In this paper, we give an overview of our previous paper concerning the investigation of the algebraic and p-adic properties of Eisenstein-Kronecker numbers using Mumford's theory of algebraic theta functions.

Abstract. In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic analogue of the result of Beilinson and Levin expressing the complex elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. Our result is valid even if the elliptic curve has supersingular reduction at p.0. Introduction 0.1. Introduction. In the paper [BL], Beilinson and Levin constructed the elliptic polylogarithm, which is an element in absolute Hodge or ℓ-adic cohomology of an elliptic curve minus the identity. This construction is a generalization to the case of elliptic curves of the construction by Beilinson and Deligne of the polylogarithm sheaf on the projective line minus three points. The purpose of this paper is to study the p-adic realization of the elliptic polylogarithm for an elliptic curve with complex multiplication, and to investigate its relation to p-adic L-functions associated to the elliptic curve, even for the case where the elliptic curve has supersingular reduction at the prime p.To achieve our goal, we first describe the de Rham realization of the elliptic polylogarithm for a general elliptic curve defined over a subfield of C. In particular, we explicitly describe the connection of the elliptic polylogarithm using rational functions. This result is of independent interest. Similar results were obtained by Levin and Racinet [LR] Section 5.1.3. A different description for this connection was also given by Besser and Solomon [BS].Using the de Rham realization of the elliptic polylogarithm, we then construct the p-adic realization of the elliptic polylogarithm as a filtered overconvergent F -isocrystal on the elliptic curve minus the identity, when the elliptic curve has complex multiplication and good reduction at a fixed prime p ≥ 5. Our main result Theorem 4.15 is an explicit description of the p-adic Using this description, we calculate the specializations of the p-adic elliptic polylogarithm to torsion points of order prime to p (more precisely, torsion points of order prime to p. See the Overview for details), and prove that the specializations give the p-adic Eisenstein-Kronecker numbers, which are special values of the p-adic distribution interpolating Eisenstein-Kronecker numbers. This result is a generalization of the result of [Ba3], where we have dealt only with the one variable case for an ordinary prime. A similar result concerning the specialization in two-variables was obtained in [BKi], again for ordinary primes, using a very different method. The result of the current paper is valid even when p is supersingular.The p-adic Eisenstein-Kronecker numbers are related to special values of p-adic L-functions which p-adically interpolate special values of Hecke L-f...

The main purpose of this paper is to construct the p-adic realization of the classical polylogarithm following the method of Beilinson and Deligne as explained by Huber and Wildeshaus. A simplicial construction of the p-adic polylogarithm was previously obtained by Somekawa. In this paper, we will give a sheaf theoretic interpretation of this construction. In particular, we will give an interpretation of the p-adic polylogarithm as an object in the p-adic analogue of the category of variation of mixed Hodge structures. We will also calculate the restriction of this object to torsion points, and will prove a result which is compatible with the results of Gros-Kurihara, Gros and Somekawa. IntroductionIn the paper [23], M. Somekawa constructed the p-adic polylogarithm in the syntomic cohomology of a simplicial scheme over the projective line minus three points. This is a p-adic analogue of the elements in absolute Hodge and l-adic cohomology de®ned by Beilinson [2] and is the image by the syntomic regulator of the motivic polylogarithm in motivic cohomology. The purpose of this paper is to develop a theory of p-adic mixed sheaves which is su½cient to give a sheaf theoretic interpretation of the construction by Somekawa. In particular, as in the classical case, the p-adic polylogarithm has an interpretation as an object in the p-adic analogue of the category of variation of mixed Hodge structures on the projective line minus three points. In Theorem 2, which is our main result, we will explicitly determine the shape of the p-adic polylogarithm.As a corollary, we will show that the specialization of this object to the roots of unity can be expressed by the p-adic polylogarithm function de®ned by Coleman. This result is compatible with the result of Somekawa, and is compatible with the results of Gros-Kurihara [14] and Gros [15] on the calculation of the image of the Beilinson element with respect to the syntomic regulator. This gives evidence that the p-adic formalism which we develop in this paper is potentially a powerful tool in the calculation of the image by the syntomic regulator of elements in motivic cohomology.

