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\documentclass[11pt]{article}
\usepackage{amsfonts}
\usepackage{amsmath}
\title{Discriminants of Hecke algebras}
\author{William A. Stein}
\newcommand{\T}{\mathbb T}
\newcommand{\F}{\mathbb F}
\DeclareMathOperator{\disc}{disc}
\begin{document}
\maketitle

Using an implementation of the modular symbols algorithm
described in Cremona's book {\em Algorithms for modular
elliptic curves} I computed, for each
prime $N$ between $2$ and $577$, an integer $D_N$
which is divisible by the discriminant of
the Hecke algebra $\T_N$ associated to weight 2 cusp forms
of level $N$ for $\Gamma_0(N)$.
For these $N$ we give a table of factored $D_N$.

The Hecke algebra $\T=\T_N$ is an order in a
product $E=\prod E_t$ of totally
real number fields.  The {\em discriminant}
$\disc(\T)$ is the product of
the discriminants of the number fields $E_i$, multiplied by the
square of the index of~$\T$ in its normalization.

Fix a prime number $N$ and let $S(N)$ be the space of
weight 2 cusp forms of level $N$ for $\Gamma_0(N)$.
For $p$ not equal to $N$ let $T_p$ be the $p$-th Hecke
operator, and let $d_p$ be the discriminant
of the characteristic polynomial of $T_p$ acting on $S(N)$.
Consider the sequence of integers
$$d_2,\quad \gcd(d_2,d_3), \quad \gcd(d_2,d_3,d_5), \quad \ldots, \quad \gcd(d_2,d_3,\ldots,d_q), \quad \ldots$$
where we omit $p$ if $p=N$.
Since each term divides its predecessor, this sequence must
eventually stabilize at some limit $\Delta_N$.
Since each term is divisible by the discriminant
of $\T$, this discriminant divides $\Delta_N$.

I wrote a program which
computes the above sequence until it repeats some value $D_N$
for 15 terms.  The result of that computation is given
in table 1, which can be found at the end of this document.

It is interesting to note that $N=389$ is the only case in our tables
for which $N|D_N$.
I have checked up to $N=14537$ and found no other
cases in which this occurs. Whether or not this ever occurs
is of interest to Ribet as this hypothesis plays a role
in his paper, {\em Torsion points on $J_0(N)$ and galois
representations}.

Another problem is to determine, for each $N$ in table 1, whether
the primes dividing $D_N$ are exactly the same as the primes dividing
$\disc(\T)$.  I have checked that this is the case for $N\leq 73$.
If the ring $\T\otimes\F_p$ is not reduced then $p|\disc(\T)$.
This ring can't be reduced if $T_q$ is not diagonalizable
(modulo $p$) for some prime $q$ not equal to $N$.
However, this sufficient condition is not always necessary as the
case $N=37$ illustrates.  Here $2$ ramifies in the Hecke algebra
even though the Hecke operators $T_q$ with $q\not=2$
act semisimply modulo $2$.
%Will this difficulty will always occur
%when there is a basis of $\Q$-rational eigenforms.

\vfill
\newpage
\begin{center}
Factored $D_N$ for $N\leq 577$.

\begin{tabular}{|r||l|}\hline
$N$	&	$D_N$\\\hline\hline
$11$	&	$1$	\\\hline
$13$	&	$0$	\\\hline
$17$	&	$1$	\\\hline
$19$	&	$1$	\\\hline
$23$	&	$5$	\\\hline
$29$	&	$2^3$	\\\hline
$31$	&	$5$	\\\hline
$37$	&	$2^2$	\\\hline
$41$	&	$2^2.37$	\\\hline
$43$	&	$2^5$	\\\hline
$47$	&	$19.103$	\\\hline
$53$	&	$2^4.37$	\\\hline
$59$	&	$2^7.31.557$	\\\hline
$61$	&	$2^4.37$	\\\hline
$67$	&	$2^4.5^4$	\\\hline
$71$	&	$3^4.257^2$	\\\hline
$73$	&	$2^4.3^2.5.13$	\\\hline
$79$	&	$2^4.83.983$	\\\hline
$83$	&	$2^8.197.11497$	\\\hline
$89$	&	$2^6.5^3.6689$	\\\hline
$97$	&	$2^6.7^2.2777$	\\\hline
$101$	&	$2^8.17568767$	\\\hline
\end{tabular}
\begin{tabular}{|r||l|}\hline
$N$	&	$D_N$\\\hline\hline
$103$	&	$2^8.5.17.411721$	\\\hline
$107$	&	$2^{12}.5.7.1667.19079$	\\\hline
$109$	&	$2^{10}.7^2.7537$	\\\hline
$113$	&	$2^{10}.3^4.7^2.11^2.107$	\\\hline
$127$	&	$2^{12}.3^4.7.86235899$	\\\hline
$131$	&	$2^{19}.5.46141.75619573$	\\\hline
$137$	&	$2^{10}.5^2.29.401.895241$	\\\hline
$139$	&	$2^{14}.3^2.7^2.997.2151701$	\\\hline
$149$	&	$2^{12}.7^2.234893.1252037$	\\\hline
$151$	&	$2^{18}.7^2.11.67^2.257.439867$	\\\hline
$157$	&	$2^{13}.61.397.48795779$	\\\hline
$163$	&	$2^{15}.3^2.65657.82536739$	\\\hline
$167$	&	$2^{16}.5.8269.5103536431379173$	\\\hline
$173$	&	$2^{14}.5^2.7.29.5608385124289$	\\\hline
$179$	&	$2^{22}.3^4.7^2.313.137707.536747147$	\\\hline
$181$	&	$2^{16}.5^2.7.61.397.595051637$	\\\hline
$191$	&	$2^8.3^3.5.382146223.319500117632677$	\\\hline
$193$	&	$2^{14}.5.11^2.17.103.401.4153.680059$	\\\hline
$197$	&	$2^{18}.5^2.61.397.35217676193989$	\\\hline
$199$	&	$2^{16}.3.5^3.29.31.71^2.347.947.37316093$	\\\hline
$211$	&	$2^{20}.3.5.7^4.41^2.43.229.52184516509$	\\\hline
$223$	&	$2^{36}.7^2.19.103.3995922697473293141$	\\\hline
\end{tabular}

