Hecke L 函数値の非消滅性について

田中, 拓弥

doi:10.51094/jxiv.902

##article.authors##

田中, 拓弥東京科学大学理学院数学系数学コース

DOI:

https://doi.org/10.51094/jxiv.902

キーワード:

Hecke L values、 Mordel-Weil rank

抄録

Rohlich, Lamplugh の研究をはじめとして, L 値の非消滅性が数多く研究されてきた. 本稿では Lamplughが虚二次体に対して得た L 値に関する結果を CM 体に拡張する. これを応用して, 適切な多重 Z_l 拡大体 J において, l 上すべての素点で通常良還元をもつ CM 型アーベル多様体 A に関するZ 加群 A(J)/A(J)_torsが有限生成 Z加群であることを示す.

利益相反に関する開示

本論文に関して，開示すべき利益相反関連事項はない。

ダウンロード *前日までの集計結果を表示します

ダウンロード実績データは、公開の翌日以降に作成されます。

引用文献

Aurelirn Galateau, Cesar Martien, "A bound for the torsion on subvariety of abelian varieties", 2017, Arxiv.

P.Deligne, Cohomologie a supports propers, expose XVII of SGA 4, Theorie de topos et Cohomologie etale des Schemas, Tome 3, Springer Verlag, Berlin 1973, p250-480, Lecture Note in Math, Vol 269.

P.Deligne, "Values of L-functions and Periods of Integrals", on seminar website. Translation

of P. Deligne, "Valeurs de Fonctions L et périodes d’intégrales", Proceedings of Symposia in Pure

Math. 33, 1979, Part 2, 313-346.

Ehud de Shalit, "The Iwasawa Theory of Elliptic Curves with Complex Multiplication", 1987

M. Hazewinkel, “FORMAL GROUPS AND APPLICATIONS", AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island ΑΓΕΩΜΕΕΙΣΙΤΩ ΤΡΗΤΟΣ ΜΗ FOUNDED 1888, AMERICAN MATHEMATICAL SOCIETY

http://dx.doi.org/10.1090/chel/375.H

Mathematics Subject Classification. Primary 14L05; Secondary 16Txx, 05Exx,

Fxx, 11Sxx, 12Fxx, 13F35, 55N22.

H. Hida, Non-vanishing modulo p of Hecke L–values, in: “Geometric

Aspects of Dwork’s Theory, II” (edited by Alan Adolphson, Francesco

Baldassarri, Pierre Berthelot, Nicholas Katz, and Francois Loeser), Walter

de Gruyter, 2004, pp. 735–784

H.Hida, "$p$ adic automorphic forms on Shimura varieties",

Springer Monographs in Mathematics (SMM).

E. Hrushovski, “The Manin-Mumford conjecture and the model theory of difference fields”. Ann. Pure Appl. Logic 112 (2001), no. 1.

K. Bannai and S. Kobayashi, Algebraic theta functions and p-adic interpolation of

Eisenstein-Kronecker numbers, preprint, arXiv:math.NT/0610163v2, 2007.

K.Bannai, S.Kobayashi, H.Furusho, p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas, Nagoya Math. J. 219 (2015), 269-302.

K. Bannai, S. Kobayashi, and T. Tsuji, On the real Hodge and p-adic realizations of the

Elliptic Polylogarithm for CM elliptic curves, preprint 2007, arXiv:0711.1701v1 [math.NT]

Kings, Sprang, "Eisenstein-Kronecker classes, integrality of critical values of Hecke $L$-functions and $p$-adic interpolation", 2024,Arxiv.

N. M. Katz, "p-adic L-functions for CM fields", Invent. Math., 49, 199–297 (1978).

A.Lei, D.Kundu "non-vanishing modulo p of Hecke L value over imaginary quadratic fields", 2023, Arxiv.

J.Lamplugh, "An analogue of the Washington–Sinnott theorem for elliptic curves with complex multiplication I",

J. London Math. Soc. 91 (2015), no. 3, 609–642G.

Laumon, “Transformation de Fourier generalisee”, arxiv:9603004, 1996.

Qing Liu, "Algebraic geometry and Arithmetic curves", 2008, Oxford University press.

M. McQuillan, Division points on semi-abelian varieties, Invent. Math. 120 (1995), no. 1,

–159

T.Ochiai, T.Hara, "The cyclotomic Iwasawa main conjecture for Hilbert cusp forms with complex multiplications", 2015, Arxiv.

N. Roby, Lois polynomes et lois formelles en theorie des modules, An. Sci. Ecole. Norm. Sup(1963), no80, 213-248.

D. Rohrlich, On L-functions of elliptic curves and cyclotomic towers. Invent. Math. 75, 409–423 (1984).

M.Raynaud, Sous-variétés d’une variété abélienne et points de torsion, Prog. Math. 35 (1983),

–352.

L. Schneps, “On the μ-invariant of p-adic L-functions attached to elliptic curves with complex

multiplication", J. Number Theory 25 (1987) 20–33.

G.Shimura, “Introduction to the arithmetic theory of automorphic functions", 1971,

Kanô Memorial Lectures, No. 1.

Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, N.J.

G.Shimura, Y.Taniyama, AMA, "Complex multiplication of abelian varieties and its applications

to number theory", Publ. Math. Soc. Japan No. 6, 1961.

Warren M. Sinnott, Γ-transforms of rational function measures on ZS, Invent. Math. 89 (1987), no. 1,

–157

J.Tate, "p-divisible group", roc. Conf. Local Fields (Driebergen, 1966), 158–183, Springer, Berlin, 1967.

A. Polishchuk , “Abelian Varieties, Theta Functions and the Fourier Transform”. Cambridge University Press; 2003.

S. Yasuda. Explicit t-expansions for the elliptic curve E : $y^2 = 4(x^3 +Ax+B)$. Proc. Japan Acad.

Ser. A Math Sci., 89(9):123–127, 2013.

Umberto Zannier. Some problems of unlikely intersections in arithmetic and geometry, volume 181 of

Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2012. With appendixes by

David Masser.