An introduction to Eisenstein measures

Eischen, Ellen

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

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.

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

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

[2] Andreatta, Fabrizio; Iovita, Adrian; Pilloni, Vincent 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] Aycock, Jon Families of differential operators for overconvergent Hilbert modular forms, 2021 (In preparation)

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

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

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

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

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

[12] Cassou-Noguès, Pierrette 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] Charollois, Pierre; Dasgupta, Samit Integral Eisenstein cocycles on ${\mathbf{GL}}_n$, 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] Clausen, Thomas Theorem, Astron. Nachr., Volume 17 (1840) no. 22, pp. 351-352 | DOI

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

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

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

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

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

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

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

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

[25] Eischen, Ellen; Wan, Xin $p$-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] Girstmair, Kurt A theorem on the numerators of the Bernoulli numbers, Am. Math. Mon., Volume 97 (1990) no. 2, pp. 136-138 | DOI | MR

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

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

[29] Goren, Eyal Z. 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] Graziani, Giacomo Modular sheaves of de Rham classes on Hilbert formal modular schemes for unramified primes (2020) | arXiv

[31] Greenberg, Ralph 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] Greenberg, Ralph 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] Greenberg, Ralph 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] Harder, Günter; Schappacher, Norbert 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] Harris, Michael 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] Hensel, Kurt Über eine neue Begründung der Theorie der algebraischen Zahlen, J. Reine Angew. Math., Volume 128 (1905), pp. 1-32 | DOI | MR

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

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

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

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

[42] Ishii, Taku; Oda, Takayuki 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] Iwasawa, Kenkichi 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] Iwasawa, Kenkichi On $p$-adic $L$-functions, Ann. Math., Volume 89 (1969), pp. 198-205 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

[62] Ribet, Kenneth A. $p$-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] Serre, Jean-Pierre 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] de Shalit, Ehud; Goren, Eyal Z. A theta operator on Picard modular forms modulo an inert prime, Res. Math. Sci., Volume 3 (2016), 28, 65 pages | DOI | MR

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

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

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

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

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

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

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

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