\begin{tabular}{|r||l|}\hline        % tex *hack*.
$N$	&	\mbox{$D_N$\hspace{3.81in}}\\\hline\hline
$227$	&	$2^{37}.3^2.5^3.7^4.13^2.29.31^2.13591.57139.273349$	\\\hline
$229$	&	$2^{32}.107.17467.39555937.53625889$	\\\hline
$233$	&	$2^{22}.3^7.53.139.653.4127.24989.8388019$	\\\hline
$239$	&	$2^{12}.7^2.2833.51817.97423.1174779433.8920940047$	\\\hline
$241$	&	$2^{23}.97.1489.20857.651474368435017$	\\\hline
$251$	&	$2^{28}.5^2.29.373.8768135668531.2006012696666681$	\\\hline
$257$	&	$2^{65}.29.479.71711.409177.654233.32354821$	\\\hline
$263$	&	$2^{20}.11.61.397.15631853.34867513.97092067.252746489$	\\\hline
$269$	&	$2^{22}.3^2.43.151.27767.65657.5550873754172978311$	\\\hline
$271$	&	$2^{24}.3^2.1367.6091.592661.1132673.14171513.172450541$	\\\hline
$277$	&	$2^{22}.5^2.19.29.37.137^2.92767.1530091.25531570859$	\\\hline
\hline\end{tabular}

