CoCalc Shared Fileswww / papers / discheckediv / _region_.texOpen in CoCalc with one click!
Author: William A. Stein
1
\message{ !name(discheckediv.tex)}\documentclass[11pt]{article} \usepackage{fancybox}
2
\hoffset=-0.1\textwidth \textwidth=1.2\textwidth
3
\voffset=-0.1\textheight \textheight=1.2\textheight
4
\include{macros}
5
\title{Discriminants of Hecke Algebras at Prime Level} \author{William
6
A. Stein\footnote{This will probably eventually be a joint paper
7
with Frank Calegari and Romyar Sharifi (?)}} \date{September 24,
8
2002} \DeclareMathOperator{\charpoly}{charpoly}
9
\DeclareMathOperator{\splittemp}{split}
10
\renewcommand{\split}{\splittemp} \DeclareMathOperator{\deriv}{deriv}
11
\newcommand{\Zpbar}{\overline{\Z}_p}
12
\newcommand{\kr}[2]{\left(\frac{#1}{#2}\right)}
13
14
15
\begin{document}
16
{\makeatletter\gdef\[email protected]#1[#2]#3{[#3#1#2]}\gdef\cite{\@ifnextchar[{\[email protected]{, }}{\[email protected]{}[]}}}
17
18
\message{ !name(discheckediv.tex) !offset(-4) }
19
20
\maketitle
21
\begin{abstract}
22
We study $p$-divisibility of discriminant of Hecke algebras
23
associated to spaces of cusp forms of prime level. By considering
24
cusp forms of weight bigger than~$2$, we are are led to make a
25
conjecture about indexes of Hecke algebras in their normalization
26
which, if true, implies that there are no mod~$p$ congruences
27
between non-conjugate newforms in $S_2(\Gamma_0(p))$.
28
\end{abstract}
29
30
\section{Introduction}
31
I started working in modular forms when Ken Ribet asked about
32
discriminants of Hecke algebras at prime level. I've recently
33
revisited this question and, with the help of Frank Calegari, have
34
made some interesting discoveries.
35
36
\section{Discriminants of Hecke Algebras}
37
Let~$R$ be a ring and let~$A$ be an~$R$ algebra that is free as an~$R$
38
module. The trace of an element of~$A$ is the trace, in the sense of
39
linear algebra, of left multiplication by that element on~$A$.
40
41
\begin{definition}[Discriminant]
42
Let $\omega_1,\ldots,\omega_n$ is a~$R$-basis for~$A$. Then the
43
{\em discriminant} of~$A$, denoted $\disc(A)$, is the determinant of
44
the $n\times n$ matrix $(\tr(\omega_i\omega_j))$, which is well
45
defined modulo squares of units in~$A$.
46
\end{definition}
47
When $R=\Z$ the discriminant is well defined, since the only units are
48
$\pm 1$.
49
50
\begin{proposition}\label{prop:separable}
51
Suppose~$R$ is a field. Then~$A$ has discriminant~$0$ if and only
52
if~$A$ is separable over~$R$, i.e., for every extension $R'$ of $R$,
53
the ring $A\tensor R'$ contains no nilpotents.
54
\end{proposition}
55
The following proof is summarized from Section~26 of Matsumura.
56
If~$A$ contains a nilpotent then that nilpotent is in the kernel of
57
the trace pairing. If~$A$ is separable then we may assume that~$R$ is
58
algebraically closed. Then~$A$ is an Artinian reduced ring, hence
59
isomorphic as a ring to a finite product of copies of~$R$, since~$R$
60
is algebraically closed. Thus the trace form on~$A$ is nondegenerate.
61
62
\subsection{The Discriminant Valuation}
63
Let $\Gamma$ be a congruence subgroup of $\SL_2(\Z)$, e.g.,
64
$\Gamma=\Gamma_0(p)$ or $\Gamma_1(p)$. For any integer $k\geq 1$, let
65
$S_k(\Gamma)$ denote the space of holomorphic weight-$k$ cusp forms
66
for $\Gamma$. Let
67
$$
68
\T = \Z[\ldots,T_n,\ldots] \subset \End(S_k(\Gamma))
69
$$
70
be the associated Hecke algebra. Then~$\T$ is a commutative ring
71
that is free and of finite rank as a $\Z$-module. Also of interest is
72
the image $\T^{\new}$ of~$\T$ in $\End(S_k(\Gamma)^{\new})$.
73
\begin{example}
74
Let $\Gamma=\Gamma_0(243)$, which is illustrated on my T-shirt.
75
Since $243=3^5$, experts will immediately deduce that $\disc(\T) =
76
0$. A computation shows that
77
$$
78
\disc(\T^{\new}) = 2^{13} \cdot 3^{40},
79
$$
80
which reflects the mod-$2$ and mod-$3$ intersections all over my
81
shirt.
82
\end{example}
83
84
85
\begin{definition}[Discriminant Valuation]
86
Let~$p$ be a prime and suppose that $\Gamma=\Gamma_0(p)$ or
87
$\Gamma_1(p)$. The {\em discriminant valuation} is
88
$$
89
d_k(\Gamma) = \ord_p(\text{the discriminant of $\T$}).
90
$$
91
%When the discriminant of $\T$ is~$0$ we define $d_k(\Gamma)$ to
92
%be~$+\infty$.
93
\end{definition}
94
95
\section{Motivation and Applications}
96
Let~$p$ be a prime and suppose that $\Gamma=\Gamma_0(p)$ or
97
$\Gamma_1(p)$. The quantity $d_k(\Gamma)$ is of interest because it
98
measures mod~$p$ congruences between eigenforms in $S_k(\Gamma)$.
99
\begin{proposition}
100
Suppose that $d_k(\Gamma)$ is finite. Then the discriminant
101
valuation $d_k(\Gamma)$ is nonzero if and only if there is a mod-$p$
102
congruence between two Hecke eigenforms in $S_k(\Gamma)$ (note that
103
the two congruent eigenforms might be Galois conjugate).
104
\end{proposition}
105
\begin{proof}
106
It follows from Proposition~\ref{prop:separable} that
107
$d_k(\Gamma)>0$ if and only if $\T\tensor \Fpbar$ is not separable.
108
The Artinian ring $\T\tensor\Fpbar$ is not separable if and only if
109
the number of ring homomorphisms $\T\tensor\Fpbar \ra \Fpbar$ is
110
less than
111
$$
112
\dim_{\Fpbar} \T\tensor\Fpbar = \dim_\C S_k(\Gamma).
113
$$
114
Since $d_k(\Gamma)$ is finite, the number of ring homomorphisms
115
$\T\tensor\Qpbar \ra \Qpbar$ equals $\dim_\C S_k(\Gamma)$. Using
116
the standard bijection between homomorphisms and normalized
117
eigenforms, we see that $\T\tensor\Fpbar$ is not separable if and
118
only if there is a mod-$p$ congruence between two eigenforms.
119
\end{proof}
120
121
\begin{example}
122
If $\Gamma=\Gamma_0(389)$ and $k=2$, then $\dim_\C S_2(\Gamma) =
123
32$. Let~$f$ be the characteristic polynomial of $T_2$. One can
124
check that~$f$ is square free and $389$ exactly divides the
125
discriminant of~$f$, so $T_2$ generated $\T\tensor \Z_{389}$ as a
126
ring. (If it generated a subring of $\T\tensor\Z_{389}$ of finite
127
index, then the discriminant of~$f$ would be divisible by $389^2$.)
128
129
Modulo~$389$ the polynomial~$f$ is congruent to
130
$$\begin{array}{l}
131
(x+2)(x+56)(x+135)(x+158)(x+175)^2(x+315)(x+342)(x^2+387)\\
132
(x^2+97x+164)(x^2 + 231x + 64)(x^2 + 286x + 63)(x^5 + 88x^4 +196x^3 + \\
133
113x^2 +168x + 349)(x^{11} + 276x^{10} + 182x^9 + 13x^8 + 298x^7 + 316x^6 +\\
134
213x^5 + 248x^4 + 108x^3 + 283x^2 + x + 101)
135
\end{array}
136
$$
137
The factor $(x+175)^2$ indicates that $\T\tensor \Fbar_{389}$ is
138
not separable since the image of $T_2+175$ is nilpotent (its square
139
is~$0$). There are $32$ eigenforms over~$\Q_2$ but only $31$
140
mod-$389$ eigenforms, so there must be a congruence. Let~$F$ be the
141
$389$-adic newform whose $a_2$ term is a root of
142
$$
143
x^2 + (-39 + 190\cdot 389 + 96\cdot 389^2 +\cdots) x + (-106 +
144
43\cdot 389 + 19\cdot 389^2 + \cdots).
145
$$
146
Then the congruence is between~$F$ and its
147
$\Gal(\Qbar_{389}/\Q_{389})$-conjugate.
148
\end{example}
149
150
\begin{example}
151
The discriminant of the Hecke algebra $\T$ associated to
152
$S_2(\Gamma_0(389))$ is
153
$$
154
2^{53} \!\cdot\! 3^{4} \!\cdot\! 5^{6} \!\cdot\! 31^{2} \!\cdot\!
155
37 \!\cdot\! 389 \!\cdot\! 3881 \!\cdot\! 215517113148241 \!\cdot\!
156
477439237737571441
157
$$
158
I computed this using the following algorithm, which was
159
suggested by Hendrik Lenstra. Using the Sturm bound I found a~$b$
160
such that $T_1,\ldots,T_b$ generate $\T$ as a $\Z$-module. I then
161
found a subset~$B$ of the $T_i$ that form a $\Q$-basis for
162
$\T\tensor_\Z\Q$. Next, viewing $\T$ as a ring of matrices acting
163
on $\Q^{32}$, I found a random vector $v\in\Q^{32}$ such that the
164
set of vectors $C=\{T(v) : T \in B\}$ is linearly independent. Then
165
I wrote each of $T_1(v),\ldots, T_b(v)$ as $\Q$-linear combinations
166
of the elements of~$C$. Next I found a $\Z$-basis~$D$ for the
167
$\Z$-span of these $\Q$-linear combinations of elements of~$C$.
168
Tracing everything back, I find the trace pairing on the elements
169
of~$D$, and deduce the discriminant by computing the determinant of
170
the trace pairing matrix. The most difficult step is computing~$D$
171
from $T_1(v),\ldots,T_b(v)$ expressed in terms of~$C$, and this
172
explains why we embed $\T$ in $\Q^{32}$ instead of viewing the
173
elements of $\T$ as vectors in $\Q^{32^2}$. This whole computation
174
takes one second on an Athlon 2000 processor.
175
\end{example}
176
177
\subsection{Literature}
178
I've seen a version of Theorem~\ref{thm:disc} referred to in the
179
following papers:
180
\begin{enumerate}
181
\item Ribet: {\em Torsion points on $J_0(N)$ and Galois
182
representations}
183
\item Lo\"\i{}c Merel and William Stein: {\em The field generated by
184
the points of small prime order on an elliptic curve}
185
\item Ken Ono and William McGraw: {\em Modular form Congruences and
186
Selmer groups} (McGraw will speak about this next week in this
187
seminar!)
188
\item Momose and Ozawa: {\em Rational points of modular curves
189
$X_{\split}(p)$}
190
\end{enumerate}
191
192
193
\section{Data About Discriminant Valuations}
194
195
\subsection{Weight Two}
196
\begin{theorem}\label{thm:disc}
197
The only prime $p<60000$ such that $d_2(\Gamma_0(p))>0$ is $p=389$.
198
(Except possibly $50923$ and $51437$, which I haven't finished
199
checking yet.)
200
\end{theorem}
201
\begin{proof}
202
This is the result of a large computer computation, and perhaps
203
couldn't be verified any other way, since I know of no general
204
theorems about $d_2(\Gamma_0(p))$. The rest of this proof describes
205
how I did the computation, so you can be convinced that there is
206
valid mathematics behind my computation, and that you could verify
207
the computation given sufficient time. The computation described
208
below took about one week using $12$ Athlon 2000MP processors. In
209
1999 I had checked the result stated above but only for $p<14000$
210
using a completely different implementation of the algorithm and a
211
200Mhz Pentium computer. These computations are nontrivial; we
212
compute spaces of modular symbols, supersingular points, and Hecke
213
operators on spaces of dimensions up to~$5000$.
214
215
The aim is to determine whether or not~$p$ divides the discriminant
216
of the Hecke algegra of level~$p$ for each $p < 60000$. If~$T$ is
217
an operator with integral characteristic polynomial, we write
218
$\disc(T)$ for $\disc(\charpoly(T))$, which also equals
219
$\disc(\Z[T])$. We will often use that
220
$$\disc(T)\!\!\!\!\mod{p} = \disc(\charpoly(T)\!\!\!\!\mod p).$$
221
222
Most levels~$p<60000$ were ruled out by computing characteristic
223
polynomials of Hecke operators using an algorithm that David Kohel
224
and I implemented in MAGMA, which is based on the Mestre-Oesterle
225
method of graphs (our implementation is ``The Modular of
226
Supersingular Points'' package that comes with MAGMA). I computed
227
$\disc(T_q)$ modulo~$p$ for several primes~$q$, and in most cases
228
found a~$q$ such that this discriminant is nonzero. The following
229
table summarizes how often we used each prime~$q$ (note that there
230
are $6057$ primes up to $60000$):
231
\begin{center}
232
\begin{tabular}{|l|l|}\hline
233
$q$ & number of $p< 60000$ where~$q$ smallest
234
s.t. $\disc(T_q)\neq 0$ mod~$p$\\\hline
235
2& 5809 times\\
236
3& 161 (largest: 59471)\\
237
5& 43 (largest: 57793)\\
238
7& 15 (largest: 58699)\\
239
11& 15 (the smallest is 307; the largest 50971)\\
240
13& 2 (they are 577 and 5417)\\
241
17& 3 (they are 17209, 24533, and 47387)\\
242
19& 1 (it is 15661 )\\\hline
243
\end{tabular}
244
\end{center}
245
246
The numbers in the right column sum to 6049, so 8 levels are missing.
247
These are
248
$$
249
389,487,2341,7057,15641,28279, 50923, \text{ and } 51437.
250
$$
251
(The last two are still being processed. $51437$ has the property
252
that $\disc(T_q)=0$ for $q=2,3,\ldots,17$.) We determined the
253
situation with the remaining 6 levels using Hecke operators $T_n$
254
with~$n$ composite.
255
\begin{center}
256
\begin{tabular}{|l|l|}\hline
257
$p$ & How we rule level~$p$ out, if possible\\\hline
258
389& $p$ does divide discriminant\\
259
487& using charpoly($T_{12}$)\\
260
2341& using charpoly($T_6$)\\
261
7057& using charpoly($T_{18}$)\\
262
15641& using charpoly($T_6$)\\
263
28279& using charpoly($T_{34}$)\\\hline
264
\end{tabular}
265
\end{center}
266
267
Computing $T_n$ with~$n$ composite is very time consuming when~$p$ is
268
large, so it is important to choose the right $T_n$ quickly. For
269
$p=28279$, here is the trick I used to quickly find an~$n$ such that
270
$\disc(T_n)$ is not divisible by~$p$. This trick might be used to
271
speed up the computation for some other levels. The key idea is to
272
efficiently discover which $T_n$ to compute. Though computing $T_n$
273
on the full space of modular symbols is quite hard, it turns out that
274
there is an algorithm that quickly computes $T_n$ on subspaces of
275
modular symbols with small dimension (see \S3.5.2 of my Ph.D. thesis).
276
Let~$M$ be the space of mod~$p$ modular symbols of level $p=28279$,
277
and let $f=\gcd(\charpoly(T_2),\deriv(\charpoly(T_2)))$. Let~$V$ be
278
the kernel of $f(T_2)$ (this takes 7 minutes to compute). If $V=0$,
279
we would be done, since then $\disc(T_2)\neq 0\in\F_p$. In fact,~$V$
280
has dimension~$7$. We find the first few integers~$n$ so that the
281
charpoly of $T_n$ on $V_1$ has distinct roots, and they are $n=34$,
282
$47$, $53$, and $89$. I then computed $\charpoly(T_{34})$ directly on
283
the whole space and found that it has distinct roots modulo~$p$.
284
\end{proof}
285
286
\subsection{Higher Weight Data}
287
\begin{enumerate}
288
\item The following are the valuations $d=d_4(\Gamma_0(p))$ at~$p$ of
289
the discriminant of the Hecke algebras associated to
290
$S_4(\Gamma_0(p))$ for $p<500$.
