An introduction to Eisenstein measures

Ellen Eischen

doi:10.5802/jtnb.1178

Ellen Eischen ¹

¹ Department of Mathematics University of Oregon Fenton Hall Eugene, OR 97403-1222, USA

Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.1, pp. 779-808.

Résumé
Abstract

Cet article fournit une introduction aux mesures d’Eisenstein, un outil puissant pour construire certaines fonctions $L$ $p$ -adiques. Vues pour la première fois dans la réalisation par Serre des fonctions zêta de Dedekind $p$ -adiques associées aux corps totalement réels, les mesures d’Eisenstein fournissent un moyen d’étendre les congruences de style kummerien, observées par Kummer pour les valeurs de la fonction zêta de Riemann (dites congruences de Kummer) à certaines autres fonctions $L$ . En plus de retracer les développements clés, nous discutons certains défis qui se posent dans des contextes plus généraux, en concluant par certains qui restent ouverts.

This paper provides an introduction to Eisenstein measures, a powerful tool for constructing certain $p$ -adic $L$ -functions. First seen in Serre’s realization of $p$ -adic Dedekind zeta functions associated to totally real fields, Eisenstein measures provide a way to extend the style of congruences Kummer observed for values of the Riemann zeta function (so-called Kummer congruences) to certain other $L$ -functions. In addition to tracing key developments, we discuss some challenges that arise in more general settings, concluding with some that remain open.

Reçu le : 2020-04-25
Révisé le : 2021-01-05
Accepté le : 2021-02-01
Publié le : 2022-01-26

DOI : 10.5802/jtnb.1178

Classification : 11F33, 11F85, 11S40, 11F30, 11F03, 11R23
Mots clés : Eisenstein measures, $p$-adic families of modular forms, $p$-adic modular forms, $p$-adic $L$-functions, $p$-adic measures

Affiliations des auteurs :

Ellen Eischen ¹

¹ Department of Mathematics University of Oregon Fenton Hall Eugene, OR 97403-1222, USA

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2021__33_3.1_779_0,
     author = {Ellen Eischen},
     title = {An introduction to {Eisenstein} measures},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {779--808},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {33},
     number = {3.1},
     year = {2021},
     doi = {10.5802/jtnb.1178},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1178/}
}

TY  - JOUR
AU  - Ellen Eischen
TI  - An introduction to Eisenstein measures
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2021
SP  - 779
EP  - 808
VL  - 33
IS  - 3.1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1178/
DO  - 10.5802/jtnb.1178
LA  - en
ID  - JTNB_2021__33_3.1_779_0
ER  -

%0 Journal Article
%A Ellen Eischen
%T An introduction to Eisenstein measures
%J Journal de théorie des nombres de Bordeaux
%D 2021
%P 779-808
%V 33
%N 3.1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1178/
%R 10.5802/jtnb.1178
%G en
%F JTNB_2021__33_3.1_779_0

Ellen Eischen. An introduction to Eisenstein measures. Journal de théorie des nombres de Bordeaux, Tome 33 (2021) no. 3.1, pp. 779-808. doi : 10.5802/jtnb.1178. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1178/

Bibliographie
Cité par

[1] Fabrizio Andreatta; Adrian Iovita Katz type $p$ -adic $L$ -functions for primes $p$ non-split in the CM field (2019) | arXiv

[2] Fabrizio Andreatta; Adrian Iovita; Vincent Pilloni On overconvergent Hilbert modular cusp forms, Arithmétique $p$ -adique des formes de Hilbert (Astérisque), Volume 382, Société Mathématique de France, 2016, pp. 163-193 | MR | Zbl

[3] Jon Aycock Families of differential operators for overconvergent Hilbert modular forms, 2021 (In preparation)

[4] Daniel Barsky Fonctions zeta $p$ -adiques d’une classe de rayon des corps de nombres totalement réels, Groupe d’Etude d’Analyse Ultramétrique (5e année: 1977/78), Secrétariat Mathématique, 1978 (Exp. No. 16, 23) | Numdam | MR

[5] Mark Behrens EISENSTEIN ORIENTATION, 2009 (Typed notes available at http://www-math.mit.edu/~mbehrens/other/coredump.pdf)

[6] Nicolas Bergeron; Pierre Charollois; Luis E. Garcia Transgressions of the Euler class and Eisenstein cohomology of ${GL}_{N} (Z)$ , Jpn. J. Math., Volume 15 (2020) no. 2, pp. 311-379 | DOI | MR

[7] Nicolas Bourbaki Elements of Mathematics. Commutative algebra. Chapters 1–7, Springer, 1998, xxiv+625 pages (Translated from the French, Reprint of the 1989 English translation) | MR

[8] George Boxer; Vincent Pilloni Higher Hida and Coleman theories on the modular curve (2020) | arXiv

