Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Iwasawa theory and the Eisenstein ideal   In this paper, we relate three objects. The first is a particular value of a
cup product in the cohomology of the Galois group of the maximal unramified
outside p extension of a cyclotomic field containing the pth roots of unity.
The second is an Iwasawa module over a nonabelian extension of the rationals, a
subquotient of the maximal pro-p abelian unramified completely split at p
extension of a certain pro-p Kummer extension of a cyclotomic field that
contains all p-power roots of unity. The third is the quotient of an Eisenstein
ideal in an ordinary Hecke algebra of Hida by the square of the Eisenstein
ideal and the element given by the pth Hecke operator minus one. For the
relationship between the latter two objects, we employ the work of Ohta, in
which he considered a certain Galois action on an inverse limit of cohomology
groups to reestablish the Main Conjecture (for p at least 5) in the spirit of
the Mazur-Wiles proof. For the relationship between the former two objects, we
construct an analogue to the global reciprocity map for extensions with
restricted ramification. These relationships, and a computation in the Hecke
algebra, allow us to prove an earlier conjecture of McCallum and the author on
the surjectivity of a pairing formed from the cup product for p < 1000. We give
one other application, determining the structure of Selmer groups of the
modular representation considered by Ohta modulo the Eisenstein ideal.

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The categories of this paper are: alg-geom
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-th
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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The categories of this paper are: solv-int