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% merel-stein_v6.tex
% 12 February 2001

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\def\prop#1{\noindent{\pc Proposition}\ #1. --- \ignorespaces}%Proposition 1.
\def\Def#1{\noindent{\pc Definition}\ #1. --- \ignorespaces}  %Definition 1.
\def\cor#1{\noindent{\pc Corollary}\ #1. --- \ignorespaces}   %Corollary 1.
\def\conj#1{\noindent{\pc Conjecture}\ #1. --- \ignorespaces} %Conjecture 1.
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\def\thp{\noindent{\pc Theorem}. --- \ignorespaces}           %Theorem 1.
\def\propp{\noindent{\pc Proposition}. --- \ignorespaces}     %Proposition 1.
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\def\corp{\noindent{\pc Corollary}. --- \ignorespaces}        %Corollairy 1.
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\def\remp{\noindent{\it Remark}: \ignorespaces}               %Remark 1.
\def\exep{\noindent{\it Example}: \ignorespaces}              %Example 1.
\def\dm{\noindent{\it Proof}. --- \ignorespaces}
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\def\Agashe{{$$}}
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\def\Merel{{$$}}
\def\Mes{{$$}}

\def\change#1{[[{\bf Change:} #1]]}

\centerline
{\titchap The field generated by the points of small}
\centerline
{\titchap prime order on an elliptic curve}
\medskip

\bigskip\bigskip\bigskip
\centerline{\pc Lo\"\i c Merel {\rm and} William A.~Stein}
\bigskip\bigskip\bigskip

\bigskip\bigskip\noindent
{\bf Introduction}
\bigskip

Let~$p$ be a prime number.
Let $\bar\B Q$ be an algebraic closure of $\B Q$, and
denote by $\B Q(\mu_p)$ the cyclotomic subfield of $\bar\B Q$
generated by the $p$th roots of unity.
Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the
points of order~$p$ of
$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.
\bigskip
\th{}{\it One has $p=2,3,5,13$ or $p>1000$.}
\bigskip

The case $p=7$ was treated by Emmanuel Halberstadt.  The
part of the theorem that concerns the case $p\equiv 3\!\!\pmod{4}$ is
given in~\Merel.  In this paper, we give the details that permit our
treating the more difficult case in which $p\equiv 1 \!\!\pmod{4}$.
We treat this last case with the aid of Proposition~2 below, which is
not present in \lc.
The case $p=13$ is currently under investigation by Marusia Rebolledo,
as part of her Ph.D.{} thesis.

\bigskip\noindent
{\bf 1. We recall the results of \Merel}
\bigskip
\change{Change the title to something like
Counterexamples define points on $X_0(p)(\B Q(\sqrt{p}))$''.}

Let $S_2(\Gamma_0(p))$ denote the space of cusp forms of weight~$2$ for
the congruence subgroup $\Gamma_0(p)$. Denote by $\B T$ the
subring of ${\rm End}\,S_2(\Gamma_0(p))$ generated by the
Hecke operators $T_n$ for all integers~$n$.
Let $f\in S_2(\Gamma_0(p))$ have $q$-expansion
$\sum_{n=1}^\infty a_nq^n$. When $\chi$ is a Dirichlet character,
denote by $L(f,\chi,s)$ the entire function which extends the
Dirichlet series $\sum_{n=1}^\infty a_n\chi(n)/n^s$.

Let $S$ be the set of isomorphism classes of supersingular elliptic
curves in characteristic~$p$.  Denote by $\Delta_S$ the group formed
by the divisors of degree~$0$ with support on~$S$.  It is equipped with
a structure of $\B T$-module (induced, for example, from the action
of the Hecke correspondences on the fiber at~$p$ of the regular minimal
model of $X_0(p)$ over $\B Z$).

Let $j\in\bar\B F_p-J_S$, where $J_S$ denotes the set of supersingular modular
invariants. We denote by  $\iota_j$ the homomorphism of
groups $\Delta_S\raw \bar\B F_p$ that associates to $\sum_E n_E[E]$
the quantity
$\sum_E n_E/(j-j(E))$, where $j(E)$ denotes the modular invariant of~$E$.

One says that an element  $j\in\B F_p$ is {\it anomalous}
if there exists an elliptic curve over $\B F_p$ with modular invariant~$j$
that possesses an $\B F_p$-rational point  of order~$p$
(then necessarily $j\notin{}J_S$).

\change{
Let~$p$ be a prime that is congruent to~$1$ modulo~$4$.
In the following proposition we prove, under a hypothesis on~$p$, that
if~$E$ is an elliptic curve over $\B Q(\mu_p)$ all of whose torsion is
$\B Q(\mu_p)$-rational, then for each subgroup $C\subset{}E(\B Q)$
of order~$p$,
the point $(E,C)$ on $X_0(p)$ is defined over $\B Q(\sqrt{p})$.  As we
will see in Proposition~2, this $\B Q(\sqrt{p})$-rationality
conclusion is contrary to fact, from which we conclude that such
elliptic curves~$E$ do not exist when the hypothesis on~$p$
is satisfied.  In Section~3 we verify this hypothesis
for $p=11$ and $13 < p < 1000$.
}

\bigskip
\prop{1}{\it Suppose that~$p$ is congruent to~$1$ modulo~$4$.
Suppose that for all anomalous
$j\in\B F_p$ and all
non-quadratic Dirichlet characters $\chi \colon \B Z/p\B Z\raw \B C$,
there exists $t_\chi\in \B T$ and
$\delta\in\Delta_S$ such that $L(f,\chi,1)\ne0$ for every newform
$f\in t_\chi S_2(\Gamma_0(p))$ and
$\iota_j(t_\chi\delta)\ne0$.

Let~$E$ be an elliptic curve over $\B Q(\mu_p)$, such that the
points of order~$p$ of
$E(\bar\B Q)$ are all $\B Q(\mu_p)$-rational.
Then for all subgroups~$C$ of order~$p$ of $E(\bar \B Q)$, there exists an
elliptic curve $E_C$ over $\B Q(\sqrt p)$ equipped with a
$\B Q(\sqrt p)$-rational subgroup $D_C$ of order~$p$, and
the pairs $(E,C)$ and $(E_C,D_C)$ are $\bar \B Q$-isomorphic.}

\dm We prove the proposition using the results of~\Merel.
The hypothesis
$\iota_j(t_\chi\delta)\ne0$ forces $t_\chi\notin p\B T$
and, {\it a fortiori}, $t_\chi\ne0$; in addition,
the non-vanishing hypothesis on the $L$-series
forces the hypothesis $H_p(\chi)$ of \lc, introduction.

\change{
{\bf (the following paragraph)}
By assumption, hypothesis $H_p(\chi)$ is satisfied for all
non-quadratic Dirichlet characters~$\chi$ of conductor~$p$.
Thus Corollary~3 of Proposition~6 of \lc{} implies that~$E$ has
potentially good reduction at the prime ideal
$\cP$ of $\B Z[\mu_p]$ that lies above~$p$.}

According to Corollary~3 of Proposition~6 of \lc,~$E$ has
potentially good reduction at the prime ideal
$\cP$ of $\B Z[\mu_p]$ that lies above~$p$
once we know that
hypothesis $H_p(\chi)$ is satisfied for all
non-quadratic Dirichlet characters~$\chi$ of conductor~$p$
(this is the case by hypothesis).

Denote by~$j$ the modular invariant of the fiber at~$\cP$ of the
N\'eron model of~$E$.
According to the corollary of Proposition~15 of \lc,
$j$ is anomalous.

Let~$C$ be a subgroup of $E(\bar\B Q)$ of order~$p$.
By assumption~$E$ is an elliptic curve over~$\B Q(\mu_p)$ whose points
of order~$p$ are all $\B Q(\mu_p)$-rational, so
the pair $(E,C)$ defines a $\B Q(\mu_p)$-rational point~$P$
of the modular curve $X_0(p)$.

Consider the morphism $\phi_{\chi}=\phi_{t_\chi}:X_0(p)\rightarrow J_0(p)$
obtained by composing the standard embedding of $X_0(p)$ into $J_0(p)$
with $t_{\chi}$.  As in section 1.3 of \lc, $\phi_{\chi}$
extends to a map from the minimal regular model of $X_0(p)$ to the
N\'eron model of $J_0(p)$.
When $\iota_j(t_\chi\delta)\ne0$, this map is a formal
immersion at the point $P_{/\B F_p}$, according to \lc,
Proposition~4.  The hypothesis that $L(f,\chi,1)\ne0$ for
every newform
$f\in t_\chi S_2(\Gamma_0(p))$, translates into $L(t_\chi J_0(p), \chi,1)\ne0$, which in turn implies that the $\chi$-isotypical
component of
$t_\chi J_0(p)(\B Q(\mu_p))$ is finite (this is Kato's theorem, see the
discussion in section 1.5 of \lc).
\change{Mazur asked us to verify that section 1.5 of Merel's other
paper properly refers to Scholl's paper.}
We can then apply Corollary~1 of Proposition~6 of \lc.  This proves
that~$P$ is
$\B Q(\sqrt p)$-rational, which translates into the conclusion of
Proposition~1.

\bigskip

\change{Delete this remark.}
\rem{1} In this proposition we content ourself with a stronger hypothesis
than the one generally used in \lc: the Hecke operator
$t_\chi$ is required to belong to $\B T$ and not to $\B T\otimes\B Z[\chi]$.

\bigskip\bigskip\noindent
{\bf 2. Elliptic curves and quadratic fields}

\bigskip
\prop{2}{\it Let~$p$ be a prime number that is congruent to~$1$
modulo~$4$. Let~$E$ be an elliptic curve over $\bar\B Q$.
\change{There exists a subgroup~$C\subset{}E(\B Q)$ of order~$p$
such that $(E,C)$ can not be defined over $\B Q(\sqrt{p})$.}
There exists a cyclic subgroup~$C$ of order~$p$ of
$E(\bar\B Q)[p]$, such that for all elliptic curves~$E'$ over
${\B Q(\sqrt{p})}$
equipped with a ${\B Q(\sqrt p)}$-rational subgroup~$C'$, the
pairs $(E,C)$ and $(E', C')$ are not $\bar\B Q$-isomorphic.
}

\dm
We procede by contradiction, i.e., we assume that for
all cyclic subgroups~$C$ of order~$p$ of $E(\bar\B Q)$,
the pair $(E,C)$ can be defined over ${\B Q(\sqrt{p})}$.
We choose such a pair $(E_0,C_0)$ over ${\B Q(\sqrt{p})}$.

Assume first that all twists of $E$ are quadratic, i.e.,
that $j(E)$ is neither~$0$ nor $1728$.
We show that the group
$\Gal(\bar\B Q/{\B Q(\sqrt p)})$ acts by scalars
on the $\B F_p$-vector space $E_0(\bar\B Q)[p]$. For this it
suffices to show that all subgroups of order~$p$ of
$E_0(\bar\B Q)[p]$ are stable by $\Gal(\bar\B Q/{\B Q(\sqrt p)})$.

\change{I reworded the following paragraph.}
Suppose $C_1$ is a cyclic subgroup of order~$p$ of $E_0(\bar\B Q)[p]$.
By assumption, there exists a quadratic twist $E_1$ of $E_0$ and
a cyclic subgroup $C_1'$ of $E_1(\bar\B Q)[p]$
that is defined over $\B Q(\sqrt{p})$, such that
the image of $C_1$ by the isomorphism $E_0\simeq E_1$ is $C'_1$.
Since $\Gal(\bar\B Q/{\B Q(\sqrt p)})$ leaves $C_1'$ stable and
the action of $\Gal(\bar\B Q/\B Q(\sqrt p))$
on $E_0(\bar\B Q)[p]$ is
a quadratic twist of the action on $E_1(\bar\B Q)[p]$,
we see that $\Gal(\bar\B Q/\B Q(\sqrt p))$ leaves $C_1$ stable.
Thus $\Gal(\bar\B Q/\B Q(\sqrt p))$ fixes all lines in
$E_0(\bar\B Q)[p]$, and hence
acts by scalars.  Denote by~$\alpha$ the corresponding character
of $\Gal(\bar\B Q/\B Q(\sqrt p))$.

