Sharedwww / tables / Notes / manin.texOpen in CoCalc
Author: William A. Stein
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\title{\bf\Huge \mbox{The Manin index of optimal new quotients}\vspace{5ex}}
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\author{\LARGE William A. Stein}
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\date{\Large October 1, 1999\vspace{2ex}\\}
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\markboth{}{W.\thinspace{}A. Stein, The Manin index of optimal new quotients}
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\begin{document}
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\Large
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\maketitle
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\tableofcontents
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\pagenumbering{Roman}
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\setcounter{page}{0}
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\Page{The Birch and Swinnerton-Dyer conjecture}
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Consider the optimal quotient $A=A_f$ associated to a newform $f$:
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$$\xymatrix{
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& *+++{IJ_0(N)}\[email protected]{^(->}[d]\\
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& J_0(N)\[email protected]{->>}[d]^{\pi}\\
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f = \sum a_n q^n \[email protected]{|~>}[r] & *+++{A}\[email protected]{^(->}[r]^{\iota}&{\cA}}$$
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Here $I := \ker (\T \onto \Z[\ldots a_n \ldots])$ is a prime ideal.
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\begin{thm}[Agash\'e,---]
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$$\frac{L(A,1)}{\Omega_{A}}\cdot c_A =
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[\pi_*(H_1(X_0(N),\Z)^+) : \pi_*(\T \e)]/c_{\infty}$$
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\end{thm}
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Here
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\begin{bulletlist}
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\item $\cA := $ N\'{e}ron model of $A$ over $\Z$.
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\item $\Omega_{A} := $ volume of $A(\R)$ wrt basis for
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$H^0(\cA,\Omega_{A})$.
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\item $c_\infty := $ number of connected components of $A(\R)$, a power of $2$.
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\item {\bf Manin index}
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$$c_A := [S_2(N;\Z)[I] : \qexp(\pi^* \iota^* H^0(\cA,\Omega_{\cA}))]
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\in\Z_{>0}$$
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\end{bulletlist}
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\begin{conj}[Birch, Swinnerton-Dyer]
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$$\displaystyle \frac{L(A,1)}{\Omega_{A}}
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= \frac{\#\Sha(A/\Q)\cdot \prod c_p}
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{\#A(\Q) \cdot \#\Adual(\Q)}.$$
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\end{conj}
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\Page{Motivation}
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{\bf Problem.}
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Verify the BSD conjecture, up to a power of~$2$,
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for specific~$A_f$ at square-free level.
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\begin{ex}
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$p=2333$
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$$J_0(2333) \isog A \cross B \cross C$$
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$\dim A = 4$, $\dim B = 89$, $\dim C=101$.
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$$L(A,1)=L(B,1)=0.$$
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$$\frac{L(C,1)}{\Omega_C} \cdot c_C
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= \frac{83341^2}{11\cdot 53}\cdot 2^?$$
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$$c_p = 11\cdot 53 = \#C(\Q) = \#C^{\vee}(\Q)= \numer((p-1)/12).$$
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If we know that $c_C$ is a power of two then,\\
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to verify BSD up to $2$-powers, must show
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that $\#\Sha=83341^2\cdot 2^?$.\\
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In this case we are lucky: $83341\mid \#(A\intersect C)$,
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which can (probably) be used to show that
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$83341\mid \#\Sha$.
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\end{ex}
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\begin{ex}
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$p=2111$ is prime.
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$$J_0(2111) \isog A \cross B$$
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with $\dim A = 64$, $\dim B = 112$.\\
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$L(A,1)=0$
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$$\frac{L(B,1)}
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{\Omega_B}\cdot c_B = \frac{211}{5}\cdot 2^?.$$
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$$c_p = 5\cdot 211 = \#B(\Q) = \#B^{\vee}(\Q) = \numer((p-1)/12).$$
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If we know that $c_B$ is a power of $2$, then
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to verify BSD, must show that $\#\Sha=211^2\cdot 2^?/c_B$.\\
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\end{ex}
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\Page{The Manin index}
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\begin{conj}[Agashe, ---, Manin when $\dim A=1$]\label{myconj}
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$$c_A = 1.$$
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\end{conj}
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{\bf Caution:} Without the condition that~$A$ is new, the conjecture
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is false. The Manin index of $J_0(33)$ is~$3$, because
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of a mod~$3$ congruence.
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A.~Logan is skeptical of Conjecture~\ref{myconj}.
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{\bf Evidence:} See Amod's talk from Wednesday.
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{\em Empirical evidence for the Birch and
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Swinnerton-Dyer conjectures for
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modular Jacobians of genus~2 curves},\\
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Flynn, Lepr\'evost, Schaeffer, ---, Stoll, Wetherell.\\
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We considered $28$ optimal quotients of dimension~$2$ that
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happened to be Jacobians.
