Let X be an arithmetic scheme (i.e., separated, of finite type over Spec ℤ) of Krull dimension one. For the associated zeta-function ζ(X, s), we write down a formula for the special value at s = n < 0 in terms of the étale motivic cohomology of X and a regulator. We prove it in the case when, for each generic point 𝜂 ∈ X with char 𝜅(𝜂) = 0, the extension 𝜅(𝜂)/ℚ is abelian. We conjecture that the formula holds for any one-dimensional arithmetic scheme. This is a consequence of the Weil-étale formalism developed by the author (arXiv preprints 2012.11034 and 2102.12114), following the work of Flach and Morin (Doc. Math. 23 (2018), 1425–1560). We also calculate the Weil-étale cohomology of one-dimensional arithmetic schemes and show that our special value formula is a particular case of the main conjecture from arXiv preprint 2102.12114.

For a two-dimensional normal surface singularity and its resolution, there are two important cycles called the maximal ideal cycle and the fundamental cycle. We are interested in whether these two cycles coincide on the minimal resolution. In this paper, for the normal surface singularity which appears triple covering branched over the analytically irreducible singular plane curves, we show that the two cycles coincide on the minimal resolution.

We study one of the multidimensional inverse scattering problems for quantum systems in time-dependent electric fields E (t ), whose leading part is represented as E₀ (1+|t |)^−𝜇, with 0 ≤ 𝜇 < 1, by utilization of the Enss–Weder time-dependent method. In our previous work, we have dealt with the case where E (t ) is a constant electric field. The present work is a continuation of that. The main purpose of this paper is to give an appropriate class of short-range potentials determined by the Enss–Weder time-dependent method, which is considered almost optimal also in terms of direct scattering problems.

In this paper, a certain two-parameter family of plane embeddings of the Edwards elliptic curve E_a ∶ x ² + y ² = a ² (1 + x ²y ²) is introduced to provide explicitly computed tropical curves corresponding to degeneration as a → 1. Applying the theta uniformization of E_a with the method of ultradiscretization given by Kajiwara et al. (Kyushu J. Math. 63 (2009), 315–338), we give a formula for the coordinate functions that traces the cycle part of the tropical elliptic curve. We also illustrate how one can recover the whole part of the tropical curve as a quotient of the Bruhat–Tits tree after Speyer’s algebraic approach in smooth cases.

Two-way relationships between transformations and quadratic forms on Wiener spaces are investigated with the help of change-of-variables formulas on Wiener spaces. Further, the evaluation of Laplace transforms of quadratic forms via Riccati or linear second-order ordinary differential equations will be shown.

A positive integer n is called a 𝜃-congruent number if there is a triangle with sides a, b and c for which the angle between a and b is equal to 𝜃 and its area is n√r ² − s ², where 0 < 𝜃 < 𝜋, cos 𝜃 = s/r and 0 ≤ |s| < r are relatively prime integers. The case 𝜃 = 𝜋/2 refers to the classical congruent numbers. It is known that the problem of classifying 𝜃-congruent numbers is related to the existence of rational points on the elliptic curve y ² = x (x + (r + s) n )( x − (r − s )n ). In this paper, we deal with a variant of the congruent number problem where the cosine of a fixed angle is ±√2/2.

Let Δ > 0 be a quadratic discriminant and Cl_∆ be the ideal class group of the quadratic order 𝒪_∆. In this paper, we introduce two subgroups Cl_∆⁽¹⁾ and Cl_∆⁽²⁾ of Cl_∆ and, by using them, give a necessary and sufficient condition for h_∆ = 1 to hold. This corresponds to a generalization of a theorem of Louboutin. Moreover, we determine an explicit form of generators of Cl_∆⁽²⁾.

For the fifth Painlevé equation it is known that a general solution is represented asymptotically by an elliptic function in cheese-like strips near the point at infinity. We present an explicit asymptotic formula for the error term of this expression, which leads to the error bound as was conjectured. An analogous formula with its bound is obtained for the error term of the correction function associated with the Lagrangian.