Because of the Weil pairing, $\alpha^2$ coincides
with the cyclotomic character modulo~$p$, and it factors through
$\Gal(\B Q(\mu_p)/\B Q(\sqrt p))$. But, when
$p\equiv 1\!\!\pmod 4$, the group $\Gal(\B Q(\mu_p)/{\B Q(\sqrt p)})$ is of
even order, and the characters modulo~$p$ form a group generated by the
reduction modulo~$p$ of the cyclotomic character, which, therefore,
can not be a square.

\change{I don't understand this argument.}
Next suppose that $j(E)=0$ or $j(E)=1728$. Indeed, in these
two cases~$E$ has
complex multiplication by an order $R_K$ of $K=\B Q[\sqrt{-1}]$ or
$\B Q[\sqrt{-3}]$.
Consider the map
$\rho : \Gal(\bar \B Q/\B Q(\sqrt p))\longrightarrow{\rm Aut}\,E_0(\bar\B Q)[p]$.
%Suppose $\rho(\Gal(\bar \B Q/K(\sqrt p)))$ contains an element of order~$p$.
Let $L_p$ be the ray class field of conductor~$p$ of~$K$.
It contains $\B Q(\sqrt p)$ since $p\equiv 1\!\!\!\pmod 4$.
By the theory
of complex multiplication,
$\rho(\Gal(\bar \B Q/L_p))$ is trivial. By class field theory, $\Gal(L_p/K)$
has no element of order~$p$, since~$K$ has class number~$1$. Therefore
$\rho(\Gal(\bar \B Q/\B Q(\sqrt p)))$ has no elements of order~$p$.
Since it is contained in the Borel subgroup of ${\rm Aut}\,E_0(\bar\B Q)[p]$
which stabilizes $C_0$, it is an abelian group. By the theory of complex
multiplication, it is the semi-direct product of $\Gal(L_p/K(\sqrt p))$ and
$\Gal(K(\sqrt p)/\B Q(\sqrt p))\simeq\Gal(\B C/\B R)$. Such a group
is not abelian since
$\Gal(L_p/K(\sqrt p))$ is not a $2$-group, hence the contradiction.

\bigskip\bigskip\noindent
{\bf 3. Verification of the hypothesis of Proposition~1}
\bigskip
Let $p$ be a prime number.  In this section we explain how we used
a computer to verify that the hypothesis of Proposition~1 are satisfied
for $p=11$ and $13 < p < 1000$.

We first list the anomalous $j$-invariants $j\in\B F_p$.  Since~$p$ is
fairly small in the range of our computations, we created this list by
simply enumerating all of the elliptic curves over $\B F_p$ and
counting the number of points on each curve.  For example, when $p=31$
the anomalous $j$-invariants are $j=10,14$.

Let~$\chi: \B Z/p\B Z\raw \B C$ be a non-quadratic
Dirichlet character, and denote by $\B Z[\chi]$ the
subring of $\B Q(\zeta_{p-1})$ generated by the image of~$\chi$.
Denote by $S_2(\Gamma_0(p);\B Z)$ be the set of modular forms
$f\in S_2(\Gamma_0(p))$ whose Fourier expansion at the cusp~$\infty$
lies in $\B Z[[q]]$.

We study the $\B T$-modules $\B T$, $\Delta_S$, and $S_2(\Gamma_0(p);\B Z)$.
After extension of scalars to~$\B Q$, these
are $\B T\otimes\B Q$-modules that are free of rank~$1$, of which the
irreducible sub-$\B T\otimes\B Q$ modules are the annihilators of the
minimal prime ideals of $\B T$.  We compute a list of the minimal
prime ideals of $\B T$ by computing appropriate kernels and
characteristic polynomials of Hecke operators of small index on
$\Delta_S$, which we find using the graph method of Mestre and
Oesterl\'e \Mes{}.

