Article contents
p–adic Families of Cohomological Modular Forms for Indefinite Quaternion Algebras and the Jacquet–LanglandsCorrespondence
Published online by Cambridge University Press: 20 November 2018
Abstract
Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.
We use the method of Ash and Stevens to prove the existence of small slope $p$-adic families of cohomological modular forms for an indefinite quaternion algebra $B$. We prove that the Jacquet–Langlands correspondence relating modular forms on $\text{G}{{\text{L}}_{\text{2}}}/\mathbb{Q}$ and cohomomological modular forms for $B$ is compatible with the formation of $p$-adic families. This result is an analogue of a theorem of Chenevier concerning definite quaternion algebras.
Keywords
- Type
- Research Article
- Information
- Copyright
- Copyright © Canadian Mathematical Society 2016
References
[1]
Andreatta, F., Iovita, A., and Pilloni, V., p-adic families of Siegel modular cuspforms.
Ann. of Math.
181(2015), no. 2, 623–697. http://dx.doi.Org/10.4007/annals.2015.181.2.5
Google Scholar
[2]
Andreatta, F., Iovita, A., and Stevens, G., Overconvergent modular sheaves and modular forms.
Israel J. Math.
201(2014), no. 1, 299–359. http://dx.doi.org/10.1007/s11856-014-1045-8
Google Scholar
[3]
Ash, A. and Stevens, G., sdecompositions.Preprint, available at http://math.bu.edu/people/ghs/research.html
Google Scholar
[4]
Ash, A., p-adic deformation of arithmetic cohomology.Preprint, available at http://math.bu.edu/people/ghs/research.html
Google Scholar
[5] S.|Bosch, U.|Guntzer, and R. |Remmert, Non-archimedean analysis.
Grundlehren der Mathematischen Wissenschaften,
261, Springer-Verlag, Berlin, 1984. http://dx.doi.org/10.1007/978-3-642-52229-1
Google Scholar
[6]
Chenevier, G., Une correspondance de Jaquet-Langlands p-adique.
Duke Math. Journal
126(2005),161–194. http://dx.doi.org/10.1215/S0012-7094-04-12615-6
Google Scholar
[7]
Coleman, R., p-adic Banach spaces and families of modular forms.
Invent. Math.
127(1997), no. 3, 417–479. http://dx.doi.org/10.1OO7/sOO222OO5O127
Google Scholar
[8]
Darmon, H., Integration of Jp x J and arithmetic applications.
Ann. of Math.
154(2001), no. 3, 589–639.
http://dx.doi.org/10.2307/3062142
Google Scholar
[9]
Dasgupta, S. and Greenberg, M., L-invariants and Shimura curves.
Algebra and Number Theory
6(2012), no. 3, 455–485. http://dx.doi.Org/10.2140/ant.2012.6.455
Google Scholar
[10]
Emerton, M., p-adic L-functions and unitary completions of representations of p-adic reductive groups.
Duke Math. J.
130(2005), no. 2, 353–392.Google Scholar
[11]
Emerton, M., On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms.
Invent. Math.
164(2006), no. 1, 1–84. http://dx.doi.org/10.1007/s00222-005-0448-x
Google Scholar
[12] M.|Greenberg, Stark-Heegner points and the cohomology of quaternionic Shimura varieties.
Duke Math. J.
147(2009), no. 3, 541–575.http://dx.doi.org/10.1215/00127094-2009-017
Google Scholar
[13]
Greenberg, M., Seveso, M. A., and Shahabi, S., Modular p-adic L-functions attached to real quadratic fields and arithmetic applications.
J. Reine Angew. Math., to appear. http://dx.doi.Org/10.1 515/crelle-2O1 4-0088
Google Scholar
[14]
Newton, J.
Completed cohomology of Shimura curves and a p-adic Jacquet-Langlands correspondence.
Math. Ann.
355(2013), no. 2, 729–763. http://dx.doi.Org/10.1007/s00208-012-0796-y
Google Scholar
[15]
Rotger, V. and Seveso, M. A., L-invariants and Darmon cycles attached to modular forms.
J. Eur.Math. Soc.
14(2012), no. 6, 1955–1999. http://dx.doi.org/10.41 71/JEMS/352
Google Scholar
[16]
Schneider, P., Nonarchimedean functional analysis.
Springer Monographs in Mathematics,Springer-Verlag, Berlin, 2002. http://dx.doi.org/10.1007/978-3-662-04728-6
Google Scholar
[17]
Schneider, P. and Teitelbaum, J., Locally analytic distributions and p-adic representation theory,with applications to GL2.
J. Amer. Math. Soc.
15(2002), no. 2, 443–468. http://dx.doi.Org/10.1090/S0894-0347-01-00377-0
Google Scholar
[18]
Serre, J.-P., Endomorphismes complètement continus des espace de Banach p-adiques.
Inst. Hautes Études Sci. Publ. Math.
12(1962), 69–85.Google Scholar
[19]
Seveso, M. A., p-adic L-functions and the rationality of Darmon cycles.
Canad. J. Math.
64(2012), no. 5, 1122–1181. http://dx.doi.org/10.4153/CJM-2011-076-8
Google Scholar
[20]
Seveso, M. A., The Teitelbaum conjecture in the indefinite setting.
Amer. J. Math.
135(2013), no. 6, 1525–1557. http://dx.doi.org/10.1353/ajm.2013.0055M
Google Scholar
[21]
Shimura, G., Introduction to the arithmetic theory of automorphic functions.
Publications of the Mathematical Society of Japan, 11, Iwanami Shoten, Publishers, Tokyo; Princeton University Press, Princeton, NJ, 1971.Google Scholar
[22]
Stevens, G., Rigid analytic modular symbols. Preprint, available at http://math.bu.edu/people/ghs/research.html
Google Scholar
You have
Access
- 3
- Cited by