Ranks of Selmer groups in an analytic family

Bellaïche, Joël

doi:10.1090/S0002-9947-2012-05504-8

Ranks of Selmer groups in an analytic family
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by Joël Bellaïche PDF

Trans. Amer. Math. Soc. 364 (2012), 4735-4761 Request permission

Abstract:

We study the variation of the dimension of the Bloch-Kato Selmer group of a $p$-adic Galois representation of a number field that varies in a refined family. We show that, if we restrict ourselves to representations that are, at every place dividing $p$, crystalline, non-critically refined, and with a fixed number of non-negative Hodge-Tate weights, then the dimension of the Selmer group varies essentially lower-semi-continuously. This allows us to prove lower bounds for Selmer groups “by continuity”, and in particular to deduce from a result of Bellaïche and Chenevier that the $p$-adic Selmer group of a modular eigenform of weight $2$ of sign $-1$ has rank at least $1$.

References

Laurent Berger and Pierre Colmez, Familles de représentations de de Rham et monodromie $p$-adique, Astérisque 319 (2008), 303–337 (French, with English and French summaries). Représentations $p$-adiques de groupes $p$-adiques. I. Représentations galoisiennes et $(\phi ,\Gamma )$-modules. MR 2493221
L. Berger, C. Breuil, P. Colmez (editors), $p$-adic representations of $p$-adic groups I: Galois representations and $(\phi ,\Gamma )$-modules, Astérisque 319 (2008).
Joël Bellaïche and Gaëtan Chenevier, Families of Galois representations and Selmer groups, Astérisque 324 (2009), xii+314 (English, with English and French summaries). MR 2656025
Bloch & Kato, Tamagawa Numbers of Motives in The Gorthendieck festschrift, vol. 1, Progress in Math 89, Birkhauser, 1990.
S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961, DOI 10.1007/978-3-642-52229-1
Henri Carayol, Formes modulaires et représentations galoisiennes à valeurs dans un anneau local complet, $p$-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991) Contemp. Math., vol. 165, Amer. Math. Soc., Providence, RI, 1994, pp. 213–237 (French). MR 1279611, DOI 10.1090/conm/165/01601
G. Chenevier, Une application des variétés de Hecke des groupes unitaires, to appear in \cite{book}, available on http://www.math.polytechnique.fr/ chenevier/pub.html.
Pierre Colmez, Représentations triangulines de dimension 2, Astérisque 319 (2008), 213–258 (French, with English and French summaries). Représentations $p$-adiques de groupes $p$-adiques. I. Représentations galoisiennes et $(\phi ,\Gamma )$-modules. MR 2493219
T. & V. Dokchitser Root numbers and parity of ranks of elliptic curves, preprint, June 2009 (available on arxiv).
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Ralph Greenberg, On the structure of certain Galois cohomology groups, Doc. Math. Extra Vol. (2006), 335–391. MR 2290593
M. Harris et al., Stabilisation de la formule des traces, variétés de Shimura, et applications arithmétiques. Book project, available on http://www.institut.math.jussieu.fr/projets/fa/bp0.html.
Uwe Jannsen, On the $l$-adic cohomology of varieties over number fields and its Galois cohomology, Galois groups over $\textbf {Q}$ (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 16, Springer, New York, 1989, pp. 315–360. MR 1012170, DOI 10.1007/978-1-4613-9649-9_{5}
K. Kedlaya, Slope filtration and $(\phi ,\Gamma )$-modules in families, preprint, available on http://math.mit.edu/˜kedlaya/papers/families.pdf
K. Kedlaya and R. Liu, On families of $(\phi ,\Gamma )$-modules, arXiv:0812.0112v2 (2009).
Mark Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math. 153 (2003), no. 2, 373–454. MR 1992017, DOI 10.1007/s00222-003-0293-8
Byoung Du Kim, The parity conjecture for elliptic curves at supersingular reduction primes, Compos. Math. 143 (2007), no. 1, 47–72. MR 2295194, DOI 10.1112/S0010437X06002569
Michel Lazard, Les zéros des fonctions analytiques d’une variable sur un corps valué complet, Inst. Hautes Études Sci. Publ. Math. 14 (1962), 47–75 (French). MR 152519
Ruochuan Liu, Cohomology and duality for $(\phi ,\Gamma )$-modules over the Robba ring, Int. Math. Res. Not. IMRN 3 (2008), Art. ID rnm150, 32. MR 2416996, DOI 10.1093/imrn/rnm150
R. Liu, Slope Filtrations in Family, arxiv:0809.0331 (2008).
R. Liu, Families of $(\phi ,\Gamma )$-modules, arxiv:0812.0112 (2008).
R. Liu, personal communication, emails to the author dated from June 2009 to July 2010.
R. Liu, On Families of Refined p-adic Representations, preprint in preparation, http://www.math.mcgill.ca/rliu/.
Sophie Morel, On the cohomology of certain noncompact Shimura varieties, Annals of Mathematics Studies, vol. 173, Princeton University Press, Princeton, NJ, 2010. With an appendix by Robert Kottwitz. MR 2567740, DOI 10.1515/9781400835393
Jan Nekovář, On the parity of ranks of Selmer groups. II, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 2, 99–104 (English, with English and French summaries). MR 1813764, DOI 10.1016/S0764-4442(00)01808-5
Jan Nekovář, Selmer complexes, Astérisque 310 (2006), viii+559 (English, with English and French summaries). MR 2333680
Jan Nekovář, On the parity of ranks of Selmer groups. III, Doc. Math. 12 (2007), 243–274. MR 2350290
Louise Nyssen, Pseudo-représentations, Math. Ann. 306 (1996), no. 2, 257–283 (French). MR 1411348, DOI 10.1007/BF01445251
Jonathan Livaudais Pottharst, Selmer growth and a “triangulordinary” local condition, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–Harvard University. MR 2711680
Raphaël Rouquier, Caractérisation des caractères et pseudo-caractères, J. Algebra 180 (1996), no. 2, 571–586 (French). MR 1378546, DOI 10.1006/jabr.1996.0083
Karl Rubin, Euler systems, Annals of Mathematics Studies, vol. 147, Princeton University Press, Princeton, NJ, 2000. Hermann Weyl Lectures. The Institute for Advanced Study. MR 1749177, DOI 10.1515/9781400865208
Christopher Skinner and Eric Urban, Sur les déformations $p$-adiques de certaines représentations automorphes, J. Inst. Math. Jussieu 5 (2006), no. 4, 629–698 (French, with English and French summaries). MR 2261226, DOI 10.1017/S147474800600003X
S. W Shin, Odd-dimensional Galois representations arising from some conpact Shiura varieties, preprint.
John Tate, Relations between $K_{2}$ and Galois cohomology, Invent. Math. 36 (1976), 257–274. MR 429837, DOI 10.1007/BF01390012