[9] Riccardo Brasca; Giovanni Rosso Hida theory over some unitary Shimura varieties without ordinary locus, Am. J. Math., Volume 143 (2021) no. 3, pp. 715-751 | DOI | MR | Zbl

[10] Yuanqing Cai; Solomon Friedberg; David Ginzburg; Eyal Kaplan Doubling constructions and tensor product $L$ -functions: the linear case, Invent. Math., Volume 217 (2019) no. 3, pp. 985-1068 | DOI | MR | Zbl

[11] Ana Caraiani; Ellen Eischen; Jessica Fintzen; Elena Mantovan; Ila Varma $p$ -adic $q$ -expansion principles on unitary Shimura varieties, Directions in number theory, Volume 3, Springer, 2016, pp. 197-243 | DOI | MR | Zbl

[12] Pierrette Cassou-Noguès Valeurs aux entiers négatifs des fonctions zêta et fonctions zêta $p$ -adiques, Invent. Math., Volume 51 (1979) no. 1, pp. 29-59 | DOI | MR | Zbl

[13] Pierre Charollois; Samit Dasgupta Integral Eisenstein cocycles on ${GL}_{n}$ , I: Sczech’s cocycle and $p$ -adic $L$ -functions of totally real fields, Camb. J. Math., Volume 2 (2014) no. 1, pp. 49-90 | DOI | MR | Zbl

[14] Thomas Clausen Theorem, Astron. Nachr., Volume 17 (1840) no. 22, pp. 351-352 | DOI

[15] John Coates On $p$ -adic $L$ -functions attached to motives over $Q$ . II, Bol. Soc. Bras. Mat., Nova Sér., Volume 20 (1989) no. 1, pp. 101-112 | DOI | MR | Zbl

[16] John Coates; Bernadette Perrin-Riou On $p$ -adic $L$ -functions attached to motives over $Q$ , Algebraic number theory (Advanced Studies in Pure Mathematics), Volume 17, Academic Press Inc., 1989, pp. 23-54 | DOI | MR

[17] John Coates; Warren Sinnott On $p$ -adic $L$ -functions over real quadratic fields, Invent. Math., Volume 25 (1974), pp. 253-279 | DOI | MR | Zbl

[18] Pierre Deligne; Kenneth A. Ribet Values of abelian $L$ -functions at negative integers over totally real fields, Invent. Math., Volume 59 (1980) no. 3, pp. 227-286 | DOI | MR

[19] Ellen Eischen $p$ -adic differential operators on automorphic forms on unitary groups, Ann. Inst. Fourier, Volume 62 (2012) no. 1, pp. 177-243 | DOI | Numdam | MR

[20] Ellen Eischen A $p$ -adic Eisenstein measure for vector-weight automorphic forms, Algebra Number Theory, Volume 8 (2014) no. 10, pp. 2433-2469 | DOI | MR | Zbl

[21] Ellen Eischen A $p$ -adic Eisenstein measure for unitary groups, J. Reine Angew. Math., Volume 699 (2015), pp. 111-142 | DOI | MR

[22] Ellen Eischen; Jessica Fintzen; Elena Mantovan; Ila Varma Differential operators and families of automorphic forms on unitary groups of arbitrary signature, Doc. Math., Volume 23 (2018), pp. 445-495 | MR

[23] Ellen Eischen; Michael Harris; Jianshu Li; Christopher Skinner $p$ -adic $L$ -functions for unitary groups, Forum Math. Pi, Volume 8 (2020), e9, 160 pages | DOI | MR

[24] Ellen Eischen; Elena Mantovan $p$ -adic families of automorphic forms in the $μ$ -ordinary setting, Am. J. Math., Volume 143 (2021) no. 1, pp. 1-51 | DOI | MR

[25] Ellen Eischen; Xin Wan $p$ -adic Eisenstein series and $L$ -functions of certain cusp forms on definite unitary groups, J. Inst. Math. Jussieu, Volume 15 (2016) no. 3, pp. 471-510 | DOI | MR

[26] Kurt Girstmair A theorem on the numerators of the Bernoulli numbers, Am. Math. Mon., Volume 97 (1990) no. 2, pp. 136-138 | DOI | MR

[27] Catherine Goldstein; Norbert Schappacher Séries d’Eisenstein et fonctions $L$ de courbes elliptiques à multiplication complexe, J. Reine Angew. Math., Volume 327 (1981), pp. 184-218 | MR

[28] Catherine Goldstein; Norbert Schappacher Conjecture de Deligne et $Γ$ -hypothèse de Lichtenbaum sur les corps quadratiques imaginaires, C. R. Math. Acad. Sci. Paris, Volume 296 (1983) no. 15, pp. 615-618 | MR

