Skip to main content Accessibility help
×
×
Home

Congruences with Eisenstein series and $\unicode[STIX]{x1D707}$ -invariants

  • Joël Bellaïche (a1) and Robert Pollack (a2)
Abstract

We study the variation of $\unicode[STIX]{x1D707}$ -invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the $p$ -adic zeta function. This lower bound forces these $\unicode[STIX]{x1D707}$ -invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When $U_{p}-1$ generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the $p$ -adic $L$ -function is simply a power of $p$ up to a unit (i.e.  $\unicode[STIX]{x1D706}=0$ ). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms.

Copyright
References
Hide All
[AS86] Ash, A. and Stevens, G., Modular forms in characteristic l and special values of their L-functions , Duke Math. J. 53 (1986), 849868.
[Bel09] Bellaïche, J., Ribet’s lemma, generalizations, and pseudocharacters, Preprint (2009),http://people.brandeis.edu/jbellaic/RibetHawaii3.pdf.
[Bel12] Bellaïche, J., Critical p-adic L-functions , Invent. Math. 189 (2012), 160.
[BD15] Bellaïche, J. and Dasgupta, S., The p-adic L-functions of evil Eisenstein series , Compos. Math. 151 (2015), 9991040.
[Col00] Colmez, P., Fonctions L p-adiques , in Séminaire Bourbaki, Vol. 1998/99, Exp. No. 851, Astérisque, vol. 266 (Société Mathématique de France, 2000), 3, 21–58.
[EPW06] Emerton, M., Pollack, R. and Weston, T., Variation of Iwasawa invariants in Hida families , Invent. Math. 163 (2006), 523580.
[FW79] Ferrero, B. and Washington, L. C., The Iwasawa invariant 𝜇 p vanishes for abelian number fields , Ann. of Math. (2) 109 (1979), 377395.
[FK12] Fukaya, T. and Kato, K., On conjectures of Sharifi, Preprint (2012),http://math.ucla.edu/sharifi/sharificonj.pdf.
[Gre89] Greenberg, R., Iwasawa theory for p-adic representations , in Algebraic number theory, Advanced Studies in Pure Mathematics, vol. 17 (Academic Press, Boston, 1989), 97137.
[Gre99] Greenberg, R., Iwasawa theory for elliptic curves , in Arithmetic theory of elliptic curves (Cetraro, 1997), Lecture Notes in Mathematics, vol. 1716 (Springer, Berlin, 1999), 51144.
[GS93] Greenberg, R. and Stevens, G., p-adic L-functions and p-adic periods of modular forms , Invent. Math. 111 (1993), 407447.
[GV00] Greenberg, R. and Vatsal, V., On the Iwasawa invariants of elliptic curves , Invent. Math. 142 (2000), 1763.
[HHO17] Hart, W., Harvey, D. and Ong, W., Irregular primes to two billion , Math. Comput. 86 (2017), 30313049.
[Hid86] Hida, H., Iwasawa modules attached to congruences of cusp forms , Ann. Sci. Éc. Norm. Supér (4) 19 (1986), 231273.
[Hid12] Hida, H., Geometric modular forms and elliptic curves, second edition (World Scientific, Hackensack, NJ, 2012).
[Kat04] Kato, K., p-adic Hodge theory and values of zeta functions of modular forms , in Cohomologies p-adiques et applications arithmétiques, III, Astérisque, vol. 295 (Société Mathématique de France, 2004), ix, 117–290.
[Kit94] Kitagawa, K., On standard p-adic L-functions of families of elliptic cusp forms , in p-adic monodromy and the Birch and Swinnerton–Dyer conjecture (Boston, MA, 1991), Contemporary Mathematics, vol. 165 (American Mathematical Society, Providence, RI, 1994), 81110.
[Kur93] Kurihara, M., Ideal class groups of cyclotomic fields and modular forms of level 1 , J. Number Theory 45 (1993), 281294.
[Mah58] Mahler, K., An interpolation series for continuous functions of a p-adic variable , J. Reine Angew. Math. 199 (1958), 2334.
[Maz78] Mazur, B., Modular curves and the Eisenstein ideal , Publ. Math. Inst. Hautes Études Sci. (1978), 33186, 1977.
[Oht99] Ohta, M., Ordinary p-adic étale cohomology groups attached to towers of elliptic modular curves , Compos. Math. 115 (1999), 241301.
[Oht00] Ohta, M., Ordinary p-adic étale cohomology groups attached to towers of elliptic modular curves. II , Math. Ann. 318 (2000), 557583.
[Oht03] Ohta, M., Congruence modules related to Eisenstein series , Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), 225269.
[Oht05] Ohta, M., Companion forms and the structure of p-adic Hecke algebras , J. Reine Angew. Math. 585 (2005), 141172.
[PS11] Pollack, R. and Stevens, G., Overconvergent modular symbols and p-adic L-functions , Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 142.
[PS13] Pollack, R. and Stevens, G., Critical slope p-adic L-functions , J. Lond. Math. Soc. (2) 87 (2013), 428452.
[Rib76] Ribet, K. A., A modular construction of unramified p-extensions of Q(𝜇 p ) , Invent. Math. 34 (1976), 151162.
[Sha11] Sharifi, R., A reciprocity map and the two-variable p-adic L-function , Ann. of Math. (2) 173 (2011), 251300.
[Shi76] Shimura, G., The special values of the zeta functions associated with cusp forms , Comm. Pure Appl. Math. 29 (1976), 783804.
[SU14] Skinner, C. and Urban, E., The Iwasawa main conjectures for GL2 , Invent. Math. 195 (2014), 1277.
[Ste94] Stevens, G., Rigid analytic modular symbols, Preprint (1994),http://math.bu.edu/people/ghs/preprints/OC-Symbs-04-94.pdf.
[Til97] Tilouine, J., Hecke algebras and the Gorenstein property , in Modular forms and Fermat’s last theorem (Boston, MA, 1995) (Springer, New York, 1997), 327342.
[Vat99] Vatsal, V., Canonical periods and congruence formulae , Duke Math. J. 98 (1999), 397419.
[Wak15] Wake, P., Hecke algebras associated to 𝛬-adic modular forms , J. Reine Angew. Math. 700 (2015), 113128.
[Wil90] Wiles, A., The Iwasawa conjecture for totally real fields , Ann. of Math. (2) 131 (1990), 493540.
[Wil95] Wiles, A., Modular elliptic curves and Fermat’s last theorem , Ann. of Math. (2) 141 (1995), 443551.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed