1,306 research outputs found
Towards a statement of the S-adic conjecture through examples
The -adic conjecture claims that there exists a condition such that a
sequence has a sub-linear complexity if and only if it is an -adic sequence
satisfying Condition for some finite set of morphisms. We present an
overview of the factor complexity of -adic sequences and we give some
examples that either illustrate some interesting properties or that are
counter-examples to what could be believed to be "a good Condition ".Comment: 2
Towards a theory of local Shimura varieties
This is a survey article that advertizes the idea that there should exist a
theory of p-adic local analogues of Shimura varieties. Prime examples are the
towers of rigid-analytic spaces defined by Rapoport-Zink spaces, and we also
review their theory in the light of this idea. We also discuss conjectures on
the -adic cohomology of local Shimura varieties.Comment: 53 page
Height one specializations of Selmer groups
We provide applications to studying the behavior of Selmer groups under
specialization. We consider Selmer groups associated to four dimensional Galois
representations coming from (i) the tensor product of two cuspidal Hida
families and , (ii) its cyclotomic deformation, (iii) the tensor product
of a cusp form and the Hida family , where is a classical
specialization of with weight . We prove control theorems to
relate (a) the Selmer group associated to the tensor product of Hida families
and to the Selmer group associated to its cyclotomic deformation and
(b) the Selmer group associated to the tensor product of and to the
Selmer group associated to the tensor product of and . On the analytic
side of the main conjectures, Hida has constructed one variable, two variable
and three variable Rankin-Selberg -adic -functions. Our specialization
results enable us to verify that Hida's results relating (a) the two variable
-adic -function to the three variable -adic -function and (b) the
one variable -adic -function to the two variable -adic -function
and our control theorems for Selmer groups are completely consistent with the
main conjectures.Comment: Incorporated changes suggested by the referee. Accepted for
publication in Annales de l'Institut Fourie
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