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\hspace<<629>>-1in<<629>>
\xymatrix@C=-7pc@R=6pc<<1075>>
;SPMamp;<<1070>>\shadowbox<<1014>>\begin<<630>>minipage<<630>>[l]<<631>>2.6in<<631>>
A Construction of Elements of $<<1226>>\mbox<<1227>><<1228>>\fontencoding<<1229>>OT2<<1229>>\fontfamily<<1230>>wncyr<<1230>>\fontseries<<1231>>m<<1231>>\fontshape<<1232>>n<<1232>>\selectfont Sh<<1228>><<1227>><<1226>>(A)$
\end<<632>>minipage<<632>><<1014>><<1070>>\ar[dl]
\ar[dr]\\
<<1074>>\shadowbox<<1071>>\begin<<633>>minipage<<633>>[l]<<634>>3.4in<<634>>
\begin<<635>>center<<635>>\underline<<636>>Birch and Swinnerton-Dyer Conjecture<<636>>
\end<<637>>center<<637>>
\begin<<638>>itemize<<638>>
\item If $L(A,1)\neq 0$, then
$$ \#<<1233>>\mbox<<1234>><<1235>>\fontencoding<<1236>>OT2<<1236>>\fontfamily<<1237>>wncyr<<1237>>\fontseries<<1238>>m<<1238>>\fontshape<<1239>>n<<1239>>\selectfont Sh<<1235>><<1234>><<1233>>(A) \stackrel<<639>>?<<639>><<640>>=<<640>> \frac<<641>>L(A,1)<<641>><<642>>\Omega_A<<642>>\cdot
\frac<<1015>>\#A(\mathbb<<1240>>Q<<1240>>)_<<643>>\tor<<643>>\cdot\#A(\mathbb<<1241>>Q<<1241>>)^<<644>>\vee<<644>>_<<645>>\tor<<645>><<1015>>
<<1016>>\prod_<<646>>p\mid N<<646>> c_<<647>>A,p<<647>><<1016>>$$
Find $A$ in nature with conjecturally nontrivial $<<1242>>\mbox<<1243>><<1244>>\fontencoding<<1245>>OT2<<1245>>\fontfamily<<1246>>wncyr<<1246>>\fontseries<<1247>>m<<1247>>\fontshape<<1248>>n<<1248>>\selectfont Sh<<1244>><<1243>><<1242>>(A)$,
and prove that $<<1249>>\mbox<<1250>><<1251>>\fontencoding<<1252>>OT2<<1252>>\fontfamily<<1253>>wncyr<<1253>>\fontseries<<1254>>m<<1254>>\fontshape<<1255>>n<<1255>>\selectfont Sh<<1251>><<1250>><<1249>>(A)$ is as big as expected.
\item
Construct $A$ such that $<<1256>>\mbox<<1257>><<1258>>\fontencoding<<1259>>OT2<<1259>>\fontfamily<<1260>>wncyr<<1260>>\fontseries<<1261>>m<<1261>>\fontshape<<1262>>n<<1262>>\selectfont Sh<<1258>><<1257>><<1256>>(A)$ is nontrivial,
then check that the BSD conjecture is not obviously
false for~$A$.
\item
Find a method for connecting the rank conjecture
about elliptic curves to the rank~$0$ formula
for abelian varieties.