291
292
\hspace{-4em}\shadowbox{\begin{minipage}[b]{1.15\textwidth}
293
\begin{tabular}{|c|ccccccccccccccccc|}\hline
294
$p$ &2& 3& 5& 7& 11& 13& 17& 19& 23& 29& 31& 37& 41& 43& 47& 53& 59\\
295
$d$ &0& 0& 0& 0& 0& 2& 2& 2& 2& 4& 4& 6& 6& 6& 6& 8& 8\\\hline
296
$p$&61& 67& 71& 73& 79& 83& 89& 97& 101& 103& 107& 109& 113& 127& 131& 137& 139\\
297
$d$ & 10& 10& 10& 12& 12& 12& 14& 16& 16& 16& 16& 18& 18& 20& 20& 22&24\\\hline
298
$p$ & 149& 151& 157& 163& 167& 173& 179& 181& 191& 193& 197& 199&
299
211& 223& 227& 229& 233\\
300
$d$ & 24& 24& 26& 26& 26&28& 28& 30& 30& 32& 32& 32& 34& 36& 36& 38& 38\\ \hline
301
$p$ & 239& 241& 251& 257& 263& 269& 271& 277&
302
281& 283& 293& 307& 311& 313& 317& 331& 337\\
303
$d$ & 38& 40& 40& 42& 42&44& 44& 46& 46& 46& 48& 50& 50& 52& 52& 54& 56\\\hline
304
$p$ & 347& 349& 353& 359& 367& 373& 379& 383& 389&397& 401& 409& 419& 421& 431& 433& 439 \\
305
$d$ & 56& 58& 58& 58& 60&62& 62& 62& 65 &66& 66& 68& 68& 70& 70& 72& 72\\\hline
306
$p$ & 443& 449& 457& 461& 463& 467& 479& 487& 491& 499 &&&&&&&\\
307
$d$ & 72& 74& 76& 76& 76& 76& 78& 80& 80& 82 &&&&&&&\\\hline
308
\end{tabular}
309
\end{minipage}}
310
311
\comment{\item For each prime~$p$, let
312
$$
313
\delta(p) = \dim S_4(\Gamma_0(p)) - \dim S_{p+3}(\Gamma_0(1)).
314
$$
315
Then $|\delta(p) - d_4(\Gamma_0(p))| \leq 2$ for each $p<500$.
316
Moreover, for every $p\neq 139$ we have that $\delta(p)\geq
317
d_4(\Gamma_0(p))$, but for $p=139$, $\delta(p)=23$ but
318
$d_4(\Gamma_0(p))=24$. }
319
\end{enumerate}
320
321
322
\section{The Discriminant is Divisible by~$p$}
323
In this section we prove that for $k\geq 4$ the discriminant of the
324
Hecke algebra associated to $S_k(\Gamma_0(p))$ is almost always
325
divisible by~$p$.
326
327
\begin{theorem}
328
Suppose~$p$ is a prime and $k\geq 4$ is an even integer. Then
329
$d_k(\Gamma_0(p))>0$ unless
330
\begin{align*}
331
(p,k) \not\in \{&(2,4),(2,6),(2,8),(2,10),\\
332
&(3,4),(3,6), (3,8),\\
333
&(5,4), (5,6), (7,4), (11,4)\},
334
\end{align*}
335
in which case $d_k(\Gamma_0(p))=0$.
336
\end{theorem}
337
\begin{proof}
338
(Romyar and William came up with this.)
339
340
Let~$p$ be a prime. For~$N$ and~$k$ integers, let
341
$$
342
S_k(\Gamma_0(N),\Zpbar) := S_k(\Gamma_0(N),\Z) \tensor \Zpbar
343
$$
344
and
345
$$
346
S_k(\Gamma_0(N),\Fpbar) := S_k(\Gamma_0(N),\Zpbar) \tensor
347
\Fpbar.
348
$$
349
350
Let $\wp$ denote the maximal ideal of $\Zpbar$. In
351
\cite[\S3]{serre:antwerp72} Serre defines a linear map
352
$$
353
t: S_k(\Gamma_0(p), \Zpbar) \ra S_{k+(p-1)}(\Gamma_0(p),\Zpbar)
354
$$
355
given by
356
$$
357
t(f) = \Tr(f\cdot{} G)
358
$$
359
where~$G$ is an Eisenstein series of weight~$p-1$ such that
360
$G\con 1\pmod{\wp}$. Serre shows (see Lemme~9 with $m=0$) that
361
$t(f) \con f \pmod{\wp}$.