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\begin{bulletlist}
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\item We computed $\Omega_A$ directly from a minimal
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proper regular models.
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\item We then computed the volume of $A(\R)$ with respect
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to $S_2(N;\Z)[I]$, and got the same answer, to several
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decimal places.
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\end{bulletlist}
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The non square-free levels treated:
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$$N=3^2\cdot 7,\quad 3^2\cdot 13,\quad 5^3,\quad 3^3\cdot 5,\quad 3\cdot 7^2,
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\quad 5^2\cdot 7,\quad 2^2\cdot 47,\quad 3^3\cdot 7.$$
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In every case, $c_A = 1$.
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\Page{Bounding the Manin index}
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We present a slight generalization of a theorem of Mazur.
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\begin{thm}[Mazur when $\dim A=1$, ---]
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Let~$m$ be the largest square dividing~$N$.
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Let $A$ be an optimal quotient of $J_0(N)^{\new}$.
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Then $c_A$ is a unit in $\Z[\frac{1}{2m}]$.
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\end{thm}
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Many ideas of the proof are in Mazur's {\em Rational isogenies
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of prime degree}.
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\begin{proof}
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We give the proof assuming that $A=A_f$.
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\begin{bulletlist}
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\item $R:=\Z[\frac{1}{2m}]$.
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\item $\cJ := $ Neron model of $J_0(N)$ over $R$.
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\item $\cA := $ ditto for $A$.
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\item $X := $ smooth locus of the minimal proper regular model of $X_0(N)$.
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\end{bulletlist}
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Consider
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$$\xymatrix{
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*+++{H^0(\cA,\Omega_{\cA/R})}\[email protected]{^(->}[rr]^{\pi^*}_{\text{pullback}}&&
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{H^0(\cJ,\Omega_{\cJ/R})}\ar[r]^{\isom}&
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{H^0(X,\Omega_X)}\ar[rr]^{\qexp}&&R[[q]].}$$
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\begin{bulletlist}
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\item $\pi^*$ is an inclusion because $H^0(\cA,\Omega_{\cA/R})$
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is free ($\cA/R$ is smooth) and $\pi^*_\C$ is an inclusion.
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\item The middle isomorphism uses Mich\'{e}le Raynaud's result:
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$$\cPic^0(X)\isom \cJ^0,$$
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induces $H^1(X,\O_X)\xrightarrow{\,\isom\,} H^1(\cJ,\O_{\cJ})$
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on tangent spaces. Now apply Grothendieck duality.
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\item $\qexp$ comes from Tate curve:
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$$\tau:\Spec R[[q]] \ra X.$$
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Formal completion of $X$ at $\bifR$ is $(\Spec R)[[q]]$.
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\end{bulletlist}
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\Page{Proof (continued)}
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{\bf To show:} Image of $H^0(\cA,\Omega_{\cA})$ saturated.
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Then $c_A\in R^*$ since $S_2(N,R)[I]\subset R[[q]]$ is saturated.
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Consider
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$$\xymatrix{
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0\ar[r]& H^0(\cA,\Omega_{\cA})\ar[rr]^{\quad\qexp}
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&&R[[q]]\ar[r]&D\ar[r]&0.}$$
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Is $D$ torsion free? Let $\ell\nmid 2m$ be a prime; tensor with $\Fl$:
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$$\xymatrix{
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&{\Tor^1(D,\Fl)}\[email protected]{=}[d]\\
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0\ar[r]&D[\ell]\ar[r]&H^0(\cA,\Omega_{\cA})\tensor\Fl\ar[r]&
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{\Fl[[q]]}\ar[r]&D\tensor\Fl\ar[r]&0}$$
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Thus
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$$D[\ell]=0 \iff
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H^0(\cA,\Omega_{\cA})\tensor\Fl\ra\Fl[[q]]\text{ injective}$$
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Since $\ell\nmid 2m$, we have $e(\Ql)=1<\ell-1$, so [Mazur, Cor.~1.1]
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gives an exact sequence
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$$
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0\ra H^0(\cA/\Zl,\Omega_{\cA/\Zl})\ra
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H^0(\cJ/\Zl,\Omega_{\cJ/\Zl})\ra
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H^0(\cB/\Zl,\Omega_{\cB/\Zl})\ra 0.$$
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Here $B=\ker(\pi:J\ra A)$.