\end<<648>>itemize<<648>>
\end<<649>>minipage<<649>>
<<1071>><<1074>>
;SPMamp; ;SPMamp;
<<1072>>\shadowbox<<1017>>\begin<<650>>minipage<<650>>[l]<<651>>3.4in<<651>>
\begin<<652>>center<<652>>
\underline<<653>>What are the possibilities for $\#<<1263>>\mbox<<1264>><<1265>>\fontencoding<<1266>>OT2<<1266>>\fontfamily<<1267>>wncyr<<1267>>\fontseries<<1268>>m<<1268>>\fontshape<<1269>>n<<1269>>\selectfont Sh<<1265>><<1264>><<1263>>(A)$?<<653>>
\end<<654>>center<<654>>
<<655>>\bf Question (Poonen, 1999 at AWS).<<655>>\\
Stoll and Poonen proved that if $A$ is a Jacobian, then $\#<<1270>>\mbox<<1271>><<1272>>\fontencoding<<1273>>OT2<<1273>>\fontfamily<<1274>>wncyr<<1274>>\fontseries<<1275>>m<<1275>>\fontshape<<1276>>n<<1276>>\selectfont Sh<<1272>><<1271>><<1270>>(A)$ is a square
or twice a square. If~$A$ is not a Jacobian, is $\#<<1277>>\mbox<<1278>><<1279>>\fontencoding<<1280>>OT2<<1280>>\fontfamily<<1281>>wncyr<<1281>>\fontseries<<1282>>m<<1282>>\fontshape<<1283>>n<<1283>>\selectfont Sh<<1279>><<1278>><<1277>>(A)$ always a square
or twice a square?\vspace<<656>>1ex<<656>>\\
<<657>>\bf Conjecture (Me, today).<<657>>\\
Let $G$ be any finite abelian group (of odd order).
Then there is an abelian variety $A$ such that
$<<1284>>\mbox<<1285>><<1286>>\fontencoding<<1287>>OT2<<1287>>\fontfamily<<1288>>wncyr<<1288>>\fontseries<<1289>>m<<1289>>\fontshape<<1290>>n<<1290>>\selectfont Sh<<1286>><<1285>><<1284>>(A) \approx G\times H$, where
$\gcd(\#G, \#H)=1$.
\end<<658>>minipage<<658>><<1017>>
<<1072>><<1075>>
$$
\par
\newpage\section<<659>>A Construction of Elements of $<<1291>>\mbox<<1292>><<1293>>\fontencoding<<1294>>OT2<<1294>>\fontfamily<<1295>>wncyr<<1295>>\fontseries<<1296>>m<<1296>>\fontshape<<1297>>n<<1297>>\selectfont Sh<<1293>><<1292>><<1291>>(A)$<<659>>
\begin<<1169>>theorem_type<<1169>>[theorem][theorem][section][plain][][]
\label<<661>>thm<<661>>
Let~$E$ be an elliptic curve over~$\mathbb<<1298>>Q<<1298>>$,
and suppose $\chi:(\mathbb<<1299>>Z<<1299>>/\ell\mathbb<<1300>>Z<<1300>>)^*\rightarrow \mathbb<<1301>>C<<1301>>^*$
is a Dirichlet character of prime modulus~$\ell\nmid N_E$ and
order~$n$ such that
\begin<<662>>itemize<<662>>
\item $L(E,\chi^a,1)\neq 0$ for $a=1,\ldots, n-1$,
\item $\displaystyle \gcd\left(n,\,\,2N_E\prod_<<663>>p\mid N_E<<663>>\#\Phi_E(\overline<<1302>>\mathbb<<1303>>F<<1303>><<1302>>_p)\right ) = 1$, and
\item $a_\ell \not\equiv \ell+1 \pmod<<664>>p<<664>>$ for all $p\mid n$.
\end<<665>>itemize<<665>>
Let~$K$ be the degree~$n$ abelian extension of $\mathbb<<1304>>Q<<1304>>$ corresponding
to~$\chi$.
