Ray class invariants over imaginary quadratic fields

Let $K$ be an imaginary quadratic field of discriminant less than or equal to -7 and $K_{(N)}$ be its ray class field modulo $N$ for an integer $N$ greater than 1. We prove that singular values of certain Siegel functions generate $K_{(N)}$ over $K$ by extending the idea of our previous work. These generators are not only the simplest ones conjectured by Schertz, but also quite useful in the matter of computation of class polynomials. We indeed give an algorithm to find all conjugates of such generators by virtue of Gee and Stevenhagen

Different Modulation of the Cortical Silent Period by Two Phases of Short Interval Intracortical Inhibition

PURPOSE: To investigate the influence of 2 phases of short interval intracortical inhibition (SICI) on the cortical silent period (SP). MATERIALS AND METHODS: Single- and paired-pulse transcranial magnetic stimulations (TMSs) at 1 and 2.5ms interstimulus intervals (ISIs) were applied to the left motor cortex in 12 healthy subjects while their right hand muscles were moderately activated. Conditioning stimulation intensity was 90% of the active motor threshold (AMT). Test stimulation intensities were 120, 140, 160, 180, 200, 220, 240, 260% of the AMT and at 100% of the maximal stimulator output, the order of which was arranged randomly. The rectified electromyography area of motor evoked potential (MEP) and duration of the SP were measured off-line using a computerized program. RESULTS: At high-test stimulation intensities, MEP areas were saturated in both single- and paired-pulse stimulations, except that saturated MEPs were smaller for the paired-pulse TMS at 1ms ISI than for the other conditions. As the test stimulation intensity increased, SP was progressively prolonged in both single- and paired-pulse stimulations but was shorter in paired-pulse than single-pulse TMS. Overall, the ratio of SP duration/MEP area was comparable between single- and paired-pulse TMS except for the paired-pulse TMS at 1 ms ISI with a test stimulation intensity at 140-180% of the AMT, in which the ratio was significantly higher than in the single pulse TMS. CONCLUSION: These results suggest that 2 phases of SICI modulate MEP saturation and SP duration differently and provide additional evidence supporting the view that 2 phases of SICI are mediated by different inhibitory mechanisms.ope

Arithmetic properties of orders in imaginary quadratic fields

Let $K$ be an imaginary quadratic field. For an order $\mathcal{O}$ in $K$ and a positive integer $N$, let $K_{\mathcal{O},\,N}$ be the ray class field of $\mathcal{O}$ modulo $N\mathcal{O}$. We deal with various subjects related to $K_{\mathcal{O},\,N}$, mainly about Galois representations attached to elliptic curves with complex multiplication, form class groups and $L$-functions for orders