362
363
By ??, there is a basis $f_1,\ldots,f_n$ of Hecke eigenforms for
364
$S_k(\Gamma_0(p),\Qpbar)$, and we may assume these $f_i$ are
365
normalized so that the leading coefficient of each~$q$-expansion
366
is~$1$. If
367
$$\dim S_{k+(p-1)}(\Gamma_0(p),\Zpbar) < \dim S_k(\Gamma_0(p),
368
\Zpbar)$$
369
then the set of $q$-expansions
370
$$
371
t(f_1) \!\!\!\!\pmod{\wp}, \,\ldots, \,t(f_n)\!\!\!\!\pmod{\wp}
372
$$
373
can not be linearly independent, which implies by Lemma~?? that
374
$d_k(\Gamma_0(p))>0$.
375
376
It follows from ?? that
377
\begin{align*}
378
\dim S_k(\Gamma_0(p))
379
&= \frac{(k-1)(p+1)}{12}\\
380
&+ \left( 1 + \kr{-4}{p}\right) \cdot \left( \frac{1-k}{4} +
381
\left\lfloor\frac{k}{4}\right\rfloor \right) + \left( 1 +
382
\kr{-3}{p}\right) \cdot \left(\frac{1-k}{3} +
383
\left\lfloor\frac{k}{3}\right\rfloor \right) - 1.
384
\end{align*}
385
Thus
386
$$
387
\dim S_k(\Gamma_0(p)) \geq \frac{(k-1)(p+1)}{12} - 3.
388
$$
389
By ??,
390
$$
391
\dim S_{k+p-1}(\Gamma_0(1)) = \begin{cases}
392
\lfloor \frac{k+p-1}{12}\rfloor - 1 & \text{ $k+p-1\con 2\pmod{12}$},\\
393
\lfloor \frac{k+p-1}{12}\rfloor & \text{otherwise.}
394
\end{cases}
395
$$
396
If
397
$$
398
\dim S_k(\Gamma_0(p)) \leq \dim S_{k+p-1}(\Gamma_0(1))
399
$$
400
then $(k-2)p\leq 36$. This reduces the assertion of the theorem
401
to a very small finite computation, which we did using the
402
algorithms described earlier in this paper.
403
\end{proof}
404
405
406
407
\begin{conjecture}
408
Suppose $p>2$ is a prime and $k\geq 3$ is an integer. If
409
\begin{align*}
410
(p,k) \not\in \{&(3,3),(3,4),(3,5),(3,6),(3,7),(3,8),\\
411
&(5,3),(5,4), (5,5), (5,6), (5,7)\\
412
&(7,3), (7,4), (7,5), (11,3), (11,4), (11,5),\\
413
&(13,3), (17,3), (19,3)\}
414
\end{align*}
415
then $d_k(\Gamma_1(p))>0$.
416
\end{conjecture}
417
418
419
\section{The Conjecture}
420
\newcommand{\tT}{\tilde{\T}}
421
422
Let~$k=2m$ be an even integer and~$p$ a prime. Let $\T$ be the Hecke
423
algebra associated to $S_k(\Gamma_0(p))$ and let $\tT$ be the
424
normalization of $\tT$ in $\T\tensor\Q$.
425
\begin{conjecture}\label{conj:big}
426
$$
427
\ord_p([\tT : \T]) = \left\lfloor\frac{p}{12}\right\rfloor\cdot
428
\binom{m}{2} + a(p,m),
429
$$
430
where
431
$$
432
a(p,m) =
433
\begin{cases}
434
0 & \text{if $p\con 1\pmod{12}$,}\\
435
3\cdot\ds\binom{\lceil \frac{m}{3}\rceil}{2} & \text{if $p\con 5\pmod{12}$,}\\
436
2\cdot\ds\binom{\lceil \frac{m}{2}\rceil}{2} & \text{if $p\con 7\pmod{12}$,}\\
437
a(5,m)+a(7,m) & \text{if $p\con 11\pmod{12}$.}
438
\end{cases}
439
$$
440
In particular, when $k=2$ we conjecture that $[\tT:\T]$ is not
441
divisible by~$p$.
442
\end{conjecture}
443
Here $\binom{x}{y}$ is the binomial coefficient ``$x$ choose $y$'',
444
and floor and ceiling are as usual. We have checked this conjecture
445
against significant numerical data. (Will describe here.)
446
447
448
\end{document}
449
450
\message{ !name(discheckediv.tex) !offset(-450) }
451