[29] Eyal Z. Goren Lectures on Hilbert modular varieties and modular forms, CRM Monograph Series, 14, American Mathematical Society, 2002, x+270 pages (With the assistance of Marc-Hubert Nicole) | MR

[30] Giacomo Graziani Modular sheaves of de Rham classes on Hilbert formal modular schemes for unramified primes (2020) | arXiv

[31] Ralph Greenberg Iwasawa theory for $p$ -adic representations, Algebraic number theory (Advanced Studies in Pure Mathematics), Volume 17, Academic Press Inc., 1989, pp. 97-137 | DOI | MR

[32] Ralph Greenberg Iwasawa theory for motives, $L$ -functions and arithmetic (Durham, 1989) (London Mathematical Society Lecture Note Series), Volume 153, Cambridge University Press, 1991, pp. 211-233 | DOI | MR

[33] Ralph Greenberg Iwasawa theory and $p$ -adic deformations of motives, Motives (Seattle, WA, 1991) (Proceedings of Symposia in Pure Mathematics), Volume 55, American Mathematical Society, 1994, pp. 193-223 | MR

[34] Günter Harder; Norbert Schappacher Special values of Hecke $L$ -functions and abelian integrals, Workshop Bonn 1984 (Bonn, 1984) (Lecture Notes in Mathematics), Volume 1111, Springer, 1985, pp. 17-49 | DOI | MR

[35] Michael Harris Special values of zeta functions attached to Siegel modular forms, Ann. Sci. Éc. Norm. Supér., Volume 14 (1981) no. 1, pp. 77-120 | DOI | Numdam | MR

[36] Kurt Hensel Über eine neue Begründung der Theorie der algebraischen Zahlen, J. Reine Angew. Math., Volume 128 (1905), pp. 1-32 | DOI | MR

[37] Haruzo Hida A $p$ -adic measure attached to the zeta functions associated with two elliptic modular forms. I, Invent. Math., Volume 79 (1985) no. 1, pp. 159-195 | DOI | MR

[38] Haruzo Hida On $p$ -adic $L$ -functions of $GL (2) \times GL (2)$ over totally real fields, Ann. Inst. Fourier, Volume 41 (1991) no. 2, pp. 311-391 | DOI | Numdam | MR

[39] Haruzo Hida $p$ -adic automorphic forms on Shimura varieties, Springer Monographs in Mathematics, Springer, 2004, xii+390 pages | DOI | MR

[40] Michael J. Hopkins Topological modular forms, the Witten genus, and the theorem of the cube, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) (1995), pp. 554-565 | DOI | MR

[41] Michael J. Hopkins Algebraic topology and modular forms, Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002) (2002), pp. 291-317 | MR

[42] Taku Ishii; Takayuki Oda A short history on investigation of the special values of zeta and $L$ -functions of totally real number fields, Automorphic forms and zeta functions, World Scientific, 2006, pp. 198-233 | DOI | MR

[43] Kenkichi Iwasawa Analogies between number fields and function fields, Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966), Belfer Graduate School of Science, 1969, pp. 203-208 | MR

[44] Kenkichi Iwasawa On $p$ -adic $L$ -functions, Ann. Math., Volume 89 (1969), pp. 198-205 | DOI | MR | Zbl

[45] Nicholas M. Katz $p$ -adic properties of modular schemes and modular forms, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) (Lecture Notes in Mathematics), Volume 350, Springer, 1973, pp. 69-190 | DOI | MR | Zbl

[46] Nicholas M. Katz $p$ -adic $L$ -functions for CM fields, Invent. Math., Volume 49 (1978) no. 3, pp. 199-297 | DOI | MR | Zbl

[47] Helmut Klingen Über die Werte der Dedekindschen Zetafunktion, Math. Ann., Volume 145 (1961/62), pp. 265-272 | DOI | MR | Zbl

[48] Daniel Kriz A New $p$ -adic Maass–Shimura operator and Supersingular Rankin–Selberg $p$ -adic $L$ -functions (2018) | arXiv | MR

[49] Tomio Kubota; Heinrich-Wolfgang Leopoldt Eine $p$ -adische Theorie der Zetawerte. I. Einführung der $p$ -adischen Dirichletschen $L$ -Funktionen, J. Reine Angew. Math., Volume 214/215 (1964), pp. 328-339 | MR

[50] Ernst E. Kummer Allgemeiner Beweis des Fermatschen Satzes, daßdie Gleichung $x^{λ} + y^{λ} = z^{λ}$ durch ganze Zahlen unlösbar ist, für alle diejenigen Potenz-Exponenten $λ$ welche ungerade Primzahlen sind und in den Zählern der ersten $1 / 2 (λ)$ Bernoullischen zahlen als Factoren nicht vorkommen, J. Reine Angew. Math., Volume 40 (1850), pp. 130-138 | DOI | MR

