CoCalc Public Fileswww / papers / 389 / 389.tex
Author: William A. Stein
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12\title{The Modular Curve $X_0(389)$:\\
13{\large Discriminants, Ranks, Shafarevich-Tate Groups, and
14Weierstrass Points}}
15\author{William Stein}
16\date{April 2002}
17\renewcommand{\T}{\mathbf{T}}
18\begin{document}
19\maketitle
20
21\section{Introduction}
22Let $N$ be a positive integer, and let $X_0(N)$ be the compactified
23coarse moduli space that classifies pairs $(E,C)$ where $E$ is an
24elliptic curve and $C$ is a cyclic subgroup of order~$N$.  The space
25$X_0(N)$ has a canonical structure of algebraic curve over~$\Q$, and
26its properties have been very well studied during the last forty years.
27For example, Breuil, Conrad, Diamond, Taylor, and Wiles proved that
28every elliptic curve over~$\Q$ is a quotient of some $X_0(N)$.
29
30The smallest~$N$ such that the Jacobian of $X_0(N)$ has positive
31Mordell-Weil rank is~$37$, and Zagier studied the genus-two curve
32$X_0(37)$ in depth in his paper~\cite{zagier:modular}.  From this
33viewpoint, the next modular curve deserving intensive investigation is
34$X_0(389)$, which is the first modular curve whose Jacobian has
35Mordell-Weil rank larger than that predicted by the signs in the
36functional equations of the $L$-series attached to simple factors of
37its Jacobian; in fact, $389$ is the smallest conductor of an elliptic
38curve with Mordell-Weil rank~$2$.  Note that $389$ is prime and
39$X_0(389)$ has genus $g=32$, which is much larger than the genus~$2$
40of $X_0(37)$, which makes explicit investigation more challenging.
41
42Work of Kolyvagin \cite{kolyvagin:weil, kolyvagin:subclass} and
43Gross-Zagier \cite{gross-zagier} has completely resolved the rank
44assertion of the Birch and Swinnerton-Dyer conjecture (see, e.g.,
45\cite{tate:bsd}) for elliptic curves~$E$ with $\ord_{s=1} L(E,s) \leq 461$.  The lowest-conductor elliptic curve~$E$ that doesn't submit to
47the work of Kolyvagin and Gross-Zagier is the elliptic curve~$E$ of
48conductor~$389$ mentioned in the previous paragraph.  At present we
49don't even have a conjectural natural construction of a finite-index
50subgroup of $E(\Q)$ analogous to that given by Gross and Zagier for
51rank~$1$ (but see Mazur's work on universal norms, which might be used
52to construct $E(\Q)\tensor\Z_p$ for some auxiliary prime~$p$).
53
54Inspired by the above observations, and with an eye towards providing
55helpful data for anyone trying to generalize the work of Gross,
56Zagier, and Kolyvagin, in this paper we compute everything we can
57about the modular curve $X_0(389)$.  Some of the computations of this
58paper have already proved important in several other papers: the
59discriminant of the Hecke algebra attached to $X_0(389)$ plays a roll
60in \cite{ribet:torsion}, the verification of condition 3 in
61\cite{merel-stein}, and the remark after Theorem~1 of
62\cite{gordon-ono:vis}; also, the arithmetic of $J_0(389)$ provides a
63key example in \cite[\S4.2]{agashe-stein:visibility}.  Finally, this
64paper serves as an entry in an encyclopaedia, atlas or hiker's
65guide to modular curves'', in the spirit of N.~Elkies
66(see \cite[pg.~22]{elkies:ffield}).
67
68We hilight several surprising firsts'' that occur at level~$389$.
69The discriminant of the Hecke algebra attached to $S_2(\Gamma_0(389))$
70has the apparently unusual property that it is divisible by~$p=389$
71(see Section~\ref{sec:disc_div}).  Also $N=389$ is the smallest integer such
72that the order of vanishing of $L(J_0(N),s)$ at $s=1$ is larger than
73predicted by the functional equations of eigenforms (see Section~\ref{sec:mwranks}).
74The author conjectures that $N=389$ is the smallest
75level such that an optimal newform factor of $J_0(N)$
76appears to have Shafarevich-Tate group with nontrivial odd
77part (see Section~\ref{sec:sha}).
78Atkin conjectures that $389$ is the largest prime such that
79the cusp of $X_0^+(389)$ fails to be a
80Weierstrass point (see Section~\ref{sec:atkin}).
81
82\noindent{\bf Acknowledgement.}
83Noam Elkies, Ken Ribet, Matt Baker.
84Hendrik Lenstra, for suggesting a method to compute discriminants
85of Hecke algebra efficiently (see Section~\ref{sec:hecke_algebra}).
