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Creators/Authors contains: "Bellaïche, Joël"

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  1. ; (, Compositio Mathematica)
    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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  2. (, Advances in mathematics)
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