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• W2896825935 abstract "For a totally real field <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=application/x-tex>F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a finite extension <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper F> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>F</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbf {F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=bold upper F Subscript p> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>F</mml:mi> </mml:mrow> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathbf {F}_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and a Galois character <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=chi colon upper G Subscript upper F Baseline right-arrow bold upper F Superscript times> <mml:semantics> <mml:mrow> <mml:mi>χ<!-- χ --></mml:mi> <mml:mo>:</mml:mo> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>F</mml:mi> </mml:msub> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>F</mml:mi> </mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>×<!-- × --></mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>chi : G_F to mathbf {F}^{times }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> unramified away from a finite set of places <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma superset-of left-brace German p bar p right-brace> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:mo>⊃<!-- ⊃ --></mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=fraktur>p</mml:mi> </mml:mrow> <mml:mo>∣<!-- ∣ --></mml:mo> <mml:mi>p</mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Sigma supset {mathfrak {p} mid p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, consider the Bloch–Kato Selmer group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H colon equals upper H Subscript normal upper Sigma Superscript 1 Baseline left-parenthesis upper F comma chi Superscript negative 1 Baseline right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>:=</mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>χ<!-- χ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>H:=H^1_{Sigma }(F, chi ^{-1})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The authors previously proved that the number <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of isomorphism classes of (nonsemisimple, reducible) residual representations <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho overbar> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mo accent=false>¯<!-- ¯ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=application/x-tex>{overline rho }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> giving rise to lines in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=application/x-tex>H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which are modular by some <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=rho Subscript f> <mml:semantics> <mml:msub> <mml:mi>ρ<!-- ρ --></mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>rho _f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (also unramified outside <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma> <mml:semantics> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:annotation encoding=application/x-tex>Sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) satisfies <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than-or-equal-to n colon equals dimension Subscript bold upper F Baseline upper H> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mi>n</mml:mi> <mml:mo>:=</mml:mo> <mml:msub> <mml:mi>dim</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>F</mml:mi> </mml:mrow> </mml:mrow> </mml:msub> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>H</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>d geq n:= dim _{mathbf {F}} H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This was proved under the assumption that the order of a congruence module is greater than or equal to that of a divisible Selmer group. We show here that if in addition the relevant local Eisenstein ideal <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper J> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding=application/x-tex>J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is nonprincipal, then <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than n> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>d >n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. When <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F equals bold upper Q> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=bold>Q</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>F=mathbf {Q}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> we prove the desired bounds on the congruence module and the Selmer group. We also formulate a congruence condition implying the nonprincipality of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper J> <mml:semantics> <mml:mi>J</mml:mi> <mml:annotation encoding=application/x-tex>J</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that can be checked in practice, allowing us to furnish examples where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than n> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>d>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W2896825935 created "2018-10-26" @default.
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• W2896825935 date "2019-06-03" @default.
• W2896825935 modified "2023-09-30" @default.
• W2896825935 title "Modularity of residual Galois extensions and the Eisenstein ideal" @default.
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• W2896825935 doi "https://doi.org/10.1090/tran/7851" @default.
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