86
87\section{Factors of $J_0(389)$}
88To each newform $f\in S_2(\Gamma_0(389))$, Shimura \cite{shimura:factors}
89associated a quotient $A_f$ of $J_0(389)$, and
90$J_0(389)$ is isogeneous to the  product $\prod A_f$,
91where the product runs over the $\Gal(\Qbar/\Q)$-conjugacy classes of newforms.
92Moreover, because $389$ is prime each factor $A_f$ cannot be decomposed further
93up to isogeny, even over $\Qbar$ (see \cite{ribet:endo}).
94
95\subsection{Newforms of Level $389$}
96There are five $\Gal(\Qbar/\Q)$-conjugacy classes of newforms in $S_2(\Gamma_0(389))$.
97The first class corresponds to the unique elliptic curve of conductor $389$, and its
98$q$-exansion begins
99$$100 f_1 = q - {2}q^{2} - {2}q^{3} + {2}q^{4} - {3}q^{5} + {4}q^{6} - {5}q^{7} + q^{9} + {6}q^{10} + \cdots. 101$$
102The second has coefficients in the quadratic field $\Q(\sqrt{2})$, and has $q$-expansion
103$$104 f_2 = q + {\sqrt{2}}q^{2}+({\sqrt{2} - {2}})q^{3} - q^{5}+({{-2}\sqrt{2} + {2}})q^{6}+ \cdots. 105$$
106The third has coefficients in the cubic field generated by a root $\alpha$ of
107$x^3-4x-2$:
108$$109 f_3 = q + {\alpha}q^{2} - {\alpha}q^{3}+({\alpha^{2} - {2}})q^{4}+({-\alpha^{2} + 1})q^{5} - {\alpha^{2}}q^{6} +\cdots. 110$$
111The fourth has coefficients that generate the degree-six field defined by
112a root $\beta$ of $x^6+3x^5-2x^4-8x^3+2x^2+4x-1$ and $q$-expansion
113$$114 f_4 = q + {\beta}q^{2}+({\beta^{5} + {3}\beta^{4} - {2}\beta^{3} - {8}\beta^{2} + \beta + {2}})q^{3}+\cdots. 115$$
116The fifth and final newform (up to conjugacy) has coefficients that generate
117the degree $20$ field defined by a root of
118\begin{eqnarray*}
119f_5 &=& x^{20} - 3x^{19} - 29x^{18} + 91x^{17} + 338x^{16} - 1130x^{15} - 2023x^{14} + 7432x^{13} \\
120&& + 6558x^{12} - 28021x^{11} - 10909x^{10} + 61267x^{9} + 6954x^8 - 74752x^7   \\
121&& + 1407x^6+ 46330x^5 - 1087x^4 - 12558x^3 - 942x^2 + 960x + 148.
122\end{eqnarray*}
123
124\subsubsection{Congruences}
125The vertices in Figure~\ref{fig:cong} correspond to the newforms $f_i$; there is an edge between
126$f_i$ and $f_j$ labeled~$p$ if there is a maximal ideal $\wp\mid p$ of the field generated
127by the Fourier coefficients of $f_i$ and $f_j$ such that $f_i \con f_j \pmod{\wp}$.
128\begin{figure}
129\begin{center}
130\psfrag{f1}{$f_1$}
131\psfrag{f2}{$f_2$}
132\psfrag{f3}{$f_3$}
133\psfrag{f4}{$f_4$}
134\psfrag{f5}{$f_5$}
135\psfrag{2          }{$2$}
136\psfrag{2       }{$2$}
137\psfrag{2,5}{$2,5$}
138\psfrag{3         }{$3$}
139\psfrag{31}{$31$}
140\psfrag{2}{$2$}
141\psfrag{2         }{$2$}
142\psfrag{2           }{$2$}
143\psfrag{2       }{$2$}
144\includegraphics{cong.epsi}
145\caption{Congruences Between Newforms\label{fig:cong}}
146\end{center}
147\end{figure}
148
149
150\comment{
151> D := SortDecomposition(NewformDecomposition(CuspidalSubspace(ModularSymbols(389,2))));
152  S := CuspidalSubspace(ModularForms(389,2));
153  f := [* *]; for i in [1..5] do Append(~f,Newform(S,i)); end for;
154  for i in [1..5] do
155    for j in [i+1..5] do
156       F := Factorization(#CongruenceGroup(Parent(f[i]),Parent(f[j]),200));
157       if #F gt 0 then
158          printf "%o--%o [label=\"", i,j;
159          for i in [1..#F] do
160             printf "%o",F[i][1];
161             if i lt #F then
162                printf ",";
163             end if;
164          end for;
165          printf "\"];\n";
166      end if;
167    end for;
168  end for;
169
170
171shell-> ps2epsi cong.ps cong.epsi
172
173}
174
175\subsection{Isogeny Structure}
176We deduce from the above determination of the newforms in $S_2(\Gamma_0(389))$
177that $J_0(389)$ is $\Q$-isogenous to a product of $\Qbar$-simple abelian varieties
178$$179 J \sim A_1\cross A_2 \cross A_3 \cross A_4 \cross A_{5}. 180$$
181
182View the duals $A_i^{\vee}$ of the $A_i$ as abelian subvarieties of $J_0(389)$.
