Note on Coleman’s formula for the absolute Frobenius on Fermat curves
Annales de l'Institut Fourier, Online first, 22 p.

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.

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.

Received:
Revised:
Accepted:
Online First:
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)
@unpublished{AIF_0__0_0_A62_0,
     author = {Kashio, Tomokazu},
     title = {Note on {Coleman{\textquoteright}s} formula for the absolute {Frobenius} on {Fermat} curves},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3615},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Kashio, Tomokazu
TI  - Note on Coleman’s formula for the absolute Frobenius on Fermat curves
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3615
LA  - en
ID  - AIF_0__0_0_A62_0
ER  - 
%0 Unpublished Work
%A Kashio, Tomokazu
%T Note on Coleman’s formula for the absolute Frobenius on Fermat curves
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3615
%G en
%F AIF_0__0_0_A62_0
Kashio, Tomokazu. Note on Coleman’s formula for the absolute Frobenius on Fermat curves. Annales de l'Institut Fourier, Online first, 22 p.

[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

Cited by Sources: