## Algebra & Number Theory

### A $p$-adic Eisenstein measure for vector-weight automorphic forms

Ellen Eischen

#### Abstract

We construct a $p$-adic Eisenstein measure with values in the space of vector-weight $p$-adic automorphic forms on certain unitary groups. This measure allows us to $p$-adically interpolate special values of certain vector-weight $C∞$ automorphic forms, including Eisenstein series, as their weights vary. This completes a key step toward the construction of certain $p$-adic $L$-functions.

We also explain how to extend our methods to the case of Siegel modular forms and how to recover Nicholas Katz’s $p$-adic families of Eisenstein series for Hilbert modular forms.

#### Article information

Source
Algebra Number Theory, Volume 8, Number 10 (2014), 2433-2469.

Dates
Revised: 22 September 2014
Accepted: 3 November 2014
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730330

Digital Object Identifier
doi:10.2140/ant.2014.8.2433

Mathematical Reviews number (MathSciNet)
MR3298545

Zentralblatt MATH identifier
06387028

#### Citation

Eischen, Ellen. A $p$-adic Eisenstein measure for vector-weight automorphic forms. Algebra Number Theory 8 (2014), no. 10, 2433--2469. doi:10.2140/ant.2014.8.2433. https://projecteuclid.org/euclid.ant/1513730330

#### References

• M. Ando, M. Hopkins, and C. Rezk, “Multiplicative orientations of $KO$-theory and of the spectrum of topological modular forms”, preprint, 2010, http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf.
• J. Cogdell, “Lectures on integral representation of $L$-functions”, unpublished notes, 2006, http://www.math.osu.edu/~cogdell/columbia-www.pdf.
• M. Courtieu and A. Panchishkin, Non-Archimedean $L$-functions and arithmetical Siegel modular forms, 2nd. ed., Lecture Notes in Mathematics 1471, Springer, Berlin, 2004.
• P. Deligne and K. A. Ribet, “Values of abelian $L$-functions at negative integers over totally real fields”, Invent. Math. 59:3 (1980), 227–286.
• E. E. Eischen, “$p$-adic differential operators on automorphic forms on unitary groups”, Ann. Inst. Fourier $($Grenoble$)$ 62:1 (2012), 177–243.
• E. E. Eischen, “A $p$-adic Eisenstein measure for unitary groups”, J. reine angew. Math. (online publication April 2013).
• E. E. Eischen, “Differential operators, pullbacks, and families of automorphic forms”, preprint, 2014.
• E. E. Eischen, M. Harris, J.-S. Li, and C. M. Skinner, “$p$-adic $L$-functions for unitary groups”, In preparation.
• S. Gelbart, I. Piatetski-Shapiro, and S. Rallis, Explicit constructions of automorphic $L$-functions, Lecture Notes in Mathematics 1254, Springer, Berlin, 1987.
• M. Harris, “Special values of zeta functions attached to Siegel modular forms”, Ann. Sci. École Norm. Sup. $(4)$ 14:1 (1981), 77–120.
• M. Harris, J.-S. Li, and C. M. Skinner, “$p$-adic $L$-functions for unitary Shimura varieties, I: Construction of the Eisenstein measure”, Doc. Math. Extra Vol. (2006), 393–464.
• H. Hida, Geometric modular forms and elliptic curves, World Scientific, River Edge, NJ, 2000.
• H. Hida, $p$-adic automorphic forms on Shimura varieties, Springer, New York, 2004.
• H. Hida, “$p$-adic automorphic forms on reductive groups”, pp. 147–254 in Automorphic forms, I, Astérisque 298, Société Mathématique de France, Paris, 2005.
• M. J. Hopkins, “Topological modular forms, the Witten genus, and the theorem of the cube”, pp. 554–565 in Proceedings of the International Congress of Mathematicians (Zürich, 1994), vol. 1, edited by S. D. Chatterji, Birkhäuser, Basel, 1995.
• M. J. Hopkins, “Algebraic topology and modular forms”, pp. 291–317 in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), edited by T. Li, Higher Ed. Press, Beijing, 2002.
• J. C. Jantzen, Representations of algebraic groups, Pure and Applied Mathematics 131, Academic Press, Boston, 1987.
• N. M. Katz, “$p$-adic properties of modular schemes and modular forms”, pp. 69–190 in Modular functions of one variable (Antwerp, 1972), vol. III, edited by W. Kuyk and J.-P. Serre, Lecture Notes in Mathematics 350, Springer, Berlin, 1973.
• N. M. Katz, “$p$-adic $L$-functions for CM fields”, Invent. Math. 49:3 (1978), 199–297.
• K.-W. Lan, “Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties”, J. Reine Angew. Math. 664 (2012), 163–228.
• K.-W. Lan, Arithmetic compactifications of PEL-type Shimura varieties, London Mathematical Society Monographs 36, Princeton University Press, Princeton, NJ, 2013.
• A. A. Panchishkin, “The Maass–Shimura differential operators and congruences between arithmetical Siegel modular forms”, Mosc. Math. J. 5:4 (2005), 883–918, 973–974.
• J.-P. Serre, “Formes modulaires et fonctions zêta $p$-adiques”, pp. 69–190 in Modular functions of one variable (Antwerp, 1972), vol. III, edited by W. Kuyk and J.-P. Serre, Lecture Notes in Mathematics 350, Springer, Berlin, 1973.
• G. Shimura, “On Eisenstein series”, Duke Math. J. 50:2 (1983), 417–476. http://msp.org/idx/mr/84k:10019MR 84k:10019
• G. Shimura, “Differential operators and the singular values of Eisenstein series”, Duke Math. J. 51:2 (1984), 261–329.
• G. Shimura, “On differential operators attached to certain representations of classical groups”, Invent. Math. 77:3 (1984), 463–488.
• G. Shimura, Euler products and Eisenstein series, CBMS Regional Conference Series in Mathematics 93, Amer. Math. Soc., Providence, RI, 1997.
• G. Shimura, Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs 82, Amer. Math. Soc., Providence, RI, 2000.
• V. Tan, “Poles of Siegel Eisenstein series on ${\rm U}(n,n)$”, Canad. J. Math. 51:1 (1999), 164–175.