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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.
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Analog Category and Complexity
We study probabilistic variants of the Lusternik–Schnirelmann category and topological complexity, which bound the classical invariants from below. We present a number of computations illustrating both wide agreement and wide disagreement with the classical notions. In the aspherical case, where our invariants are group invariants, we establish a counterpart of the Eilenberg– Ganea theorem in the torsion-free case, as well as a contrasting universal upper bound in the finite case.
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- Award ID(s):
- 2105553
- PAR ID:
- 10625959
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- SIAM journal on applied algebra and geometry
- ISSN:
- 2470-6566
- Subject(s) / Keyword(s):
- Algebraic Topology, Robotics, Complexity, group theory
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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