Then there exists a $K$-twist~$A$ of $E^<<666>>\oplus (n-1)<<666>>$ of rank~$0$
such that $L(A,s) = \prod_<<667>>a=1<<667>>^<<668>>n-1<<668>> L(E,\chi^a,s)$
and $$E(\mathbb<<1305>>Q<<1305>>)/n E(\mathbb<<1306>>Q<<1306>>)\subset <<1307>>\mbox<<1308>><<1309>>\fontencoding<<1310>>OT2<<1310>>\fontfamily<<1311>>wncyr<<1311>>\fontseries<<1312>>m<<1312>>\fontshape<<1313>>n<<1313>>\selectfont Sh<<1309>><<1308>><<1307>>(A/\mathbb<<1314>>Q<<1314>>).$$\end<<1170>>theorem_type<<1170>>
\par
\begin<<1171>>theorem_type<<1171>>[remark][theorem][][remark][][]
Note that $K$ is contained in the totally real subfield
$\mathbb<<1315>>Q<<1315>>(\mu_\ell)^+$ of $\mathbb<<1316>>Q<<1316>>(\mu_\ell)$ because the order
of $\chi(-1)$ divides the odd number~$n$.\end<<1172>>theorem_type<<1172>>
\par
\begin<<672>>proof<<672>>[Sketch of Proof]
Let $R=\Res_<<673>>K/\mathbb<<1317>>Q<<1317>><<673>>(E_K)$ be the Weil restriction of scalars
of $E_K$ down to~$\mathbb<<1318>>Q<<1318>>$. For any $\mathbb<<1319>>Q<<1319>>$-scheme~$S$,
we have $R(S)=E_K(S\times _\mathbb<<1320>>Q<<1320>>K)$, and
as $\Gal(\overline<<1321>>\mathbb<<1322>>Q<<1322>><<1321>>/\mathbb<<1323>>Q<<1323>>)$-modules
$$
R(\overline<<1324>>\mathbb<<1325>>Q<<1325>><<1324>>) = E(\overline<<1326>>\mathbb<<1327>>Q<<1327>><<1326>>\otimes K) \cong E(\overline<<1328>>\mathbb<<1329>>Q<<1329>><<1328>>)\otimes _<<674>>\mathbb<<1330>>Z<<1330>><<674>> \mathbb<<1331>>Z<<1331>>[\Gal(K/\mathbb<<1332>>Q<<1332>>)],
$$
where $\tau\in \Gal(\overline<<1333>>\mathbb<<1334>>Q<<1334>><<1333>>/\mathbb<<1335>>Q<<1335>>)$ acts on
$\sum P_\sigma\otimes \sigma \in E(\overline<<1336>>\mathbb<<1337>>Q<<1337>><<1336>>)\otimes _<<675>>\mathbb<<1338>>Z<<1338>><<675>>\mathbb<<1339>>Z<<1339>>[\Gal(K/\mathbb<<1340>>Q<<1340>>)]$ by
$$\tau\left(\sum P_\sigma\otimes \sigma\right) =
\sum \tau(P_\sigma)\otimes \sigma\tau_<<676>>|K<<676>>.
$$
The $L$-series of~$R$ is $\prod_<<677>>a=1<<677>>^<<678>>n<<678>> L(E,\chi^a,s)$, and $R$ has
good reduction at all $p\nmid \ell\cdot N$.
\par
Let $\Delta : E\hookrightarrow R$ be the diagonal embedding, which sends~$P$
to $\sum_<<679>>\sigma\in\Gal(K/\mathbb<<1341>>Q<<1341>>)<<679>> P\otimes \sigma$,
and let $\Sigma : R\rightarrow E$ be the summation map, which sends
$\sum P_\sigma \otimes \sigma$ to $\sum P_\sigma$.
Note that both $\Delta$ and $\Sigma$ are defined over~$\mathbb<<1342>>Q<<1342>>$
and that $\Sigma\circ \Delta = [n]$.
If $A=\ker(\Sigma)$ then
$$
A_<<680>>\overline<<1343>>\mathbb<<1344>>Q<<1344>><<1343>><<680>> = \ker \left(+: E_<<681>>\overline<<1345>>\mathbb<<1346>>Q<<1346>><<1345>><<681>>^<<682>>\oplus n<<682>> \rightarrow E_<<683>>\overline<<1347>>\mathbb<<1348>>Q<<1348>><<1347>><<683>>\right)
\cong E^<<684>>\oplus (n-1)<<684>>,
$$
the isomorphism being the one that sends
$(P_1,\ldots,P_<<685>>n-1<<685>>)$ to $(P_1,\ldots,P_<<686>>n-1<<686>>,-(\sum P_i))$.