183Using modular symbols as in \cite[\S3.4]{agashe-stein:bsd}
184we find that, for $i\neq j$, a prime~$p$
185divides $\#(A_i^{\vee}\intersect A_j^{\vee})$ if and only if $f_i\con f_j\pmod{\wp}$
186for some prime $\wp\mid p$ (recall that the congruence primes
187are given in Figure~\ref{fig:cong} above).
188
189
190\subsection{Mordell-Weil Ranks}\label{sec:mwranks}
191Suppose $f\in S_2(\Gamma_0(N))$ is a newform of some level~$N$.
192The functional equation for $L(f,s)$ implies that $\ord_{s=1}L(f,s)$
193is odd if and only if the sign of the eigenvalue of the Atkin-Lehner
194involution $W_N$ on~$f$ is $+1$.
195\begin{proposition}\label{prop:minanrank}
196If $f\in S_2(\Gamma_0(N))$ is a newform of level $N<389$, then
197$\ord_{s=1}L(f,s)$ is either$0$ or~$1$.
198\end{proposition}
199\begin{proof}
200The proof amounts to a large computation, which divides into two parts:
201\begin{enumerate}
202\item Verify, for each newform~$f$ of level $N<389$ such that
203$W_N(f) = -f$, that $L(f,1) = *\int_{0}^{i\infty} f(z) dz$ (for some nonzero $*$)
204is nonzero.  This is a purely algebraic computation involving modular symbols.
205\item Verify, for each newform~$f$ of level $N<389$ such that
206$W_N(f) = f$, that $L'(f,1)\neq 0$ (see \cite[\S4.1]{empirical},
207which points to \cite[\S2.11,\S2.13]{cremona:algs}).
208We do this by approximating an infinite
209series that converges to $L'(f,1)$ and noting that the value we
210get is far from~$0$.
211\end{enumerate}
212\end{proof}
213
214Thus $N=389$ is the smallest level such that the $L$-series of
215some factor $A_f$ of $J_0(N)$ has order of vanishing higher than
217
218%The elliptic curve~$A_1$ of rank~$2$ is the lowest-conductor elliptic
219%curve having rank~$>1$, because every elliptic curve over~$\Q$ is modular
221%any elliptic curve of conductor $<389$ having rank~$>1$.
222
223%The $L$-function corresponding to~$A_{5}$ does not vanish at $s=1$,
224%so $A_{5}$ has analytic rank~$0$, and hence algebraic rank~$0$, by
225%the theorem of Kolyvagin and Logachev.
226
227%The genus of $X^+=X/w_{389}$ is $g^{+} =11$, and $J_0(389)^{-}$
228%is isogeneous to $A_{1} \cross A_{5}.$
229
230\begin{proposition}\label{prop:anrank}
231The following table summarizes the dimensions and Mordell-Weil ranks
232(over the image of the Hecke ring) of the newform factors of $J_0(N)$:
233\begin{center}
234\begin{tabular}{|l|c|c|c|c|c|}\hline
235\text{\rm }& $A_1$ & $A_2$ & $A_3$ & $A_4$ & $A_{5}$\\\hline
236\text{\rm Dimension}& $1$ & $2$ & $3$ & $6$ & $20$\\\hline
237\text{\rm Rank} & $2$ & $1$ & $1$ & $1$ & $0$\\\hline
238\end{tabular}
239\end{center}
240\end{proposition}
241\begin{proof}
242The elliptic curve $A_1$ is $389A$ in Cremona's tables, which is
243the elliptic curve of smallest conductor having rank $2$.
244For $A_{5}$ we directly compute whether or not the $L$-function
245vanishes using modular symbols, by taking an inner product with
246the winding element $e_w=-\{0,\infty\}$.  We find that the $L$-function
247does not vanish.  By Kolyvagin-Logachev, it follows that $A_{5}$ has
248Mordell-Weil rank $0$.
249
250For each of the other three factors, the sign of the functional
251equation is odd, so the analytic ranks are odd.
252As in the proof of Proposition~\ref{prop:minanrank},
253we verify that the analytic rank is~$1$ in each case.  By work of
254Gross, Zagier, and Kolyvagin it follows that the ranks are~$1$.
255\end{proof}
256
257%The elliptic curve $A_1$ has minimal Weierstrass model
258%             $$y^2+y=x^3+x^2-2x.$$
259%It satisfies $A_1(\Q)=\Z\oplus \Z$. The modular
260%degree is $40$.
