Two-cover descent on plane quartics with rational bitangents

We implement two-cover descent for plane quartics over Q with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the relevant moduli space and use it to generate examples. We observe that local obstructions are quite rare for such curves, and only seem to occur in practice at primes of good reduction. In particular, having good reduction at 11 implies having no rational points. We also gather numerical data on two-Selmer ranks of Jacobians of these curves, which suggests that these often have non-trivial Tate-Shafarevich groups. We implement two-cover descent for plane quartics over Q with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the relevant moduli space and use it to generate examples. We observe that local obstructions are quite rare for such curves and only seem to occur in practice at primes of good reduction. In particular, having good reduction at 11 implies having no rational points. We also gather numerical data on two-Selmer ranks of Jacobians of these curves, providing evidence these behave differently from those of general abelian varieties due to the frequent presence of an everywhere locally trivial torsor.Comment: 15 pages; Some minor improvements to algorithm and rank data analysi

Direct evidence for two-band superconductivity in MgB_2 single crystals from directional point-contact spectroscopy in magnetic fields

We present the results of the first directional point-contact spectroscopy experiments in high-quality MgB_2 single crystals. Due to the directionality of the current injection into the samples, the application of a magnetic field allowed us to separate the contributions of the sigma and pi bands to the total conductance of our point contacts. By using this technique, we were able to obtain the temperature dependency of each gap independent of the other. The consequent, strong reduction of the error on the value of the gap amplitude as function of temperature allows a stricter test of the predictions of the two-band model for MgB_2.Comment: 4 pages, 5 eps figures. References added, Fig.1 changed, some text adde

Independent determination of the two gaps by directional point-contact spectroscopy in MgB_2 single crystals

Directional point-contact spectroscopy measurements were performed for the first time in state-of-the-art MgB_2 single crystals. The selective suppression of the superconductivity in the "pi" band by means of a suitable magnetic field allowed separating the partial contribution of each band to the total point-contact conductance. By fitting the partial conductance curves sigma_sigma(V) and sigma_pi(V), we got an independent determination of the two gaps, Delta_sigma and Delta_pi, with a strong reduction of the experimental uncertainty. Their temperature dependence was found to agree well with the predictions of the two-band models for MgB_2.Comment: 6 pages, 4 eps figures. References added, abstract rewritten, text slightly changed. Proceedings of the BOROMAG Conference, June 17-19, Genoa, Ital