In particular,~$A$ is a twist of $E^<<687>>\oplus (n-1)<<687>>$.
We summarize this information in the following diagram:
\begin<<688>>equation<<688>>\label<<689>>eqn:d1<<689>>
\xymatrix@=3pc<<1018>>
<<690>>E[n]<<690>>\ar[d] \ar[r] ;SPMamp; <<691>>E<<691>>\ar@<<692>>^(-;SPMgt;<<692>>[d]^<<693>>\Delta<<693>> \ar[r]^<<694>>[n]<<694>> ;SPMamp; <<695>>E<<695>>\ar@<<696>>=<<696>>[d]\\
A \ar@<<697>>^(-;SPMgt;<<697>>[r] ;SPMamp; R \ar@<<698>>-;SPMgt;;SPMgt;<<698>>[r]^<<699>>\Sigma<<699>> ;SPMamp; E.
<<1018>>
\end<<700>>equation<<700>>
\par
Now pass to $\mathbb<<1349>>Q<<1349>>$-rational points in diagram (\ref<<701>>eqn:d1<<701>>) and
rearrange things to obtain the following diagram:
$$
\xymatrix@=3pc<<1019>>
0 \ar[r];SPMamp; <<702>>E(\mathbb<<1350>>Q<<1350>>)<<702>>\ar[r]^<<703>>[n]<<703>>\ar[d] ;SPMamp; <<704>>E(\mathbb<<1351>>Q<<1351>>)<<704>>\ar@<<705>>=<<705>>[d] \ar[r] ;SPMamp; <<706>>E(\mathbb<<1352>>Q<<1352>>)/n E(\mathbb<<1353>>Q<<1353>>)<<706>>\ar[r]\ar[d]^<<707>>\iota<<707>> ;SPMamp; 0\\
0 \ar[r];SPMamp; <<708>>R(\mathbb<<1354>>Q<<1354>>)/A(\mathbb<<1355>>Q<<1355>>)<<708>>\ar[r] ;SPMamp; <<709>>E(\mathbb<<1356>>Q<<1356>>)<<709>>\ar[r];SPMamp;<<710>>\ker(H^1(\mathbb<<1357>>Q<<1357>>,A)\rightarrow H^1(\mathbb<<1358>>Q<<1358>>,R))<<710>>\ar[r] ;SPMamp; 0.\\
<<1019>>
$$
Here we have used that $E(\mathbb<<1359>>Q<<1359>>)[n]=0$, since $E[p]$ is irreducible for
$p\mid n$, and we've included the beginning of the long exact sequence of
Galois cohomology associated to $0\rightarrow A\rightarrow <<711>><<711>>R\rightarrow <<712>><<712>>E\rightarrow 0$.
Using the snake lemma, we see that~$\iota$ is surjective and has kernel
a subgroup of $R(\mathbb<<1360>>Q<<1360>>)/(A(\mathbb<<1361>>Q<<1361>>)+E(\mathbb<<1362>>Q<<1362>>))$.
One can use that $a_\ell\not\equiv \ell+1\pmod<<713>>p<<713>>$ for any $p\mid n$ and that
$A(\mathbb<<1363>>Q<<1363>>)$ is finite (which follows from Kato's Euler system work!) to
show that $R(\mathbb<<1364>>Q<<1364>>)/(A(\mathbb<<1365>>Q<<1365>>)+E(\mathbb<<1366>>Q<<1366>>))$ contains no $p$-torsion for $p\mid n$,
hence $\ker(\iota)=0$.
\par
To show that the image of~$\iota$ lies in the
subgroup $<<1367>>\mbox<<1368>><<1369>>\fontencoding<<1370>>OT2<<1370>>\fontfamily<<1371>>wncyr<<1371>>\fontseries<<1372>>m<<1372>>\fontshape<<1373>>n<<1373>>\selectfont Sh<<1369>><<1368>><<1367>>(A/\mathbb<<1374>>Q<<1374>>)$ of $H^1(\mathbb<<1375>>Q<<1375>>,A)$, uses that $\gcd(n,2\cdot
N_E\cdot c)=1$, where~$c$ is the product of all Tamagawa numbers
of~$E$ and~$A$. These last steps are fairly technical and use some
nontrivial machinery.