Having computed the minimal prime ideals of $\B T$, we verify that
some nontrivial ideal $\cI$ of $\B T$ (always a minimal prime
ideal in the range of our computations) simultaneously satisfies
the following three conditions:
\vskip 2ex

1)
For each anomalous $j$-invariant, there exists $x\in\Delta_S$ such that
$\cI x=0$ and  $\iota_j(x)\ne 0$.\vskip 1ex

2) Each of the newforms~$f\in S_2(\Gamma_0(p))$ with
$\cI f=0$ satisfies $L(f,\chi,1)\ne 0$.
\vskip 1ex

3) The image of~$\cI$ in the $\B T$-module $\B T/p\B T$
is a direct factor.\vskip 2ex

Let $\cI$ be an ideal of $\B T$. Here is how we verify these conditions
for $\cI$.

\bigskip
{\it \noindent Verification of condition 1.}

We verified that $\cI$ satisfies the first condition by
finding a $\B T$-eigenvector~$v$ of $\Delta_S\otimes \bar\B Z$ that is
annihilated by $\cI$ and satisfies $\iota_j(v)\neq 0$ for all anomalous $j$-invariants.  Because $\iota_j$
is a homomorphism, this implies the existence of~$x$ as in condition 1.

\bigskip
{\it \noindent Verification of condition 2.}

We verified the second condition using modular symbols.
Our method is purely algebraic, so we do not perform
any approximate computation of integrals.
Using the algorithm described in \Cremona, we compute the action of
the Hecke algebra $\B T$ on the space
$\Hom(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$.   By intersecting the kernels
of appropriate elements of $\B T$, we find a basis
$\varphi_1,\ldots,\varphi_n$ for the subspace of
$\Hom(H_1(X_0(p);\B Q[\chi]),\B Q[\chi])$ that is annihilated by~$\cI$.
Let~$\Phi_{\cI}=\varphi_1\times \cdots \times \varphi_n$ denote the linear map
$H_1(X_0(p);\B Q[\chi])\raw \B Q[\chi]^n$
defined by the $\varphi_i$.

Let $\B T_{\B Q[\chi]} = \B T \otimes \B Q[\chi]$, where $\B Q[\chi]$
is the number field generated the image of~$\chi$.
The {\it $\chi$-twisted winding element} (denoted $\theta_\chi$ in
\Merel)
$$\B e_\chi = \sum_{a\in (\B Z/p\B Z)^*} \bar\chi(a) \Big\{\infty,{a \over p}\Big\}$$
generates the {\it $\chi$-twisted winding submodule}
$\B T_{\B Q[\chi]}\cdot \B e_\chi$.  To compute this submodule,
we use that $\B T$ is generated, even as a $\B Z$-module,
by $T_1,T_2,\ldots, T_b$, for any $b\geq (p+1)/6$
(see \Agashe).

\bigskip
\lem  3
{\it Let $\cI$ be a minimal prime ideal of~$\B T$, and
let $\chi:(\B Z/N\B Z)^*\raw \B C^*$
be a nontrivial Dirichlet character.
Then the dimension of $\Phi_{\cI}(\B T_{\B Q[\chi]} \cdot \B e_\chi)$ is
equal to the cardinality of the set of newforms~$f$ such that
$\cI f=0$ and $L(f,\chi,1) \neq 0$.
}

