A study of the local components of the Hecke algebra mod 𝑙

Abstract

We use information about modular forms <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mod l"> <mml:semantics> <mml:mrow> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mi>l</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\bmod l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to study the local structure of the Hecke ring. In particular, we find nontrivial lower bounds for the dimensions of the Zariski tangent spaces of the local components of the Hecke ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mod l"> <mml:semantics> <mml:mrow> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mi>l</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\bmod l</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These results suggest that the local components of the Hecke ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mod l"> <mml:semantics> <mml:mrow> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mi>l</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\bmod l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are more complex than originally expected. We also investigate the inverse limits of the Hecke rings of weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k mod l"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mi>l</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">k\bmod l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> varies within a fixed congruence class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mod l minus 1"> <mml:semantics> <mml:mrow> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mi>l</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\bmod l - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As an immediate corollary to some of the above results, we show that when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is sufficiently large, an arbitrary prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l"> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding="application/x-tex">l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> must divide the index of the classical Hecke ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper T Subscript k"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">T</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {T}}_k}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the ring of integers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper T Subscript k Baseline circled-times bold upper Q"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">T</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">Q</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\mathbf {T}}_k} \otimes {\mathbf {Q}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

• Transactions of the American Mathematical Society - View - PDF