[51] Ernst E. Kummer Bestimmung der Anzahl nicht äquivalenter Classen für die aus $λ^{t} e n$ Wurzeln der Einheit gebildeten complexen Zahlen und die idealen Factoren derselben, J. Reine Angew. Math., Volume 40 (1850), pp. 93-116 | DOI | MR

[52] Ernst E. Kummer Über eine allgemeine Eigenschaft der rationalen Entwickelungscoefficienten einer bestimmten Gattung analytischer Functionen, J. Reine Angew. Math., Volume 41 (1851), pp. 368-372 | DOI | MR

[53] Zheng Liu $p$ -adic $L$ -functions for ordinary families on symplectic groups, J. Inst. Math. Jussieu, Volume 19 (2020) no. 4, pp. 1287-1347 | DOI | MR

[54] Zheng Liu; Giovanni Rosso Non-cuspidal Hida theory for Siegel modular forms and trivial zeros of $p$ -adic $L$ -functions, Math. Ann., Volume 378 (2020) no. 1-2, pp. 153-231 | DOI | MR

[55] David Loeffler; Vincent Pilloni; Christopher Skinner; Sarah Livia Zerbes Higher Hida theory and $p$ -adic L-functions for GSp(4) (2019) | arXiv

[56] Barry Mazur; Peter Swinnerton-Dyer Arithmetic of Weil curves, Invent. Math., Volume 25 (1974), pp. 1-61 | DOI | MR

[57] Barry Mazur; Andrew Wiles Class fields of abelian extensions of $ℚ$ , Invent. Math., Volume 76 (1984) no. 2, pp. 179-330 | DOI | MR

[58] Alexei A. Panchishkin Two variable $p$ -adic $L$ functions attached to eigenfamilies of positive slope, Invent. Math., Volume 154 (2003) no. 3, pp. 551-615 | DOI | MR

[59] Vincent Pilloni Higher coherent cohomology and $p$ -adic modular forms of singular weight, Duke Math. J., Volume 169 (2020) no. 9, pp. 1647-1807 | DOI | MR

[60] Robert A. Rankin The scalar product of modular forms, Proc. Lond. Math. Soc., Volume 2 (1952), pp. 198-217 | DOI | MR

[61] Michael Rapoport Compactifications de l’espace de modules de Hilbert–Blumenthal, Compos. Math., Volume 36 (1978) no. 3, pp. 255-335 | MR

[62] Kenneth A. Ribet $p$ -adic interpolation via Hilbert modular forms, Algebraic geometry (Humboldt State Univ., Arcata, Calif., 1974) (Proceedings of Symposia in Pure Mathematics), Volume 29, American Mathematical Society, 1975, pp. 581-592 | DOI | MR

[63] Jean-Pierre Serre Formes modulaires et fonctions zêta $p$ -adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972) (Lecture Notes in Mathematics), Volume 350, Springer, 1973, pp. 191-268 | DOI | MR

[64] Ehud de Shalit; Eyal Z. Goren A theta operator on Picard modular forms modulo an inert prime, Res. Math. Sci., Volume 3 (2016), 28, 65 pages | DOI | MR

[65] Ehud de Shalit; Eyal Z. Goren Theta operators on unitary Shimura varieties, Algebra Number Theory, Volume 13 (2019) no. 8, pp. 1829-1877 | DOI | MR

[66] Goro Shimura On some arithmetic properties of modular forms of one and several variables, Ann. Math., Volume 102 (1975) no. 3, pp. 491-515 | DOI | MR

[67] Goro Shimura The special values of the zeta functions associated with cusp forms, Commun. Pure Appl. Math., Volume 29 (1976) no. 6, pp. 783-804 | DOI | MR

[68] Goro Shimura Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, 82, American Mathematical Society, 2000, x+302 pages | MR

[69] Carl Ludwig Siegel Berechnung von Zetafunktionen an ganzzahligen Stellen, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., Volume 1969 (1969), pp. 87-102 | MR

[70] Carl Ludwig Siegel Über die Fourierschen Koeffizienten von Modulformen, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl., Volume 1970 (1970), pp. 15-56 | MR

[71] Karl G. C. von Staudt Beweis eines Lehrsatzes, die Bernoullischen Zahlen betreffen, J. Reine Angew. Math., Volume 21 (1840), pp. 372-374 | DOI | MR

[72] Karl G. C. von Staudt De numeris Bernoullianis, Universitatsschrift, Erlangen, 1845

[73] Lawrence C. Washington Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer, 1997, xiv+487 pages | DOI | MR

[74] André Weil Elliptic functions according to Eisenstein and Kronecker, Classics in Mathematics, Springer, 1999, viii+93 pages (Reprint of the 1976 original) | MR

Cité par Sources :