\dm
We have
$$\dim_{\B Q[\chi]} \Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_\chi) = \dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi).$$
This dimension is invariant upon changing the basis
$\varphi_1,\ldots, \varphi_n$ used to define $\Phi_{\cI}$.
In particular, over $\B C$ there is a basis
$\varphi_1',\ldots, \varphi_n'$ so that the resulting
map $\Phi_{\cI}'$ satisfies
$$\Phi_{\cI}'(x) = \Bigl(\rp(\int_x f^{(1)}), \ip(\int_x f^{(1)}), \ldots, \rp(\int_x f^{(d)}),\ip(\int_x f^{(d)})\Bigr),$$
where $f^{(1)}, \ldots, f^{(d)}$ are the Galois conjugates
of a newform~$f^{(1)}=\sum a_n^{(1)} q^n$ such that $\cI f^{(1)}=0$.
Furthermore, $\Phi_{\cI}'$ is a $\B T_{\B C}$-module homomorphism
if we declare that $\B T_{\B C}$ as acts on $\B R^{2d} = \B C^d$ via
$$T_n(x_1,y_1, \ldots, x_d, y_d) = T_n(z_1,\ldots,z_d) = (a_n^{(1)} z_1,\ldots, a_n^{(d)}z_d),$$
where $z_j = x_j + i y_j$ and
the $a_n^{(j)}$ are Fourier coefficients of the $f^{(j)}$.

As explained in Section 2.2 of~\Merel,
$\int_{\B e_\chi} f = *\cdot L(f,\chi,1)$, where~$*$
is some nonzero real or pure-imaginary complex number,
according to whether $\chi(-1)$ equals~$1$ or~$-1$,
respectively.
Combining this observation with the equality
$$\dim_{\B C} \Phi_{\cI}(\B T_{\B C} \cdot \B e_\chi) = \dim_{\B C} (\B T_{\B C}\cdot \Phi_{\cI}(\B e_\chi)),$$
and that the image of $\B T_{\B C}$ in $\End(\B C^d)$ is
equal to the diagonal matrices, proves the asserted equality.

\bigskip
\rem{2} The dimension of $\Phi_{\cI}(\B T_{\B Q[\chi]}\cdot \B e_{\chi})$
is unchanged if~$\chi$ is
replaced by a Galois-conjugate character.

\bigskip

In practice, computations over the cyclotomic field $\B Q[\chi]$ are
extremely expensive.  Fortunately, for our application it suffices to
give a lower bound on the dimension appearing in the lemma.  Such a
bound can be efficiently obtained by instead computing the reductions
of~$\Phi$,~$\chi$, and the $\chi$-twisted winding submodule modulo a
suitable maximal ideal of the ring of integers of $\B Q[\chi]$ that
splits completely; this amounts to performing the above linear algebra
over a relatively small prime finite field $\B F_\ell$ such
that~$\ell$ is congruent to~$1$ modulo $p-1$.

\bigskip

\rem{3} For every newform~$f$ in $S_2(\Gamma_0(p))$, with $p\leq 1000$,
and every mod~$p$ Dirichlet character~$\chi$, we found that
$L(f,\chi,1)\neq 0$ if and only if
$L(f^{\sigma},\chi,1)\neq 0$ for all conjugates $f^{\sigma}$
of~$f$.
More generally, for any~$f$ and~$\chi$, this equivalence holds if
$\B Q[\chi]$ is linearly disjoint from the
field $K_f=(\B T/\cI)\otimes\B Q$.
The first few primes
for which there is a form~$f$ and a mod~$p$ character~$\chi$
such that the linear disjointness hypothesis fails are
$p=31, 113, 127$, and $191$.
The analogue of this nonvanishing observation is false if we instead consider
newforms on $\Gamma_1(p)$ and allow~$\chi$ to be arbitrary.
For example, let~$f$
be one of the two Galois-conjugate newforms in $S_2(\Gamma_1(13))$.
Then there is a character $\chi:(\B Z/7\B Z)^*\raw \B C^*$ of
order~$3$ such that $L(f,\chi,1) = 0$ and $L(f^{\sigma},\chi,1)\neq 0$.

\bigskip

{\it \noindent Verification of condition 3.}

The third condition is satisfied for all $p<10000$, except possibly
$p = 389$,
because we have verified that the discriminant of $\B T$ is
prime to~$p$ for all such $p\neq 389$,
so the ring $\B T/p\B T$ is semisimple.
The discriminant computation was carried out by the second author
as follows.
Using the method of \Mes{}, we computed discrimininants of characteristic
polynomials mod~$p$ of the Hecke operators $T_2$, $T_3$, $T_5$, and $T_7$.
In the few cases when all four of these characteristic polynomials had
discriminant equal to~$0$ mod~$p$, we resorted to modular symbols to
compute several more characteristic polynomials until we found one
having nonzero discriminant modulo~$p$.

