Irreducible components of the eigencurve of finite degree are finite over the weight space

<jats:title>Abstract</jats:title> <jats:p>Let <jats:italic>p</jats:italic> be a rational prime and <jats:italic>N</jats:italic> a positive integer which is prime to <jats:italic>p</jats:italic>. Let <jats:inline-formula id="j_crelle-2018-0030_ineq_9999_w2aab3b7b1b1b6b1aab1c14b1b7Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒲</m:mi> </m:math> <jats:tex-math>{\mathcal{W}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the <jats:italic>p</jats:italic>-adic weight space for <jats:inline-formula id="j_crelle-2018-0030_ineq_9998_w2aab3b7b1b1b6b1aab1c14b1c11Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>GL</m:mi> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>ℚ</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{{\mathrm{GL}}_{2,\mathbb{Q}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Let <jats:inline-formula id="j_crelle-2018-0030_ineq_9997_w2aab3b7b1b1b6b1aab1c14b1c13Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝒞</m:mi> <m:mi>N</m:mi> </m:msub> </m:math> <jats:tex-math>{\mathcal{C}_{N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the <jats:italic>p</jats:italic>-adic Coleman–Mazur eigencurve of tame level <jats:italic>N</jats:italic>. In this paper, we prove that any irreducible component of <jats:inline-formula id="j_crelle-2018-0030_ineq_9996_w2aab3b7b1b1b6b1aab1c14b1c19Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝒞</m:mi> <m:mi>N</m:mi> </m:msub> </m:math> <jats:tex-math>{\mathcal{C}_{N}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> which is of finite degree over <jats:inline-formula id="j_crelle-2018-0030_ineq_9995_w2aab3b7b1b1b6b1aab1c14b1c21Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒲</m:mi> </m:math> <jats:tex-math>{\mathcal{W}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is in fact finite over <jats:inline-formula id="j_crelle-2018-0030_ineq_9994_w2aab3b7b1b1b6b1aab1c14b1c23Aa"> <jats:alternatives> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒲</m:mi> </m:math> <jats:tex-math>{\mathcal{W}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.</jats:p> <jats:p>Combined with an argument of Chenevier and a conjecture of Coleman–Mazur–Buzzard–Kilford (which has been proven in special cases, and for general quaternionic eigencurves) this shows that the only finite degree components of the eigencurve are the ordinary components.</jats:p>