(That~$n$ is odd is only used to show that~$\iota$
maps into $<<1376>>\mbox<<1377>><<1378>>\fontencoding<<1379>>OT2<<1379>>\fontfamily<<1380>>wncyr<<1380>>\fontseries<<1381>>m<<1381>>\fontshape<<1382>>n<<1382>>\selectfont Sh<<1378>><<1377>><<1376>>(A/\mathbb<<1383>>Q<<1383>>)$.)
\end<<714>>proof<<714>>
\par
\section<<715>>Data Collection<<715>>
Next we collect some data that both gives evidence for the Birch and
Swinnerton-Dyer conjecture and for my conjecture that if~$G$ is an
abelian group then there is an abelian variety~$A$ such that
$<<1384>>\mbox<<1385>><<1386>>\fontencoding<<1387>>OT2<<1387>>\fontfamily<<1388>>wncyr<<1388>>\fontseries<<1389>>m<<1389>>\fontshape<<1390>>n<<1390>>\selectfont Sh<<1386>><<1385>><<1384>>(A)\approx G\times H$ with $\gcd(\#H,\#G)=1$.
We will always choose~$E$ below so that $N_E$ is prime,~$E$
is isolated in its isogeny class (hence $\rho_<<716>>E,p<<716>>$ is
surjective for all~$p$), and $c_<<717>>E,p<<717>>=1$ for all $p\mid N$.
\par
Let $\#<<1391>>\mbox<<1393>><<1395>>\fontencoding<<1396>>OT2<<1396>>\fontfamily<<1397>>wncyr<<1397>>\fontseries<<1398>>m<<1398>>\fontshape<<1399>>n<<1399>>\selectfont Sh<<1395>><<1393>><<1391>>_<<1392>>\mbox<<1394>>\tiny \rm an<<1394>><<1392>>(A)^*$ denote the prime-to-$2\ell$ part of
$$
\frac<<718>>L(A,1)<<718>><<719>>\Omega_A<<719>>\cdot
\frac<<1020>>\#A(\mathbb<<1400>>Q<<1400>>)_<<720>>\tor<<720>>\cdot\#A^<<721>>\vee<<721>>(\mathbb<<1401>>Q<<1401>>)_<<722>>\tor<<722>><<1020>>
<<1021>>\prod_<<723>>p\mid \ell N_E<<723>> c_<<724>>A,p<<724>><<1021>>.
$$
Remark 5.4 of Edixhoven's <<725>>\em N\'eron models and tame ramification<<725>> can
be used to show that
$$
\Phi_<<726>>A,\ell<<726>>(\overline<<1402>>\mathbb<<1403>>F<<1403>><<1402>>_\ell)=E(\overline<<1404>>\mathbb<<1405>>F<<1405>><<1404>>_\ell)[n]\approx (\mathbb<<1406>>Z<<1406>>/n\mathbb<<1407>>Z<<1407>>)^2,
$$
so $c_<<727>>A,\ell<<727>>=1$, since $E(\mathbb<<1408>>F<<1408>>_\ell)[p]=0$ for all $p\mid n$.
Since~$K$ is only ramified at~$\ell$
and the formation of N\'eron models commutes with unramified base change,
$c_<<728>>A,p<<728>> = c_<<729>>E,p<<729>>^<<730>>n-1<<730>>=1$ for $p\mid N_E$.
Since $A(\mathbb<<1409>>Q<<1409>>)\subset A(K) \approx E(K)^<<731>>\oplus (n-1)<<731>>$,
and $E(K)_<<732>>\tor<<732>>=0$ (since all $\rho_<<733>>E,p<<733>>$ are surjective),
we have $\#A(\mathbb<<1410>>Q<<1410>>)_<<734>>\tor<<734>> = \#A^<<735>>\vee<<735>>(\mathbb<<1411>>Q<<1411>>)_<<736>>\tor<<736>>=1$.