261
262\comment{\subsection{Characteristic polynomials}
263The characteristic polynomials of the first two Hecke operators are
264as follows:
265\begin{eqnarray*}
266T_2&=&(x + 2)({x^{2} - {2}})({x^{3} - {4}x - {2}})(x^{6} + {3}x^{5} - {2}x^{4} - {8}x^{3} + {2}x^{2} + {4}x - 1)\\
267 &&(x^{20} - {3}x^{19} - {29}x^{18} + {91}x^{17} + {338}x^{16} - {1130}x^{15} - {2023}x^{14}+ {7432}x^{13}+ {6558}x^{12}\\
268 &&- {28021}x^{11} - {10909}x^{10} + {61267}x^{9} + {6954}x^{8}- {74752}x^{7} + {1407}x^{6} + {46330}x^{5}\\
269 && - {1087}x^{4} - {12558}x^{3} - {942}x^{2} + {960}x + {148})\\
270T_3&=&(x + 2)(x^2 + 4x + 2)(x^3 -4x + 2)(x^6 + 5x^5 + 4x^4 -13x^3 -21x^2 -6x + 1)\\
271 &&(x^{20} -11x^{19} + 19x^{18} + 204x^{17} -845x^{16} -781x^{15} + 8883x^{14} -6177x^{13} -40916x^{12}\\
272 &&+ 63058x^{11} + 85034x^{10} -215618x^9 -46920x^8 + 342529x^7 -84612x^6 -241030x^5 \\
273 &&+ 112365x^4 + 51018x^3 -28526x^2 + 3560x -100)
274\end{eqnarray*}
275}
276
277
278\section{The Hecke algebra}\label{sec:hecke_algebra}
279\subsection{The Discriminant is Divisible By $p$}
280\label{sec:disc_div}
281Let $N$ be a positive integer.
282The Hecke algebra $\T\subset\End(S_2(\Gamma_0(N)))$ is
283the subring generated by all Hecke operators $T_n$ for $n=1,2,3,\ldots$.
284We are concerned with the {\em discriminant} of the
285trace pairing $(t,s)\mapsto \Tr(ts)$.
286
287When~$N$ is prime, $\T_\Q=\T\tensor_\Z\Q$ is a product
288$K_1\cross\cdots\cross K_n$ of  totally real number fields.
289Let $\tilde{\T}$ denote the integral closure of~$\T$ in $\T_\Q$;
290note that $\tilde{\T}=\prod \O_i$ where~$\O_i$ is the ring
291of integers of $K_i$.
292Then $\disc(\T)=[\tilde{\T}:\T]\cdot \prod_{i=1}^n\disc(K_i)$.
293%One can view the primes dividing $\prod_{i=1}^n\disc(K_i)$ as being
294%associated to singularities of the irreducible components of $\Spec(\T)$,
295%and the primes dividing $[\tilde{\T}:\T]$ are associated to intersections
296%of irreducible components, or, equivalently, congruences between eigenforms.
297
298\begin{proposition}
299The discriminant of the Hecke algebra associated to
300$S_2(\Gamma_0(389))$ is
301$$302 2^{53}\cdot{}3^4\cdot{}5^6\cdot{}31^2\cdot{}37\cdot{}389 303 \cdot{}3881\cdot{}215517113148241\cdot{}477439237737571441. 304$$
305\end{proposition}
306\begin{proof}
307By \cite{agashe-stein:schoof-appendix}, the Hecke algebra~$\T$
308is generated as a $\Z$-module by $T_1,T_2,\ldots T_{65}$.
309%Then $\disc(\T)$ can be quickly computed once one finds
310%a $\Z$-basis for~$\T$, which can be accomplished as follows.
311To compute $\disc(\T)$, we proceed as follows.
312First, compute the space $\sS_2(\Gamma_0(389))$ of
313cuspidal modular symbols, which is a faithful $\T$-module.
314Choose a random element $x\in \sS_2(\Gamma_0(389))_+$ of the $+1$-quotient
315of the cuspidal modular symbols, then compute the
316images $v_1=T_1(x), v_2=T_2(x), \ldots, v_{65}=T_{65}(x)$.  If these don't span
317a space of dimension $32=\rank_\Z \T$ choose a new random element~$x$
318and repeat.  Using the Hermite Normal Form, find a $\Z$-basis $b_1,\ldots, b_{32}$ for
319the $\Z$-span of $v_1,\ldots, v_{65}$.   The trace pairing on $\T$ induces a trace
320pairing on the $v_i$, and hence on the $b_i$.  Then $\disc(\T)$ is the discriminant
321of this pairing on the $b_i$.   The reason we embed $\T$ in $\sS_2(\Gamma_0(389))_+$
322as $\T{}x$ is because directly finding a $\Z$-basis for $\T$ would involve computing
323the Hermite Norm Form of a list of $65$ vectors in a $1024$-dimensional space, which
324is unnecessarily difficult (though possible).
