On central LL-values and the growth of the 33-part of the Tate-Shafarevich group

Given any cube-free integer $\lambda>0$, we study the $3$-adic valuation of the algebraic part of the central $L$-value of the elliptic curve $$X^3+Y^3=\lambda Z^3.$$ We give a lower bound in terms of the number of distinct prime factors of $\lambda$, which, in the case $3$ divides $\lambda$, also depends on the power of $3$ in $\lambda$. This extends an earlier result of the author in which it was assumed that $3$ is coprime to $\lambda$. We also study the $3$-part of the Tate-Shafarevich group for these curves and show that the lower bound is as expected from the conjecture of Birch and Swinnerton-Dyer, taking into account also the growth of the Tate-Shafarevich group.Comment: 15 pages. A 3-descent argument has been added to Section

Tamagawa number divisibility of central LL-values of twists of the Fermat elliptic curve

Given any integer $N>1$ prime to $3$, we denote by $C_N$ the elliptic curve $x^3+y^3=N$. We first study the $3$-adic valuation of the algebraic part of the value of the Hasse-Weil $L$-function $L(C_N,s)$ of $C_N$ over $\mathbb{Q}$ at $s=1$, and we exhibit a relation between the $3$-part of its Tate-Shafarevich group and the number of distinct prime divisors of $N$ which are inert in the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-3})$. In the case where $L(C_N,1)\neq 0$ and $N$ is a product of split primes in $K$, we show that the order of the Tate-Shafarevich group as predicted by the conjecture of Birch and Swinnerton-Dyer is a perfect square.Comment: 21 pages. To appear in the Journal de Th\'{e}orie des Nombres de Bordeaux (Iwasawa 2019 special issue

Analysis of the Pathogenesis of Experimental Autoimmune Optic Neuritis

Optic neuritis associated with multiple sclerosis has a strong association with organ-specific autoimmune disease. The goal of our research is to establish an optimal organ-specific animal model to elucidate the pathogenetic mechanisms of the disease and to develop therapeutic strategies using the model. This paper is divided into five sections: (1) clinical picture of optic neuritis associated with multiple sclerosis, (2) elucidation of pathogenesis using animal models with inflammation in optic nerve and spinal cord, (3) clinical relevance of concurrent encephalomyelitis in optic neuritis model, (4) retinal damage in a concurrent multiple sclerosis and optic neuritis model, and (5) development of novel therapies using mouse optic neuritis model. Advanced therapies using biologicals have succeeded to control intractable optic neuritis in animal models. This may ultimately lead to prevention of vision loss within a short period from acute onset of optic neuritis in human. By conducting research flexibly, ready to switch from the bench to the bedside and from the bedside to the bench as the opportunity arises, this strategy may help to guide the research of optic neuritis in the right direction

On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication

The conjecture of Birch and Swinnerton-Dyer is unquestionably one of the most important open problems in number theory today. Let $E$ be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers of $K$. Let us assume the complex $L$-series $L(E/K,s)$ of $E$ over $K$ does not vanish at $s=1$. K. Rubin showed, using Iwasawa theory, that the $p$-part of Birch and Swinnerton-Dyer conjecture holds for $E$ for all prime numbers $p$ which do not divide the order of the group of roots of unity in $K$. In this thesis, we discuss extensions of this result. In Chapter $2$, we study infinite families of quadratic and cubic twists of the elliptic curve $A = X_0(27)$, so that they have complex multiplication by the ring of integers of $\mathbb{Q}(\sqrt{-3})$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by Birch and Swinnerton-Dyer. In the remaining chapters, we let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$. Denote by $H$ the Hilbert class field of $K$. \mbox{B. Gross} proved the existence of an elliptic curve $A(q)$ defined over $H$ with complex multiplication by the ring of integers of $K$ and minimal discriminant $-q^3$. We consider twists $E$ of $A(q)$ by quadratic extensions of $K$. In the case $q=7$, we have $A(q)=X_0(49)$, and Gonzalez-Aviles and Rubin proved, again using Iwasawa theory, that if $L(E/\mathbb{Q},1)$ is nonzero then the full Birch--Swinnerton-Dyer conjecture holds for $E$. Suppose $p$ is a prime number which splits in $K$, say $p=\mathfrak{p}\mathfrak{p}^*$, and $E$ has good reduction at all primes of $H$ above $p$. Let $H_\infty=HK_\infty$, where $K_\infty$ is the unique $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. We establish in this thesis the main conjecture for the extension $H_\infty/H$. Furthermore, we provide the necessary ingredients to state and prove the main conjecture for $E/H$ and $p$, and discuss its relation to the main conjecture for $H_\infty/H$ and the $p$-part of the Birch--Swinnerton-Dyer conjecture for $E/H$

Studies of the Genus Bupleurum (Umbelliferae) from Nepal: (1) A Histological Study of Leaves and the Botanical Origin of Tibetan Crude Drug Tunak Chunga

ネパール産のセリ科ミシマサイコ属（Bupleurum）8分類群の葉を比較組織学的に検討し，本属の組織分類学的な要素を明らかにするとともに，ネパールの高山帯で薬用として利用されているチベット薬物 TUNAK CHUNGA の基源解明を試みた．その結果，組織学的には茎の中央部付近の葉において，横切面における主脈部や葉縁部の形，厚角組織の発達状態，油道の存在数，乳状突起の有無や上面の気孔の分布数などの形質で全種を分類することが可能であった（Table 1）．また TUNAK CHUNGA の基源は，ネパール高山帯の本属植物では資源的にもっとも豊富なB. falcatum subsp. falcatum var. gracillimum の全草であることが明らかになった．本種はチベット薬物の原植物としての初めての記録である

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