Multi-objective optimization problems require simultaneously optimizing two or more objective functions. Many studies have reported that the solution set of an M -objective optimization problem often forms an (M − 1)-dimensional topological simplex (a curved line for M = 2, a curved triangle for M = 3, a curved tetrahedron for M = 4, etc.). Since the dimensionality of the solution set increases as the number of objectives grows, an exponentially large sample size is needed to cover the solution set. To reduce the required sample size, this paper proposes a Bézier simplex model and its fitting algorithm. These techniques can exploit the simplex structure of the solution set and decompose a high-dimensional surface fitting task into a sequence of low-dimensional ones. An approximation theorem of Bézier simplices is proven. Numerical experiments with synthetic and real-world optimization problems demonstrate that the proposed method achieves an accurate approximation of high-dimensional solution sets with small samples. In practice, such an approximation will be conducted in the postoptimization process and enable a better trade-off analysis. PreliminariesLet us introduce notations for defining simplicial problems and review an existing method of Bézier curve fitting. Simplicial ProblemA multi-objective optimization problem is denoted by its objective map f = (f 1 , . . . , f M ) : X → R M . Let I := { 1, . . . , M } be the index set of objective functions and ∆ M −1 := (t 1 , . . . , t M ) ∈ R M 0 ≤ t m , m∈M t m = 1 be the standard simplex in R M . For each non-empty subset J ⊆ I, we call∆ J := (t 1 , . . . , t M ) ∈ ∆ M −1 t m = 0 (m ∈ J) the J-face of ∆ M −1 andThe problem class we are interested in is as follows:We call such φ and f • φ a triangulation of the Pareto set X * (f ) and the Pareto front f X * (f ), respectively. For each non-empty subset J ⊆ I, we call X * (f J ) the J-face of X * (f ) and f X * (f J ) the J-face of f X * (f ). For each 0 ≤ m ≤ M − 1, we callthe m-skeleton of X * (f ) and f X * (f ), respectively.By definition, any subproblem of a simplicial problem is again simplicial. As shown in Figure 1b, the Pareto sets forms a simplex. The second condition asserts that f | X * (f ) : X * (f ) → R M is a C 0 -embedding. This means that the Pareto front of each subproblem is homeomorphic to its Pareto set as shown in Figure 1c. Therefore, the Pareto set/front of an M -objective simplicial problem can be identified with a curved (M − 1)-simplex. We can find its J-face by solving the J-subproblem.

The purpose of this paper is to interpret rigid syntomic cohomology, defined by Amnon Besser [Bes], as a p-adic absolute Hodge cohomology. This is a p-adic analogue of a work of Beilinson [Be1] which interprets Beilinson-Deligne cohomology in terms of absolute Hodge cohomology. In the process, we will define a theory of p-adic absolute Hodge cohomology with coefficients, which may be interpreted as a generalization of rigid syntomic cohomology to the case with coefficients. (2000):14F30, 14G20 Mathematics Subject Classification IntroductionIn the paper "Notes on absolute Hodge cohomology" [Be1], Beilinson gave an interpretation of Beilinson-Deligne cohomology as an absolute Hodge cohomology; in other words, as extension groups in the derived category of the category of mixed Hodge structures. It is widely believed that the p-adic analogue of Beilinson-Deligne cohomology is syntomic cohomology. The purpose of this paper is to give the p-adic analogue of the result of Beilinson mentioned above. Namely, we will show that rigid syntomic cohomology, first defined by M. Gros [G] then fully developed by Amnon Besser [Bes], is expressed as extension groups in the derived category of weakly admissible filtered Frobenius modules MF f K . In the process, we will define a theory of p-adic absolute Hodge cohomology with coefficients. This may be interpreted as a generalization to the case with coefficients of rigid syntomic cohomology. This is also a generalization of absolute syntomic cohomology with coefficients defined in our previous paper [Ba1] to the case when the base field is ramified over Q p . We remark that a cohomology theory with coefficients for proper and smooth 444 K. Bannai schemes has been considered by W. Nizioĺ [Ni]. The philosophy of this paper played an important role in the papers [Ba1] [Ba2].Let K be a finite extension of Q p with ring of integers O K and residue field k. We let K 0 be the maximum unramified extension of Q p in K and W its ring of integers. Let σ be the lifting to K 0 of the Frobenius automorphism on k.The Philosophy. For a scheme X smooth and of finite type over O K , there should exist a Tannakian category MHM p (X) of the p-adic mixed Hodge modules. This category should be the p-adic analogue of the category of mixed Hodge modules defined by Morihiko Saito [Sa]. In particular, on the derived category D b (X) of MHM p (X), we should have the formalism of six Grothendieck functors.Although the category MHM p (X) still has not been constructed, in the special case X = Spec O K , the category MHM p (Spec O K ), which we denote by MHM p (O K ), should be the category MF f K of weakly admissible filtered modules defined by Fontaine ([Fon1] 4.1.4), consisting of objects (M 0 , ϕ, F • ) where (i) M 0 is a finite dimensional K 0 -vector space with a σ-linear isomorphism ϕ : M 0 → M 0 , which we call the Frobenius automorphism. (ii) F • is a descending exhaustive separated filtration on M = M 0 ⊗ K, which we call the Hodge filtration. (iii) M with the above structure is weakly admissible.