\begin{tabular}{|r||l|}\hline
$N$	&	$D_N$\\\hline\hline
$281$	&	$2^{22}.3.5.181.857.8388019.2647382149.1778899342669$	\\\hline
$283$	&	$2^{46}.349.1297.413713.73199099.5832488839$	\\\hline
$293$	&	$2^{26}.3^2.29.233.2351^2.69763.42711913589792108923$	\\\hline
$307$	&	$2^{50}.3^6.5^5.11^2.13^3.107^2.457.3697.21577.974513.568380457$	\\\hline
$311$	&	$2^{16}.5^2.29.3013091897.2106873009119126062143259000543887593$	\\\hline
$313$	&	$2^{24}.5.41^2.8619587.9614923.130838023.2164322751511$	\\\hline
$317$	&	$2^{26}.7.367.3217.660603043.14989400036918065702697531$	\\\hline
$331$	&	$2^{38}.3^2.53^2.229.1399.21911.205493.6363601.584461573862449$	\\\hline
$337$	&	$2^{28}.113.593.2791.2963615537.747945736667.4122851467451$	\\\hline
$347$	&	$2^{61}.5.7^2.19^2.331.349.479.617.1797330450291217.918291275915301361$	\\\hline
$349$	&	$2^{28}.13.103.1118857.72318613.6771977049413.1313981654817031$	\\\hline
$353$	&	$2^{34}.3^2.5.127^2.229.114641.551801.12611821.7779730837.24314514437$	\\\hline
$359$	&	$2^{36}.3^6.2777.16512254293.64542630435970307.2171776478013633068927$	\\\hline
$367$	&	$2^{44}.7.81421.251387.418175501.15354151381.13144405392643360366681$	\\\hline
$373$	&	$2^{32}.7.11^3.23.199.673.2143.1542194372227.72819251148518000363297$	\\\hline
$379$	&	$2^{34}.59.317.421.278329.5698591.2117788336277.2851210737989187265253$	\\\hline
$383$	&	$2^{32}.5.11^2.13.72893.3151861.16141144314299.$\\
&	$178236551484825400362837637090811$	\\\hline
$389$	&	$2^{63}.3^4.5^6.31^2.37.389.3881.215517113148241.477439237737571441$	\\\hline
$397$	&	$2^{57}.23^2.31^2.97.317.761^2.302609750073209.83566618884497478937$	\\\hline
$401$	&	$2^{96}.5^2.19.163.293^2.811.1218675071.71742740351.388881803749.$\\
&	$34393898968391$	\\\hline
$409$	&	$2^{32}.3^3.17.1667.1741.2341.537071.14884451.18631199.$\\
&	$1334964067081334453235547$	\\\hline
$419$	&	$2^{55}.17.43.113.151.167.971.493657.20375986548898473293.$\\
&	$53097073649092855361102575237$	\\\hline
$421$	&	$2^{34}.3.31.557.4729.825403.857459.144211946777593109.$\\
&	$2328579379136648917067$	\\\hline
$431$	&	$2^{91}.3^4.5^6.11.19^2.29.31.43.197.257.6947^2.37619.$\\
&	$29252013842927.806505757406715084824003$	\\\hline
$433$	&	$2^{68}.3^7.7^2.37^2.101.379.1439.3613.18719.2792477.77087971.$\\
&	$5830108671536745647$	\\\hline
$439$	&	$2^{66}.3^2.5.31^2.173.84179.85667.16794662617.$\\
&	$513841517138871835091506167235408934202857$	\\\hline
$443$	&	$2^{88}.3^2.7^2.31^2.499.6899.48508479390300197.$\\
&	$2817219327571188909266947704801865987$	\\\hline
$449$	&	$2^{40}.3.7^2.101.44933757980789.188247485945671.$\\
&	$653016225615601.1431966252229376199841$	\\\hline
$457$	&	$2^{36}.5.31^2.653.3169.38983093.52621913.33122975406370693.$\\
&	$5653726203394180386934181$	\\\hline
\hline\end{tabular}
\begin{tabular}{|r||l|}\hline
$N$	&	$D_N$\\\hline\hline
$461$	&	$2^{73}.5.7^2.19^3.97^2.80750473.3104029729.607263139073.$\\
&	$3729490905341009668647473177$	\\\hline
$463$	&	$2^{62}.113.311.9929.568201.132502583.1474412920219.$\\
&	$2770309905285622039024420194209857723$	\\\hline
$467$	&	$2^{71}.17^2.1212648089519.32432206859088781.$\\
&	$6296651104824906148358708614333895055221783$\\\hline
$479$	&	$2^{32}.13.17.1861.4021.28745083.41556253.1202203127423.$\\
&	$201529385024397103.7037463122648759781611869895003$\\\hline
$487$	&	$2^{72}.3^{16}.5^4.13^2.17^2.19^4.59.103^2.109.257.$\\
&	$623519211698413571686763.15408475904697077364866629$\\\hline
$491$	&	$2^{104}.5^6.19^2.43.131.479.887.5650859.$\\
&	$54796097920639362740205317747356273097682333252495603721$	\\\hline
$499$	&	$2^{69}.3^{11}.5.71^2.167.495613.25224990196319.$\\
&	$573452584782809.277143583167463430555979797274731$	\\\hline
$503$	&	$2^{78}.3^2.5^4.11^2.19^3.257.821.2003^2.13597.45587.$\\
&	$384479819.8659024393.20115672029938390602701696607766073563$	\\\hline
$509$	&	$2^{71}.3^3.13.157.971.1277.4567.3691783.42330311.$\\
&	$1157039662523351992921397.6331071860925306189417509$\\\hline
$521$	&	$2^{42}.23.53.67.929.13877.531096383.19526270957.1089951135204631559833.$\\
&	$14340527343875384245648725589439$	\\\hline
$523$	&	$2^{91}.3.5.41^3.59.149^2.1201.279121937.8371971617.$\\
&	$9059602909494267071628228952878552757512056969593$	\\\hline
$541$	&	$2^{46}.3^2.5.13.277.307.591581.1940573213.221136462575339.$\\
&	$1453183329662653.18044474614550745414465332996771$	\\\hline
$547$	&	$2^{105}.7^3.73.103^2.5501.11783.16097.43781.1152631.146768003.$\\
&	$9959758037.91268351929.102277460687.106666343972273$	\\\hline
$557$	&	$2^{46}.7^4.13^2.4787.252163.16849164271275021852893.$\\
&	$53296770296923102812608983.2381022539751738307256162767$	\\\hline
$563$	&	$2^{139}.5^2.13^4.37^2.61^2.37591.52667.155083.301703.938251.$\\
&	$46706589087295134421.299128314984453465128592656821021$	\\\hline
$569$	&	$2^{46}.73.449531828286229614392569.189316003.$\\
&	$257022598600391962761793946239.2294643649486046267496627432517$	\\\hline
$571$	&	$2^{166}.3^{12}.5^8.7^4.13^2.17.37^3.41^2.79^2.127^2.181.211.293.709.$\\
&	$1579^2.1667^2.12030433.807024744595934649052018211$	\\\hline
$577$	&	$2^{131}.3^{12}.5^4.13^3.59^2.61^2.257.163753.$\\
&	$41340850017998228328234516909328723846661.$\\
&	$85934741209775683850815667$	\\\hline
\end{tabular}

\end{center}

\end{document}