=2,3,4$ and we show that these are the only families with $g'=1$ satisfying this sufficient condition. We show that among these examples two families yield new Shimura subvarieties of $\mathsf{A}_g$, while the other examples arise from certain Shimura subvarieties of $\mathsf{A}_g$ already obtained as families of Galois coverings of $\mathbb{P}^1$ in [FGP]. Finally we prove that if a family satisfies this sufficient condition with $g'\geq 1$, then $g \leq 6g'+1$.Comment: 18 pages, to appear in Geometriae Dedicat