no. 3
Note on Coleman’s formula for the absolute Frobenius on Fermat curves
[Note sur la formule de Coleman pour la Frobenius absolue sur les courbes de Fermat]
Kashio, Tomokazu1
1 Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510 (Japan)
Annales de l'Institut Fourier, Tome 74 (2024) no. 3, pp. 1229-1250.

Coleman a calculé explicitement le Frobenius absolu sur les courbes de Fermat. Dans cet article, nous montrons qu’une sorte de continuité p-adique implique une grande partie de sa formule. Pour ce faire, nous étudions une relation entre les équations fonctionnelles de la fonction gamma, les relations monomiales sur CM-périodes, et leurs analogues p-adiques.

Coleman calculated the absolute Frobenius on Fermat curves explicitly. In this paper we show that a kind of p-adic continuity implies a large part of his formula. To do this, we study a relation between functional equations of the gamma function, monomial relations on CM-periods, and their p-adic analogues.

Reçu le :
Révisé le :
Accepté le :
Première publication :
Publié le :
DOI : 10.5802/aif.3615
Classification : 11M35, 11S80, 14F30, 14H45, 14K20, 14K22, 33B15
Keywords: Absolute Frobenius, Fermat curves, Gross–Koblitz formula, $p$-adic gamma function, CM-periods, $p$-adic periods.
Mot clés : Frobenius absolu, courbes de Fermat, formule de Gross–Koblitz, fonction gamma $p$-adique, CM-périodes, périodes $p$-adiques.

Kashio, Tomokazu 1

1 Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510 (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2024__74_3_1229_0,
     author = {Kashio, Tomokazu},
     title = {Note on {Coleman{\textquoteright}s} formula for the absolute {Frobenius} on {Fermat} curves},
     journal = {Annales de l'Institut Fourier},
     pages = {1229--1250},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {74},
     number = {3},
     year = {2024},
     doi = {10.5802/aif.3615},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3615/}
}
TY  - JOUR
AU  - Kashio, Tomokazu
TI  - Note on Coleman’s formula for the absolute Frobenius on Fermat curves
JO  - Annales de l'Institut Fourier
PY  - 2024
SP  - 1229
EP  - 1250
VL  - 74
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3615/
DO  - 10.5802/aif.3615
LA  - en
ID  - AIF_2024__74_3_1229_0
ER  - 
%0 Journal Article
%A Kashio, Tomokazu
%T Note on Coleman’s formula for the absolute Frobenius on Fermat curves
%J Annales de l'Institut Fourier
%D 2024
%P 1229-1250
%V 74
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3615/
%R 10.5802/aif.3615
%G en
%F AIF_2024__74_3_1229_0
Kashio, Tomokazu. Note on Coleman’s formula for the absolute Frobenius on Fermat curves. Annales de l'Institut Fourier, Tome 74 (2024) no. 3, pp. 1229-1250. doi : 10.5802/aif.3615. https://aif.centre-mersenne.org/articles/10.5802/aif.3615/

[1] Artin, Emil The gamma function, Athena Series: Selected Topics in Mathematics, Holt, Rinehart and Winston, New York-Toronto-London, 1964, vii+39 pages (translated by Michael Butler) | MR | Zbl

[2] Bouyer, Florian; Streng, Marco Examples of CM curves of genus two defined over the reflex field, LMS J. Comput. Math., Volume 18 (2015) no. 1, pp. 507-538 | DOI | MR | Zbl

[3] Coleman, Robert F. The Gross-(Koblitz formula, Galois representations and arithmetic algebraic geometry (Kyoto, 1985/Tokyo, 1986) (Advanced Studies in Pure Mathematics), Volume 12, North-Holland, 1987, pp. 21-52 | DOI | MR | Zbl

[4] Coleman, Robert F. On the Frobenius matrices of Fermat curves, p-adic analysis (Trento, 1989) (Lecture Notes in Mathematics), Volume 1454, Springer, 1990, pp. 173-193 | DOI | MR | Zbl

[5] Gross, Benedict H. On the periods of abelian integrals and a formula of Chowla and Selberg, Invent. Math., Volume 45 (1978) no. 2, pp. 193-211 (with an appendix by David E. Rohrlich) | DOI | MR | Zbl

[6] Gross, Benedict H.; Koblitz, Neal Gauss sums and the p-adic Γ-function, Ann. Math., Volume 109 (1979) no. 3, pp. 569-581 | DOI | MR | Zbl

[7] Kashio, Tomokazu Fermat curves and a refinement of the reciprocity law on cyclotomic units, J. Reine Angew. Math., Volume 741 (2018), pp. 255-273 | DOI | MR | Zbl

[8] Kashio, Tomokazu On a common refinement of Stark units and Gross-Stark units (2018) (https://arxiv.org/abs/1706.03198)

[9] Kashio, Tomokazu; Yoshida, Hiroyuki On p-adic absolute CM-periods. I, Am. J. Math., Volume 130 (2008) no. 6, pp. 1629-1685 | DOI | MR | Zbl

[10] Kashio, Tomokazu; Yoshida, Hiroyuki On p-adic absolute CM-periods. II, Publ. Res. Inst. Math. Sci., Volume 45 (2009) no. 1, pp. 187-225 | DOI | MR | Zbl

[11] Koblitz, Neal p-adic numbers, p-adic analysis, and zeta-functions, Graduate Texts in Mathematics, 58, Springer, 1984, xii+150 pages | DOI | MR

[12] Otsubo, Noriyuki On the regulator of Fermat motives and generalized hypergeometric functions, J. Reine Angew. Math., Volume 660 (2011), pp. 27-82 | DOI | MR | Zbl

[13] Shimura, Goro Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, 46, Princeton University Press, 1998, xvi+218 pages | DOI | MR | Zbl

[14] Yoshida, Hiroyuki Absolute CM-periods, Mathematical Surveys and Monographs, 106, American Mathematical Society, 2003, x+282 pages | DOI | MR | Zbl

Cité par Sources :