325\end{proof}
326
327
328
329We compute this discriminant by applying the definition of
330discriminant to a matrix representation of the first~$65$ Hecke
331operators $T_1,\ldots, T_{65}$.  Matrices representing these
332Hecke operators were computed using the modular symbols algorithms
333described in \cite{cremona:algs}.
334[Sturm, {\em On the congruence of modular forms}].
335%\end{proof}
336
337%It might be possible to choose a
338%bound smaller than~$65$ by directly considering a basis of
339%modular forms.
340%\comment{  A \pari{} program which can usually compute the discriminant
341%of a commutative $\Z$-module represented by matrices is included as an
342%appendix. }
343
344\comment{
345We should also note that by computing successive greatest common divisors of
346discriminants of characteristic polynomials of Hecke operators $T_p$, one
347can get a fairly good multiplicative upper bound on the discriminant.
348If $D$ is the bound computed by computing successive greatest common divisors of
349discriminants of characteristic polynomials of Hecke operators $T_p$, until the
350gcd stabilized $15$ times in a row, then $D$ is $2^{10}$ times the correct
351discriminant (in the case $N=389$, of course).}
352
353
354In the case of $X_0(389)$,
355$\T\tensor \Q = K_1\cross K_2\cross K_3\cross K_6\cross K_{20},$
356where~$K_d$ has degree~$d$ over~$\Q$.
357We have
358\begin{eqnarray*}
359K_1&=&\Q, \\
360K_2&=&\Q(\sqrt{2})\\
362K_6&=&\Q(\gamma),
367-1087\delta^4-12558\delta^3-942\delta^2+960\delta+148=0.
368\end{eqnarray*}
369
370\comment{Note that the given generators of these fields are the eigenvalues
371$a_2$ of the corresponding eigenforms.}
372
373The discriminants of the $K_i$ are
374\begin{center}
375\begin{tabular}{|c|c|c|c|}\hline
376    $K_1$ & $K_2$ & $K_3$ & $K_6$ \\\hline
377 1 & $2^3$  & $\quad 2^2\cdot 37$ & $\quad 5^3\cdot 3881$ \\\hline
378\end{tabular}
379\end{center}
380and
381$$382 \disc(K_{20}) = 2^{14}\cdot 5\cdot 389 \cdot 215517113148241\cdot 477439237737571441. 383$$
384Observe that the discriminant of $K_{20}$ is divisible by~$389$.
385The product of the discriminants is
386$$2^{19}\cdot 5^4\cdot 37\cdot 389\cdot 3881\cdot 215517113148241\cdot 387 477439237737571441.$$
388This differs from the exact discriminant by a factor
389of $2^{34}\cdot 3^4\cdot 5^2\cdot 31^2$, so
390the index of $\T$ in its normalization is
391$$[\tilde{\T}:\T]=2^{17}\cdot 3^2\cdot 5\cdot 31.$$
392Notice that $389$ does not divide this index, and that $389$ is not
393a congruence prime'', so~$389$ does not divide any
394modular degrees.
395
396\begin{question}
397Is there a newform optimal quotient $A_f$ of $J_0(p)$
398such that~$p$ divides the modular degree of $A_f$?
399(No, if $p<14000$.)
400\end{question}
401
402\comment{% this is wrong since the Hecke algebra doesn't split as a product!
403Away from the primes $211$ and $65011$, $a_2$ actually generates the
404ring $\Z(f_{20})=\Z[a_1,a_2,\ldots]$ generated by the Fourier
405coefficients of one of the degree $20$ eigenforms. The discriminant
406of the order generated by $a_2$ divided by the discriminant of the
407maximal order in $K_{20}$ is $2^{44}\cdot 5^2\cdot 211^2\cdot 65011^2$.
408Thus the discriminant of $\T$ must be divisible by
409$$2^{63}\cdot 5^6\cdot 37\cdot 389\cdot 3881\cdot 215517113148241\cdot 410 477439237737571441.$$
411}
412
413\subsection{Congruences Primes in $S_{p+1}(\Gamma_0(1))$}
414K.~Ono asked the following question, in connection with
415Theorem~1 of~\cite{gordon-ono:vis}.
416\begin{question}
417Let~$p$ be a prime.  Is~$p$ ever a congruence prime on $S_{p+1}(\Gamma_0(1))$?
418More precisely, if~$K$ is the number field generated by all the
419eigenforms of weight $p+1$ on $\Gamma_0(1)$, can there be a prime
420ideal~$\wp\mid p$ for which
421$422f \con g \pmod{\wp} 423$
424for distinct eigenforms $f, g \in S_{p+1}(\Gamma_0(1))$?
425\end{question}
426
427The answer is yes''.
