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In this talk, we introduce the concept of comprehensive Groebner systems, which is a tool for manipulating parameters in computer algebra. I start with a survey of the known results and then describe a new case of indecomposability when the modular Galois representation is a deformation of an induced representation. 楕円曲線のp進L関数が例外零点を持つには, pで分裂乗法的還元を持つことが必要かつ十分である. Since I have already presented the above result at several universities, I'll also explain about a few problems in mathematical crystallography that could expand the boundary of applied algebra and number theory. I intend to survey some of these topics. They are considered as Groebner bases with parameters. 本講演では包括的グレブナー基底系に関する性質を述べるとともに, それに基づいた"複素数領域における限量子消去"や"実数領域における限量子消去"についても述べる. 私は今回F, G, HのうちG, Hを, Hida familyをより一般的な保型形式のp進族に拡張した場合についても, unbalancedな場合に限り, 三変数のp進三重積L関数を構成することに成功した. If time allows I will also give some insides into the proof for the dynamical systems results using the transfer operator method. I constructed three-variable p-adic triple product L-functions for p-adic families of modular forms F, G, H where only F must be a Hida family but G and H are allowed to be more general p-adic families of modular forms and I proved an interpolation formula only in the unbalanced case. val w に関する既知の性質として, 金子昌信教授により年に予想されBengoechea-Imamogluにより年に解決された, wの連分数表示に関するある種の連続性が挙げられる. In this talk, we discuss arithmetic behavior of Hecke eigenvalues of elliptic modular forms as well as Siegel modular forms of degree two. Speiser in but nothing is known for the case of higher order derivatives. The values val w when w goes over all real quadratic points satisfy some continuity with respect to the continued fraction expansions of real quadratic points. In the case of Siegel modular forms of degree 2, there is a famous result of Igusa what stated that such the ring is generated by 15 modular forms. このとき 複素L関数の中心値が 0 でなくても p進L関数の中心値は 0 になってしまう. The precise relation between derivative of the p-adic L-function and the algebraic part of the central value was conjectured by Mazur-Tate-Teitelbaum and proved by Greenberg-Stevens. 本講演では, 三つのp通常楕円曲線に付随する三重積p進L関数の例外零点を決定し, その 高 階 微分値と複素L関数の値の関係式を証明する. removing the largest summands of a sum of identically distributed iid random variables, has a long tradition to prove limit theorems which are not valid if one considers the untrimmed sum - one example is the strong law of large numbers for random variables with an infinite mean or in case of ergodic transformations Birkhoff's ergodic theorem. Ge and I hope to be able to introduce necessary details for the improvement we obtained.{/INSERTKEYS}{/PARAGRAPH} {PARAGRAPH}{INSERTKEYS}In this talk, we give answers to their questions on this new object. Many answers to this question are known in the literature. Finally, we indicate how these properties can be used to answer the above question. より詳細に言えば, 我々は包括的グレブナー基底系を利用することで, 分割されたパラメータ空間と各空間に対応するパラメータ付きグレブナー基底を得ることができる. Moreover, the limit formulas for three types of the Eisenstein series bring new perspective to them. This is a very striking and interesting result proven by A. Computer algebra is a scientific area devoted to the development of algorithms for manipulating mathematical expressions. This is a joint work with Ming-Lun Hsieh. We discuss the injectivity of the Prym maps under a suitable condition on the dimension of the moduli spaces. This is a joint work with Masanori Asakura. In this talk, we will introduce a reformulation of Perrin-Riou's conjecture for elliptic curves using syntomic regulators and we will give some numerical evidences for Perrin-Riou's conjecture including critical slope case. Motivated by the question why the other infinite families were not detected in the search, we prove that if two pairs of ternary quadratic forms have the identical simultaneous representations over Q, their constant multiples are equivalent over Q. We then show explicit formulas of mod 2 arithmetic Chern-Simons and Dijkgraaf-Witten invariants for certain real quadratic fields. This is joint work with Chan-Ho Kim. In this talk I will determine the exceptional zeros of cyclotomic p-adic L-functions associated to three ordinary elliptic curves and prove an identity between double or triple derivatives of the p-adic L-function and central L-values. In the proof, the parametrization of quartic rings and their resolvent rings by Bhargava is used to discuss pairs of ternary quadratic forms. Their exponents often are important to study, too relating to e. In this talk I will first give a background about trimmed strong laws of large numbers for iid random variables and compare these results with some random variables obtained from ergodic transformations, for example piecewise expanding interval maps and subshifts of finite type. Then the p-adic L-function trivially vanish at the point, and such a zero is called an exceptional zero. In this talk, we state that the Prym maps are injective for double coverings of elliptic curves, and explain how the covering is reconstructed from the polarized abelian variety. We discuss properties of comprehensive Groebner systems, then introduce a quantifier elimination method based on the properties. This is joint work with Kazuto Ota. Speiser 氏により示されたが、高階導関数の場合への拡張がまだ困難 である。それでも、 年代にリーマンゼータ関数の導関数の零点に関する研究が盛り、リー マンゼータ関数自身の零点の分布との関係もたくさん導かれてきた。この講演では、この研究 の背景となるいくつかの重要な研究成果および、この話題における講演者が行ってきた研究を 紹介する。特に、F. This kind of continuity was conjectured by Professor Masanobu Kaneko in and proved by Bengoechea-Imamoglu in As a result, it turns out that val is away from the continuity in the Euclidean topology. I will talk about the construction of these three-variable p-adic triple product L-functions. This conjecture is widely known as the Riemann hypothesis and a lot of equiv- alent statements are known. The result of our exhaustive search shows that some strong limitation is imposed to existence of such a pair, although a small number of non-regular forms is still contained in the obtained list of such quadratic forms. The construction defines the Prym map from the moduli space of coverings to the moduli space of polarized abelian varieties. It is known that the Prym map is generically injective, but it is not injective in general. 講演者はBengoechea-Imamogluの手法を整理することでより一般的な状況での連続性を証明し, その結果valがEuclid的でない連続性を持つことが分かった. We will explain ideas and key ingredients of the proof. The p-adic L-function of an elliptic curve E has an exceptional zero if and only if E has split multiplicative reduction at p. Greenberg asserts that a modular 2-dimensional p-adic Galois representation of a cusp form of weight larger than or equal to 2 is indecomposable over the p-inertia group. このときのp進L関数の微分値と複素L関数の中心値の関係式が, Mazur-Tate-Teitelbaum に予想され, 肥田理論を使って Greenberg-Stevens に証明された. 興味深いことに, この修正項が関数等式の中心において零点を持つことがある. I would like to focus on the recent progress I made with F. In this talk, I will introduce the results as mentioned above. p-adic L-functions involve modified p-factors which measure the discrepancy between the p-adic and classical special values in the interpolation formula. We will discuss the proof of the main theorem on a quantitative level lowering. A lot of studies have been conducted on the distribution of zeros of the derivatives of the Riemann zeta function and I would like to introduce some important results, including my contribution to this study. In this talk, the speaker introduces a result for the case of Hermitian modular forms of degree 2 with integral Fourier coefficients over the Gaussian number field, that the ring of them whose weights are multiples of 4 is generated by 24 modular forms. このような零点は例外 零点と呼ばれている. We also show a topological analogue of our results. It is a puzzling fact that this factor can vanish at the central point. これは台湾国立大学の Ming-Lun Hsieh 教 授との共同研究である.