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Kazuma Morita

Publications and source records attributed to Kazuma Morita.

8 recordsLinked to original sources
arXiv2022

Birch and Swinnerton-Dyer conjecture in the complex multiplication case and the congruent number problem

For an elliptic curve $E$ over $K$, the Birch and Swinnerton-Dyer conjecture predicts that the rank of Mordell-Weil group $E(K)$ is equal to the order of the zero of $L(E_{/ K},s)$ at $s=1$. In this paper, we shall give a proof for elliptic curves with complex multiplications. The key method of the proof is to reduce the Galois action of infinite order on the Tate module of an elliptic curve to that of finite order by using the $p$-adic Hodge theory. As a corollary, we can determine whether a given natural number is a congruent number (congruent number problem). This problem is one of the oldest unsolved problems in mathematics.

math.NT↗
arXiv2011

Crystalline and semi-stable representations in the imperfect residue field case

Let K be a p-adic local field with residue field k such that [k:k^p]=p^e<\infty and V be a p-adic representation of Gal(\bar{K}/K). Then, by using the theory of p-adic differential modules, we show that V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(\bar{K}/K) if and only if V is a potentially crystalline (resp. potentially semi-stable) representation of Gal(\bar{K^{pf}}/K^{pf}) where K^{pf}/K is a certain p-adic local field whose residue field is the smallest perfect field k^{pf} containing k. As an application, we prove the p-adic monodromy theorem of Fontaine in the imperfect residue field case.

math.NT↗
arXiv2011

Generalization of the theory of Sen in the semi-stable representation case

For a semi-stable representation V, we will construct a subspace D_{π-Sen}(V) of C_p\otimes_{Q_p}V endowed with a linear derivation \nabla^{(π)}. The action of \nabla^{(π)} on D_{π-Sen}(V) is closely related to the action of the monodromy operator N on D_{st}(V). Furthermore, in the geometric case, the action of \nabla^{(π)} on D_{π-Sen}(V) describes an analogy of the infinitesimal variations of Hodge structures and satisfies formulae similar to the Griffiths transversality and the local monodromy theorem.

math.NT↗
arXiv2011

On the topological aspects of arithmetic elliptic curves

In this short note, we shall construct a certain topological family which contains all elliptic curves over Q and, as an application, show that this family provides some geometric interpretations of the Hasse-Weil L-function of an elliptic curve over Q whose Mordell-Weil group is of rank $\leq1$.

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