428There is a standard relationship between
429$S_{p+1}(\Gamma_0(1))$ and $S_2(\Gamma_0(p))$.  As noted
430in Section~\ref{sec:disc_div},  $p=389$ is a
431congruence prime for $S_2(\Gamma_0(389))$, so we investigate
432$S_{389+1}(\Gamma_0(1))$.
433
434\begin{proposition}
435There exist distinct newforms $f, g\in S_{389+1}(\Gamma_0(1))$
436and a prime $\wp$ of residue characteristic $389$ such that
437$f\con g \pmod{\wp}$.
438\end{proposition}
439\begin{proof}
440We compute the characteristic polynomial~$f$ of the Hecke operator $T_2$
441on $S_{389+1}(\Gamma_0(1))$ using nothing
442more than \cite[Ch.~VII]{serre:arithmetic}.
443We find that~$f$ factors modulo $389$ as follows:
444\begin{eqnarray*}
445\fbar&=&(x + 2)(x + 56)(x + 135)(x + 158)(x + 175)^2(x + 315)(x + 342)(x^2 + 387)\\
446&&(x^2 + 97x + 164)(x^2 + 231x + 64)(x^2 + 286x + 63)\\
447&&(x^5 + 88x^4 + 196x^3 + 113x^2 + 168x + 349)\\
448&&(x^{11} + 276x^{10} + 182x^9 + 13x^8 + 298x^7 + 316x^6 + 213x^5 \\
449&&\qquad+ 248x^4 + 108x^3 + 283x^2 + x + 101)
450\end{eqnarray*}
451Moreover,~$f$ is irreducible and $389\mid\mid \disc(f)$, so
452the square factor $(x+175)^2$ implies that~$389$
453is ramified in the degree-$32$ field $L$ generated by a single root of $f$.
454Thus there are exactly $31$ distinct homomorphisms from the ring of integers of $L$ to
455$\Fbar_{389}$.  That is, there are exactly $31$ ways to reduce the $q$-expansion of a
456newform in $S_{390}(\Gamma_0(1))$ to obtain a
457$q$-expansion in $\Fbar_{389}[[q]]$.
458Let~$K$ be the field generated by all eigenvalues of the $32$
459newforms $g_1, \ldots g_{32} \in S_{390}(\Gamma_0(1))$, and let $\wp$
460be a prime of $\O_K$ lying over $389$.
461Then the subset
462$\{g_1 \pmod{\wp}, g_2 \pmod{\wp}, \ldots, g_{32}\pmod{\wp}\}$
463of $\Fbar_{389}[[q]]$ has cardinality at most $31$, so
464there exists $i \neq j$ such that
465$g_i \con g_j \pmod{\wp}$.
466\end{proof}
467
468
469\section{Supersingular Points in Characteristic $389$}
470\subsection{The Supersingular $j$-invariants In Characteristic $389$}
471Let $\alpha$ be a root of $\alpha^2+95\alpha+20$.  Then the
472$33=g(X_0(389))+1$ supersingular $j$-invariants in $\F_{389^2}$ are
473$$474\begin{array}{l} 4750, 7, 16, 17, 36, 121, 154, 220, 318, 327, 358, 60\alpha + 22, 68\alpha + 166, 80\alpha + 91, 86\alpha + 273, \\ 47693\alpha + 333, 123\alpha + 350, 123\alpha + 375, 129\alpha + 247, 131\alpha + 151, 160\alpha + 321, 176\alpha + 188,\\ 477213\alpha + 195, 229\alpha + 292, 258\alpha + 154, 260\alpha + 51, 266\alpha + 335, 266\alpha + 360, 296\alpha + 56, \\ 478303\alpha + 272, 309\alpha + 271, 321\alpha + 319, 329\alpha + 157. 479\end{array} 480$$
481Figure~\ref{fig:graph_t2} contains the graph of the Hecke operator $T_2$ computed
482using method of Mestre and Oesterl\'{e} \cite{mestre:graphs}.
483Here, if $T_2(j) = \sum_{i} j_i$, then there is an edge from
484the node~$j$ to each of the nodes~$j_i$.