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Since $H^0(\cB/\Zl,\Omega_{\cB/\Zl})$ is torsion free,
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$$\xymatrix{
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*+++{H^0(\cA/\Zl,\Omega_{\cA/\Zl})\tensor\Fl}
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\[email protected]{^(->}[r]&
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{H^0(\cJ/\Zl,\Omega_{\cJ/\Zl})\tensor\Fl}}$$
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Fact (see e.g., pg 183 of Eisenbud's book):
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$\Zl/R$ is flat.\\
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``Cohomology commutes with flat base change'':
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$$H^0(\cA,\Omega_{\cA})\tensor\Zl \isom H^0(\cA/\Zl,\Omega_{\cA/\Zl})$$
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\Page{Proof (continued)}
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$$\xymatrix{
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*+++{H^0(\cA,\Omega_{\cA})\tensor\Fl}
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\[email protected]{^(->}[rrr]\ar[dddrrr]_{\txt{injective?}\quad}&&&
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H^0(\cJ,\Omega_{\cJ})\tensor\Fl\ar[d]^{\isom}\\
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&&&
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H^0(X,\Omega_{X})\tensor\Fl\ar[d]^{\isom}\\
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&&&
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H^0(X_{\Fl},\Omega_{X_{\Fl}})\ar[d]^{\qexp}\\
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&&&
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{\Fl[[q]].}
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}$$
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The isomorphism
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$H^0(X,\Omega_{X})\tensor\Fl\isom
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H^0(X_{\Fl},\Omega_{X_{\Fl}})$
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uses cohomological trickery and that $k=2$, $N\geq 5$.
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If $\ell\nmid N$, then $\qexp$ is injective
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(see Deligne-Rapoport, Theorem 3.9).
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Thus $H^0(\cA,\Omega_{\cA})\tensor\Fl \hookrightarrow \Fl[[q]]$,
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so $D[\ell]=0$, as required.
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If $\ell\mid N$ then $\qexp$ is not necessarily injective.
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However, $\ell$ exactly divides $N$. Deligne-Rapoport model
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of $X_{\Fl}$:
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(175,951)(203,937)(229,923)
468
(253,910)(275,897)(296,884)
469
(316,871)(335,858)(352,844)
470
(370,830)(387,814)(401,801)
471
(416,787)(431,773)(446,758)
472
(461,742)(477,725)(493,707)
473
(510,689)(527,670)(545,651)
474
(563,631)(581,611)(600,591)
475
(619,571)(638,551)(657,531)
476
(676,512)(696,493)(715,474)
477
(734,456)(753,439)(773,422)
478
(792,405)(812,389)(831,375)
479
(850,361)(870,347)(890,333)
480
(912,319)(934,306)(957,292)
481
(980,279)(1005,265)(1030,253)
482
(1055,240)(1082,228)(1108,216)
483
(1135,204)(1162,194)(1189,183)
484
(1216,174)(1243,165)(1270,156)
485
(1296,148)(1323,141)(1348,135)
486
(1374,129)(1399,123)(1424,119)
487
(1450,114)(1475,111)(1500,108)
488
(1526,105)(1552,102)(1579,101)
489
(1606,99)(1634,99)(1662,98)
490
(1691,99)(1719,99)(1749,101)
491
(1778,103)(1807,105)(1836,108)
492
(1864,112)(1893,115)(1920,120)
493
(1947,125)(1974,130)(1999,136)
494
(2024,142)(2048,148)(2071,155)
495
(2094,162)(2116,169)(2137,177)
496
(2162,187)(2186,197)(2210,208)
497
(2235,220)(2259,232)(2283,245)
498
(2307,259)(2331,273)(2355,288)
499
(2378,304)(2402,320)(2424,336)
500
(2446,352)(2468,368)(2488,384)
501
(2508,400)(2527,416)(2546,431)
502
(2563,446)(2580,461)(2596,475)
503
(2612,489)(2629,505)(2646,520)
504
(2663,536)(2680,552)(2698,568)
505
