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

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

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.

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Compositio Mathematica
  • ISSN: 0010-437X
  • EISSN: 1570-5846
  • URL: /core/journals/compositio-mathematica
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