In this paper we generalize work of Amice and Lazard from the early sixties. Amice determined the dual of the space of locally Qp-analytic functions on Zp and showed that it is isomorphic to the ring of rigid functions on the open unit disk over Cp. Lazard showed that this ring has a divisor theory and that the classes of closed, flnitely generated, and principal ideals in this ring coincide. We study the space of locally L-analytic functions on the ring of integers in L, where L is a flnite extension of Qp. We show that the dual of this space is a ring isomorphic to the ring of rigid functions on a certain rigid variety X. We show that the variety X is isomorphic to the open unit disk over Cp, but not over any discretely valued extension fleld of L; it is a twisted form of the open unit disk. In the ring of functions on X, the classes of closed, flnitely generated, and invertible ideals coincide, but unless L=Qp not all flnitely generated ideals are principal. The paper uses Lubin-Tate theory and results on p-adic Hodge theory. We give several applications, including one to the construction of p- adic L-functions for supersingular elliptic curves.
Abstract Although there is strong evidence for the positive effects of musical training on auditory perception, processing, and training‐induced neuroplasticity, there is still little knowledge on the auditory and neurophysiological short‐term plasticity through listening training. In a sample of 37 adolescents (20 musicians and 17 nonmusicians) that was compared to a control group matched for age, gender, and musical experience, we conducted a 2‐week active listening training (AULOS: Active IndividUalized Listening OptimizationS). Using magnetoencephalography and psychoacoustic tests, the short‐term plasticity of auditory evoked fields and auditory skills were examined in a pre‐post design, adapted to the individual neuro‐auditory profiles. We found bilateral, but more pronounced plastic changes in the right auditory cortex. Moreover, we observed synchronization of the auditory evoked P1, N1, and P2 responses and threefold larger amplitudes of the late P2 response, similar to the reported effects of musical long‐term training. Auditory skills and thresholds benefited largely from the AULOS training. Remarkably, after training, the mean thresholds improved by 12 dB for bone conduction and by 3–4 dB for air conduction. Thus, our findings indicate a strong positive influence of active listening training on neural auditory processing and perception in adolescence, when the auditory system is still developing.
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