485Thus, for example,
486$$487 T_2([318]) = [0] + [93\alpha+333] + [296\alpha+56]. 488$$
489\begin{center}
490\begin{figure}
491\psfrag{1}{\small\!\!\!\!\!\!\!\!\!\!\!$0$}
492\psfrag{2}{\small\!\!\!\!\!\!\!\!\!\!\!$7$}
493\psfrag{3}{\small\!\!\!\!\!\!\!\!\!\!\!$16$}
494\psfrag{4}{\small\!\!\!\!\!\!\!\!\!\!\!$17$}
495\psfrag{5}{\small\!\!\!\!\!\!\!\!\!\!\!$36$}
496\psfrag{6}{\small\!\!\!\!\!\!\!\!\!\!\!$121$}
497\psfrag{7}{\small\!\!\!\!\!\!\!\!\!\!\!$154$}
498\psfrag{8}{\small\!\!\!\!\!\!\!\!\!\!\!$220$}
499\psfrag{9}{\small\!\!\!\!\!\!\!\!\!\!\!$318$}
500\psfrag{10}{\small\!\!\!\!\!\!\!\!\!\!\!$327$}
501\psfrag{11}{\small\!\!\!\!\!\!\!\!\!\!\!$358$}
502\psfrag{12}{\small\!\!\!\!\!\!\!\!\!\!\!$60\alpha + 22$}
503\psfrag{13}{\small\!\!\!\!\!\!\!\!\!\!\!$68\alpha + 166$}
504\psfrag{14}{\small\!\!\!\!\!\!\!\!\!\!\!$80\alpha + 91$}
505\psfrag{15}{\small\!\!\!\!\!\!\!\!\!\!\!$86\alpha + 273$}
506\psfrag{16}{\small\!\!\!\!\!\!\!\!\!\!\!$93\alpha + 333$}
507\psfrag{17}{\small\!\!\!\!\!\!\!\!\!\!\!$123\alpha + 350$}
508\psfrag{18}{\small\!\!\!\!\!\!\!\!\!\!\!$123\alpha + 375$}
509\psfrag{19}{\small\!\!\!\!\!\!\!\!\!\!\!$129\alpha + 247$}
510\psfrag{20}{\small\!\!\!\!\!\!\!\!\!\!\!$131\alpha + 151$}
511\psfrag{21}{\small\!\!\!\!\!\!\!\!\!\!\!$160\alpha + 321$}
512\psfrag{22}{\small\!\!\!\!\!\!\!\!\!\!\!$176\alpha + 188$}
513\psfrag{23}{\small\!\!\!\!\!\!\!\!\!\!\!$213\alpha + 195$}
514\psfrag{24}{\small\!\!\!\!\!\!\!\!\!\!\!$229\alpha + 292$}
515\psfrag{25}{\small\!\!\!\!\!\!\!\!\!\!\!$258\alpha + 154$}
516\psfrag{26}{\small\!\!\!\!\!\!\!\!\!\!\!$260\alpha + 51$}
517\psfrag{27}{\small\!\!\!\!\!\!\!\!\!\!\!$266\alpha + 335$}
518\psfrag{28}{\small\!\!\!\!\!\!\!\!\!\!\!$266\alpha + 360$}
519\psfrag{29}{\small\!\!\!\!\!\!\!\!\!\!\!$296\alpha + 56$}
520\psfrag{30}{\small\!\!\!\!\!\!\!\!\!\!\!$303\alpha + 272$}
521\psfrag{31}{\small\!\!\!\!\!\!\!\!\!\!\!$309\alpha + 271$}
522\psfrag{32}{\small\!\!\!\!\!\!\!\!\!\!\!$321\alpha + 319$}
523\psfrag{33}{\small\!\!\!\!\!\!\!\!\!\!\!$329\alpha + 157$}
524\includegraphics[height=0.98\textheight,width=1.1\textwidth]{graph_t2.epsi}
525%\caption{Figure~\ref{fig:graph_t2}: Graph of Hecke Operator $T_2$}
526\caption{Graph of the Hecke Operator $T_2$\label{fig:graph_t2}}
527\end{figure}
528\end{center}
529\comment{
530> S := SupersingularInvariants(389);
531> k<a> := GF(389^2);
532> P := Sort([Reverse(Eltseq(a)) : a in S]);
533> B := [GF(389^2)!Reverse(a) : a in P];
534> X := SupersingularModule(389);
535> Xss_points := [<x,x> : x in B];
536> T2 := HeckeOperator(X,2);
537> Attach("graph.m");
538> Digraph(T2);
539...
540
541shell-> dot -Tps t2.dot -o graph_t2.ps; ps2epsi graph_t2.ps graph_t2.epsi
542}
543
545
546\section{Miscellaneous}
547
548\subsection{The Shafarevich-Tate Group}\label{sec:sha}
549Using visibility theory \cite[\S4.2]{agashe-stein:visibility},
550one sees that $\#\Sha(A_5)$ is divisible by an odd prime, because
551$$552 (\Z/5\Z)^2 \ncisom A_1(\Q)/5 A_1(\Q) \subset \Sha(A_{5}). 553$$
554Additional computations suggest the following conjecture.
555\begin{conjecture}
556$N=389$ is the smallest level such that there is an optimal newform
557quotient $A_f$ of $J_0(N)$ with $\#\Sha(A_f)$
558divisible by an odd prime.
