Let F be a totally real number field. Let p be a prime number and for any integer n let Fun denote the group of nth roots of unity. Let 41 be a p-adic valued Artin character for F and let F,, be the extension of F attached to 4, i.e., so that 4 is the character of a faithful representation of Gal(F,,/F). We will assume that F,, is also totally real. For a number field K let K., denote the cyclotomic Zp-extension of K. Following Greenberg we say that 4 is of type S if F., n Fc, = F and of type W if 4 is one-dimensional with F.,p c Fcc. Deligne and Ribet (in [DR], following Kubota and Leopoldt for the case F= Q) have proved the existence of a p-adic L-function associated to a one-dimensional Artin character 4 with F,, totally real. This function Lp(s, 4) is continuous for s e Zp{ 1}, and even at s = 1 if 4, is not trivial, and satisfies the following interpolation property:

The Iwasawa conjecture for totally real fields

i=1 (cf. [8]). In this paper we will prove that their conjecture is true. Moreover, we obtain a simple Fourier coefficient formula for F(Z). Ibukiyama [23] also made a similar conjecture which was formulated in terms of the Koecher-Maass series which we will not consider here. Let us describe our results. Throughout this paper, we fix positive integers k and n such that k = n mod 2, and put e = (-1)k = (-1)n. For any positive rational number N, we denote the absolute value of the discriminant of Q( VN)/Q by DN. The primitive Dirichlet character corresponding to Q(V/N)/Q is denoted by XN. Let fN be the positive rational number such that N = Nf2NLet B E M2n(Q) be a positive definite half-integral symmetric matrix. Put DB = det(2B), DB = D(DB), fB = f(DB), and XB = X(DB). Note that EDB 0, 1 mod 4 and that fB is an integer. For each prime p, let p : Qp -- CX be the unique additive character such that bp(x) = exp(-27ir-lTx) for x E Z[p-1]. Recall that the Siegel series for B is defined by

On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n

One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on ‘elliptic Artin’ are due to Alina Cojocaru. Wojciec Gajda wrote a section on ‘Artin for K-theory of number fields’, and Hester Graves (together with me) on ‘Artin’s conjecture and Euclidean

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Artin's primitive root conjecture -a survey -

On the local factors of representations of classical groups

Combining induction formulas for local densities with a functional equation for the Siegel series, we give an explicit formula for the Siegel series. By this formula, we also give an explicit formula for the Fourier coefficients of the Siegel Eisenstein series.

An explicit formula for Siegel series

We are concerned with Dirichlet series which appear in the Fourier expansion of the non-analytic Eisenstein series on the Siegel upper half space Hm of degree m. In the case of m = 2 Kaufhold [1] evaluated them. Here we treat the general cases by a different method.

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Dirichlet series in the theory of Siegel modular forms

In this paper, we report our development of an analytical index on stabilometry to diagnose a decrease in the cerebellar equilibrium function controlling vestibulo-spinal reflex. We measured the sway of the center-of-gravity of the body with a stabilometer for seven healthy men aged 19 to 26 years, and also recorded statokinesigrams after alcoholic intake. We propose classification of these statokinesigrams under two groups, statokinesigrams before the intake and after it. Suzuki et al. (1996) standardized indices, area of sway, total locus length and other parameters, on stabilometry, however, it was difficult to classify statokinesigrams with those indices. Here, we introduce a new index, sparse density, into stabilometry. We show that sparse density is a useful and stable index for classification in accordance with non-parametric statistics. Although we expected a similar effect on the circulatory system by the alcoholic load, variation of these indices was more complex than the variation derived from the organs for the recovery from the standpoint of time. We also analyzed data by paying attention to the force acting on them, which generated the statokinesigrams. We especially turned our attention to singular points where statistically tiny or huge forces are exerted. We could pursue the sparse density with sum of local force that acted on the center-of-gravity. We concluded that sparse density depends upon the function for the control of the center-of-gravity of the body.

Physical Meaning on Geometrical Index for Stabilometry

We measured the sway of the center-of-gravity of the body with a stabirometer for five men aged of 19 to 25 years. There are standardized parameters on stabirometry (observation with the stabirometer) by SUZUKI et al. (1996). We have analyzed the measured data by paying attention to force acting on it. We could admit polarity and rhythm on the force and the kind of effect of visual information. We also constructed a mathematical model of the sway of the center-of-gravity of the body on some conditions. We obtained a stochastic dynamical equation system as a mathematical model corresponding to the sway of the center-of-gravity of the body. We compared the statokinesigram obtained from the numerical simulation of this SDE with that obtained from the measured data.

http://www.scipress.org/journals/forma/pdf/1601/16010017.pdf

Mathematical Index and Model in Stabirometry

In this note we study quaternary even positive definite quadratic forms of prime discriminant. In § 1 we classify quaternary even positive definite quadratic forms of prime discriminant p = 1 mod 4 (called simply nice quaternary lattices in this note) which represent two. We note that the class number of such forms is closely related to the dimension of the space of certain automorphic forms. (Remark 4 in the text). By using the classification in § 1 and the theory of integral representations of cyclic groups we show that the orthogonal group of a nice quaternary lattice is generated by ±1 and symmetries (of the lattice). In § 3, we calculate the class number of nice quaternary lattices. Notations and terminologies will generally be those of O'Meara [5]. Any exceptions to this convention will be stated explicitly. Through this note Q(x) and B(x, y) denote quadratic forms and corresponding bilinear forms (i.e., 2B(x,y) = Q(x + y) — Q(x) — Q(y)), and p denotes a fixed prime number Ξlmod4. § 1. We say that a quadratic lattice N over the ring of rational integers Z is even if and only if Q(x) = 0 mod 2 for any element in N. For brevity, a quadratic lattice N is called nice in this note if and only if N is an even positive definite quadratic lattice over Z, its discriminant d(N) is p, 2p or 4p according as N is quarternary, ternary or binary respectively, and moreover the Hasse invariant S2(N) at the prime two of N is — (— i when N is binary.

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Quaternary even positive definite quadratic forms of prime discriminant

Local densities of quadratic forms are important invariants in the theory of quadratic forms and they appear in Fourier coefficients of Eisenstein series. But it is not easy to evaluate them. To study their properties, it is desirable to look for relations among them, and it is known that there are many relations [3], but they are not concise. We consider a different kind of relations here and improve a result of Zharkovskaja [8, 9] in the case of Eisenstein series as an application.

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Yoshiyuki Kitaoka

Papers

Dirichlet series in the theory of Siegel modular forms

Physical Meaning on Geometrical Index for Stabilometry

Mathematical Index and Model in Stabirometry

Quaternary even positive definite quadratic forms of prime discriminant

Local densities of quadratic forms and Fourier coefficients of Eisenstein series