We consider the remaining case $p=389$ in detail.  There are exactly
five minimal prime ideals of $\B T$, which we denote $\cP_1$, $\cP_2$,
$\cP_3$, $\cP_6$, and $\cP_{20}$, where the quotient field of $\B T/\cP_i$ has dimension~$i$.  The discriminant of the characteristic
polynomial of $T_2$ is exactly divisible by $389$.  Since the field of
fractions of $\B T/\cP_{20}$ has discriminant divisible by $389$, we
see that $389$ is not the residue characteristic of any congruence
prime.  Let $\cO_i = \B T/\cP_{i}$.  The natural map $\B T \rightarrow \prod \cO_i$ has finite kernel and cokernel each of order coprime to
$389$, so $\B T / 389 \B T \cong \prod \cO_i/389 \cO_i$.  The
nonquadratic characters $\chi:(\B Z/p\B Z)^*\rightarrow \B C^*$ have
orders $1, 4, 97, 193, 388$.  We must verify that for each of these
degrees, one of the ideals $\cP_i$ satisfies conditions 1--3.  We
check as above that conditions 1--3 for~$\chi$ of order~$4$ are
satisfied by $\cP_2$ and conditions 1--3 for~$\chi$ of order greater
than~$4$ are satisfied by $\cP_1$.  When~$\chi$ is the trivial
character, conditions~1--3 are satisfied only by $\cP_{20}$.

\bigskip
{\it \noindent Summary.}

For each prime $p<1000$ different than $2,3,5,7, 13$, we
verified the existence of an ideal that satisfies the three conditions
given above, as follows.  For each~$p$, we consider each Galois conjugacy class of
non-quadratic characters~$\chi$.  We find a single newform~$f$ such
that $L(f,\chi,1)\ne 0$ for all conjugates of~$f$ and of~$\chi$.  Then
we let $\cI$ be the annihilator of~$f$, and try to verify condition~1
for {\it all} of the anamolous $j$-invariants in $\B F_p$.
When the three conditions are satisfied for an ideal~$\cI$ of~$\B T$,
there exists $t_\chi\in\B T$ that is annihilated by $\cI$ and is the
inverse image of a projector of $\B T/p\B T$ on the complement of
$\cI+p\B T$.  Putting $\delta=x$, one has
$\iota_j(t_\chi \delta)=\iota_j(\delta)\ne0$
(because $\iota_j$ takes its values in
characteristic~$p$, it follows that $\delta$ is annihilated by~$\cI$ and
$t_\chi\in 1+p\B T+\cP$).
Every newform $f\in t_\chi S_2(\Gamma_0(p))$ satisfies
$\cI f=0$, and therefore, by our second condition, $L(f,\chi,1)\ne0$.
The pair $(t_\chi,\delta)$ then satisfies the conditions required by
Proposition~1.

\bigskip\bigskip

\vskip .5in

\centerline{\pc Bibliography}
\bigskip

\item{\Agashe}{\pc A. Agashe},
{\it On invisible elements of the Tate-Shafarevich group},
C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), no. 5, 369--374.
\vskip 1ex

\item{\Cremona}{\pc J. Cremona},
{\it Algorithms for modular elliptic curves},
second ed., Cambridge University Press, Cambridge,
{\oldstyle 1997}.
\vskip 1ex

\item{\Merel}{\pc L. Merel},
{\it Sur la nature non cyclotomique des points d'ordre fini des courbes
elliptiques},
To appear in Duke Math. Journal.
\vskip 1ex

\item{\Mes}{\pc J.-F. Mestre}, {\it La m\'ethode des graphes.
Exemples et  applications}, Proceedings of the international
conference on class numbers and fundamental units of algebraic number
fields (Katata), 217--242, {\oldstyle 1986}.
\end

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