I think (but have not proven, yet!) that
$$
\Omega_<<737>>A/\mathbb<<1412>>Q<<1412>><<737>> = \left(\frac<<738>>1<<738>><<1022>>\sqrt<<739>>\ell<<739>><<1022>>\cdot \Omega_<<740>>E/\mathbb<<1413>>Q<<1413>><<740>>\right)^<<741>>n-1<<741>>.
$$
To prove this, it would (mostly) suffice to show that
$
\Omega_<<742>>A/K<<742>> = \Omega_<<743>>A/\mathbb<<1414>>Q<<1414>><<743>>^n\cdot \ell^<<1023>>\binom<<744>>n<<744>><<745>>2<<745>><<1023>>,
$
where $\binom<<746>>n<<746>><<747>>2<<747>> = n(n-1)/2$. Assume this formula for
$\Omega_<<748>>A/\mathbb<<1415>>Q<<1415>><<748>>$, we can very quickly compute $<<1416>>\mbox<<1418>><<1420>>\fontencoding<<1421>>OT2<<1421>>\fontfamily<<1422>>wncyr<<1422>>\fontseries<<1423>>m<<1423>>\fontshape<<1424>>n<<1424>>\selectfont Sh<<1420>><<1418>><<1416>>_<<1417>>\mbox<<1419>>\tiny \rm an<<1419>><<1417>>(A)^*$ using
modular symbols.
\par
The elliptic curves <<749>>\bf 61A<<749>> of rank~$1$, <<750>>\bf 389A<<750>> of rank~$2$, and
<<751>>\bf 5077A<<751>> of rank~$3$ each have prime conductor, trivial torsion subgroup,
and Tamagawa number~$c_p=1$. In the table below,
$p_d$ denotes a $d$-digit prime number (where $d$ is written in
Roman numerals), and
a $-$ means that some hypothesis of Theorem~\ref<<752>>thm<<752>> is <<753>>\em not<<753>> satisfied.
(This table took under ten minutes to compute on a Pentium III 933.)
\par
$$\begin<<754>>array<<754>><<755>>|c|c||c|c|c|<<755>>\hline
n ;SPMamp; \ell ;SPMamp; \#<<1425>>\mbox<<1427>><<1429>>\fontencoding<<1430>>OT2<<1430>>\fontfamily<<1431>>wncyr<<1431>>\fontseries<<1432>>m<<1432>>\fontshape<<1433>>n<<1433>>\selectfont Sh<<1429>><<1427>><<1425>>_<<1426>>\mbox<<1428>>\tiny \rm an<<1428>><<1426>>^*\text<<756>> for <<756>><<757>>\bf 61A<<757>>;SPMamp; \#<<1434>>\mbox<<1436>><<1438>>\fontencoding<<1439>>OT2<<1439>>\fontfamily<<1440>>wncyr<<1440>>\fontseries<<1441>>m<<1441>>\fontshape<<1442>>n<<1442>>\selectfont Sh<<1438>><<1436>><<1434>>_<<1435>>\mbox<<1437>>\tiny \rm an<<1437>><<1435>>^*\text<<758>> for <<758>><<759>>\bf 389A<<759>> ;SPMamp; \#<<1443>>\mbox<<1445>><<1447>>\fontencoding<<1448>>OT2<<1448>>\fontfamily<<1449>>wncyr<<1449>>\fontseries<<1450>>m<<1450>>\fontshape<<1451>>n<<1451>>\selectfont Sh<<1447>><<1445>><<1443>>_<<1444>>\mbox<<1446>>\tiny \rm an<<1446>><<1444>>^*\text<<760>> for <<760>><<761>>\bf 5077A<<761>>\\ \hline\hline