559\end{conjecture}
560
561
562\subsection{Weierstrass Points on $X_0^+(p)$}\label{sec:atkin}
563Oliver Atkin has conjectured that $389$ is the largest prime such that
564the cusp on $X_0^+(389)$ fails to be a Weierstrass point.  He verified
565that the cusp of $X_0^+(389)$ is not a Weierstrass point but
566that the cusp of $X_0^+(p)$ is a Weierstrass point for all primes~$p$
567such that $389<p\leq 883$ (see, e.g., \cite[pg.39]{elkies:ffield}).
568In addition, the author has extended the verification of Atkin's conjecture
569for all primes $<3000$.    Explicitly, this involves computing a reduced-echelon
570basis for the subspace of $S_2(\Gamma_0(p))$ where the Atkin-Lehner involution
571$W_p$ acts as $+1$, and comparing the largest valuation of an element of this
572basis with the dimension of the subspace.  These numbers differ exactly when
573the cusp is a Weierstrass point.
574
575
576\comment{
577function IsWP(p)
578   M := CuspidalSubspace(ModularSymbols(p,2,+1));
579   W := AtkinLehnerSubspace(M,p,+1);
580   d := Dimension(W);
581   Q := qExpansionBasis(W,d+5);
582   return Coefficient(Q[#Q],d) eq 0;
583end function;
584}
585
586\subsection{A Property of the Plus Part of the Integral Homology}
587For any positive integer~$N$, let $H^+(N) = H_1(X_0(N),\Z)^+$ be the
588$+1$ eigen-submodule for the action of complex conjugation on the
589integral homology of $X_0(N)$.  Then $H^+(N)$ is a module over the
590Hecke algebra~$\T$.  Let
591$$592 F^+(N) = \coker\left(H^+(N) \cross \Hom(H^+(N),\Z) \ra \Hom(\T,\Z)\right) 593$$
594where the map sends $(x,\vphi)$ to the homomorphism
595$t\mapsto \vphi(tx)$.  Then $\#F^+(p)\in \{1,2,4\}$ for
596all primes $p<389$, but $\#F^+(389) = 8$.
597
598\subsection{The Field Generated By Points of
599   Small Prime Order On an Elliptic Curve}
600The prime $389$ arises in a key way in the verification of condition~3
601in \cite{merel-stein}.
602
603
604\bibliography{biblio}
605
606\end{document}
607
608
609\section{PARI code}
610The following is a \pari{} program that might be helpful
611to the reader interested in computing discriminants of $\Z$-modules.
612In retrospect it looks unfinished''.
613\begin{verbatim}
614\\ disc.gp -- Usually compute the discriminant of a finite commutative
615\\            Z-module represented by matrices T1,...,Tn, such that
616\\            the inner product is <Ti,Tj> = tr(Ti*Tj).
617\\ The word usually'' is used in the description because the program
618\\ can fail if the matrices aren't generic enough.''  See
619\\ the coords'' function.
620{nrows(M)=matsize(M)[1];}
621{ncols(M)=matsize(M)[2];}
622\\ coords: make a matrix whose columns are the col's columns of all of the Ti
623{coords(col,   i,j)=matrix(nrows(T[1]),ncols(T),i,j,  T[j][i,col]);}
624\\ Find an integral basis for the module.
625\\ The columns of the matrix returned give the
626\\ linear combination a1*T1+a2*T2+...+an*Tn, ai in Z,
627\\ which give a system of generators for the module.
628{intbasis(
629  i,j,n,C,h,U,M)=
630  n=nrows(T[1]); i=1; C=coords(i);
631  while(matrank(C)<n,
632    i++; if(i>ncols(T), print("Error, not enough generators."); return(0););
633    C=coords(i););
634  print("The ", i,"th columns coordinatize the module.");
635  print("Computing mathnf.");
636  U=mathnf(C,2)[2];
637  print("Extracting integral basis.");
638  \\ take exactly those columns of U, so that when multiplied
639  \\ by C, the result is =/= 0.
640  M=matrix(nrows(U),n,i,j,0);
641  j=1;
642  for(i=1,ncols(U),
643    if(C*U[,i]!=0,
644      if(j>n,print("ERROR: PARI's mathnf is incorrect.");return(0););
645      M[,j]=U[,i]; j++)
646  );
647  if(j<n,print("ERROR: PARI's mathnf is producing invalid results."));
648  M;
649}
650{Bmat(     i=0,j=0)=
651  B=matrix(ncols(T),ncols(T),i,j,if(i>=j,trace(T[i]*T[j]),0));
652  for(i=1,nrows(B),for(j=i+1,ncols(B),B[i,j]=B[j,i]));
653  B;}
654\\ A = computed using intbasis; an integral basis of the Hecke algebra,
655\\     expressed on the T[i].
656\\ B = computed using Bmat; matrix of inner products (tr(T[i]*T[j])).
657{DiscMat(A,B)=mattranspose(A)*B*A;}
658\end{verbatim}
659
660
`