(2715,584)(2733,600)(2752,617)
506
(2770,634)(2789,650)(2808,667)
507
(2826,683)(2845,699)(2864,715)
508
(2882,731)(2901,746)(2919,760)
509
(2937,774)(2956,788)(2975,802)
510
(2992,815)(3010,827)(3029,840)
511
(3048,853)(3069,866)(3090,878)
512
(3112,891)(3135,904)(3159,917)
513
(3183,930)(3208,942)(3234,953)
514
(3259,964)(3285,974)(3311,984)
515
(3336,993)(3362,1001)(3387,1008)
516
(3412,1014)(3437,1019)(3462,1023)
517
(3487,1027)(3510,1030)(3534,1031)
518
(3557,1032)(3582,1033)(3607,1033)
519
(3633,1032)(3659,1030)(3686,1027)
520
(3714,1024)(3742,1020)(3770,1015)
521
(3798,1010)(3826,1003)(3854,997)
522
(3881,989)(3908,981)(3934,973)
523
(3960,964)(3985,954)(4009,945)
524
(4033,935)(4056,924)(4078,913)
525
(4100,902)(4121,890)(4142,878)
526
(4163,865)(4184,852)(4205,838)
527
(4227,823)(4248,808)(4270,792)
528
(4291,775)(4313,758)(4334,741)
529
(4355,724)(4376,706)(4396,688)
530
(4417,671)(4436,653)(4455,636)
531
(4474,619)(4492,603)(4509,587)
532
(4526,571)(4543,556)(4559,541)
533
(4575,527)(4592,511)(4609,496)
534
(4626,481)(4644,466)(4662,450)
535
(4680,435)(4699,419)(4718,404)
536
(4737,389)(4757,374)(4776,359)
537
(4796,344)(4816,330)(4836,317)
538
(4855,304)(4874,291)(4894,279)
539
(4912,268)(4931,257)(4950,246)
540
(4968,237)(4987,227)(5006,218)
541
(5025,209)(5045,200)(5066,191)
542
(5087,183)(5109,175)(5132,167)
543
(5156,160)(5180,153)(5205,147)
544
(5230,141)(5255,136)(5280,131)
545
(5306,128)(5331,125)(5356,123)
546
(5381,121)(5405,121)(5429,121)
547
(5453,122)(5476,124)(5500,127)
548
(5521,130)(5543,134)(5565,139)
549
(5588,144)(5611,150)(5634,157)
550
(5658,165)(5682,174)(5707,183)
551
(5731,193)(5756,203)(5780,214)
552
(5804,226)(5828,238)(5851,250)
553
(5874,262)(5895,275)(5916,288)
554
(5937,300)(5956,313)(5974,326)
555
(5992,339)(6008,352)(6025,364)
556
(6043,380)(6061,396)(6078,412)
557
(6096,429)(6112,447)(6129,465)
558
(6145,483)(6161,502)(6177,522)
559
(6193,541)(6208,561)(6222,581)
560
(6236,600)(6250,619)(6263,638)
561
(6276,657)(6288,675)(6301,692)
562
(6313,710)(6325,727)(6337,744)
563
(6349,762)(6362,779)(6375,797)
564
(6389,815)(6404,834)(6419,852)
565
(6435,871)(6451,889)(6467,908)
566
(6484,925)(6501,943)(6518,959)
567
(6535,975)(6552,990)(6569,1004)
568
(6586,1018)(6603,1030)(6620,1041)
569
(6637,1052)(6653,1061)(6669,1070)
570
(6686,1078)(6703,1085)(6721,1092)
571
(6740,1099)(6760,1104)(6780,1109)
572
(6801,1113)(6822,1116)(6844,1119)
573
(6866,1120)(6888,1120)(6911,1120)
574
(6933,1118)(6956,1115)(6978,1111)
575
(7000,1106)(7022,1100)(7044,1094)
576
(7065,1086)(7087,1077)(7106,1069)
577
(7125,1060)(7144,1049)(7164,1039)
578
(7184,1027)(7205,1014)(7227,1000)
579
(7249,986)(7272,971)(7295,955)
580
(7319,938)(7343,921)(7367,903)
581
(7392,885)(7416,866)(7441,848)
582
(7465,829)(7490,810)(7514,792)
583
(7537,773)(7561,755)(7584,737)
584
(7607,720)(7630,703)(7652,685)
585
(7675,669)(7697,652)(7720,635)
586
(7743,619)(7767,602)(7791,585)
587
(7815,568)(7840,551)(7866,534)
588
(7892,517)(7918,500)(7944,483)
589
(7971,466)(7998,449)(8024,433)
590
(8051,417)(8077,402)(8103,387)
591
(8128,373)(8153,359)(8178,346)
592
(8201,334)(8225,322)(8247,311)
593
(8269,301)(8291,291)(8312,281)
594
(8337,271)(8362,260)(8387,251)
595
(8412,242)(8439,233)(8466,224)
596
(8495,215)(8526,207)(8559,198)
597
(8593,189)(8629,180)(8666,171)
598
(8702,162)(8737,154)(8770,147)
599
(8799,141)(8823,135)(8840,132)
600
(8852,129)(8859,128)(8862,127)
601
\end{picture}
602
}
603
\vspace{2ex}
604
605
The intersections are supersingular points; the Atkin-Lehner
606
involution~$w_{\ell}$ swaps~$\pmb{0}$ and$\pmb{\infty}$.
607
608
Consider $g\in H^0(\cA,\Omega_{\cA})\tensor\Fl$ such that
609
$\qexp(g)=0$.
610
Then~$g$ vanishes on the component of~$X_{\Fl}$ containing~$\infty$.
611
Since $\cA$ is new and corresponds to a single eigenform,
612
$w_\ell(g) = \pm g$, so $\qexp(w_\ell(g))=0$; hence~$g$
613
vanishes on the other component of $X_{\Fl}$. This forces $g=0$,
614
and hence $D[\ell]=0$.
615
\end{proof}
616
617
618
619
\end{document}
620
621
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623
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629
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