Mar 2024· Memoirs of the American Mathematical Society· 3 citations· ⚡ 1 influential· 65 references
Mathematics
Abstract
<p>
Our objective in the present work is to develop a fairly complete arithmetic theory of critical
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-adic
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-functions on the eigencurve. To this end, we carry out the following tasks:
</p>
<p>
We give an “étale” construction of Bellaïche’s
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-adic
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-functions at a
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-critical point on the cuspidal Coleman–Mazur–Buzzard eigencurve.
</p>
<p>
We introduce the algebraic counterparts of these objects (which arise as appropriately defined Selmer complexes) and develop Iwasawa theory in this context, including a definition of an Iwasawa theoretic
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-invariant
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.
</p>
<p>We formulate the (punctual) critical main conjectures, and study its relationship with its slope-zero counterpart. Along the way, we also develop descent theory (paralleling Perrin-Riou’s work).</p>
<p>
We introduce what we call
<italic>thick</italic>
(Iwasawa theoretic) fundamental line and the
<italic>thick</italic>
Selmer complex to counter Bellaïche’s secondary
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-adic
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-functions. This allows us to formulate an infinitesimal thickening of the Iwasawa main conjecture, and we observe that it implies both slope-zero and punctual critical main conjectures, but it seems stronger than both.
</p>
<p>
We establish an
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-adic leading term formula for the two-variable
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-adic
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-function over the affinoid neighborhood
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<p>
We combine the relative trace formula with analytic methods to obtain zero density estimates for
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-functions in various families of automorphic representations for
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that show their strength close to the critical line. Applications include strong bounds for the average analytic rank of these
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-functions at the central point and average equidistribution results for the imaginary parts of the zeros.
</p>
Valentin Blomer, Jesse Thorner· Transactions of the American...· 0 citations
<jats:p>
We explore numbered Boolean algebras over classes
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of arithmetical and analytical hierarchies. We show the existence and uniqueness (up to computable isomorphism) of universal Boolean
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-algebras, determine the classes in which such algebras exist, and classify the universal algebras up to isomorphism. As applications, we characterize Tarski–Lindenbaum algebras of four semantic classes of models of a given finite rich signature, namely, the class of all countable saturated models having decidable
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-stable theories, the class of prime models with decidable theories and first-order definable elements, the class of models with decidable non-finitely axiomatizable theories, and the class of models with finitely axiomatizable theories; it is shown that the first three of these algebras are universal Boolean algebras over hierarchy classes
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,
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, and
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, respectively, and the fourth one is a countable atomic Boolean
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-algebra whose quotient modulo the Frèchet ideal is a
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-universal Boolean
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-algebra.
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