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Congruences with Eisenstein series and
$\unicode[STIX]{x1D707}$-invariants
- Part of
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- Journal:
- Compositio Mathematica / Volume 155 / Issue 5 / May 2019
- Published online by Cambridge University Press:
- 10 April 2019, pp. 863-901
- Print publication:
- May 2019
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- Article
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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.