3 ;SPMamp; 487 ;SPMamp; 3;SPMamp; 3^<<762>>4<<762>>;SPMamp; 3^<<763>>3<<763>>\\ \hline
9 ;SPMamp; 487 ;SPMamp; 3^<<764>>2<<764>> \!\cdot\! 19^<<765>>2<<765>>;SPMamp; 3^<<766>>8<<766>>;SPMamp; 3^<<767>>6<<767>> \!\cdot\! 17^<<768>>2<<768>>\\ \hline
27 ;SPMamp; 487 ;SPMamp; 3^<<769>>3<<769>> \!\cdot\! 19^<<770>>2<<770>> \!\cdot\! p_<<1024>>\setcounter<<1452>>myroman<<1452>><<1453>>6<<1453>>\roman<<1454>>myroman<<1454>><<1024>>^<<772>>2<<772>>;SPMamp; 3^<<773>>12<<773>> \!\cdot\! 163^<<774>>2<<774>>;SPMamp; 3^<<775>>9<<775>> \!\cdot\! 17^<<776>>2<<776>> \!\cdot\! 433^<<777>>2<<777>> \!\cdot\! p_<<1025>>\setcounter<<1455>>myroman<<1455>><<1456>>6<<1456>>\roman<<1457>>myroman<<1457>><<1025>>^<<779>>2<<779>>\\ \hline
81 ;SPMamp; 487 ;SPMamp; 3^<<780>>4<<780>> \!\cdot\! 19^<<781>>2<<781>> \!\cdot\! p_<<1026>>\setcounter<<1458>>myroman<<1458>><<1459>>4<<1459>>\roman<<1460>>myroman<<1460>><<1026>>^<<783>>2<<783>> \!\cdot\! p_<<1027>>\setcounter<<1461>>myroman<<1461>><<1462>>6<<1462>>\roman<<1463>>myroman<<1463>><<1027>>^<<785>>2<<785>> \!\cdot\! p_<<1028>>\setcounter<<1464>>myroman<<1464>><<1465>>7<<1465>>\roman<<1466>>myroman<<1466>><<1028>>^<<787>>2<<787>>;SPMamp; 3^<<788>>16<<788>> \!\cdot\! 163^<<789>>2<<789>> \!\cdot\! p_<<1029>>\setcounter<<1467>>myroman<<1467>><<1468>>19<<1468>>\roman<<1469>>myroman<<1469>><<1029>>^<<791>>2<<791>>;SPMamp; 3^<<792>>12<<792>> \!\cdot\! 17^<<793>>2<<793>> \!\cdot\! 433^<<794>>2<<794>> \!\cdot\! p_<<1030>>\setcounter<<1470>>myroman<<1470>><<1471>>4<<1471>>\roman<<1472>>myroman<<1472>><<1030>>^<<796>>2<<796>> \!\cdot\! p_<<1031>>\setcounter<<1473>>myroman<<1473>><<1474>>5<<1474>>\roman<<1475>>myroman<<1475>><<1031>>^<<798>>2<<798>> \!\cdot\! p_<<1032>>\setcounter<<1476>>myroman<<1476>><<1477>>6<<1477>>\roman<<1478>>myroman<<1478>><<1032>>^<<800>>2<<800>> \!\cdot\! p_<<1033>>\setcounter<<1479>>myroman<<1479>><<1480>>7<<1480>>\roman<<1481>>myroman<<1481>><<1033>>^<<802>>2<<802>> \!\cdot\! p_<<1034>>\setcounter<<1482>>myroman<<1482>><<1483>>9<<1483>>\roman<<1484>>myroman<<1484>><<1034>>^<<804>>2<<804>>\\ \hline
5 ;SPMamp; 251 ;SPMamp; 5;SPMamp; 5^<<805>>2<<805>>;SPMamp; -\\ \hline
25 ;SPMamp; 251 ;SPMamp; 5^<<806>>2<<806>> \!\cdot\! 151^<<807>>2<<807>> \!\cdot\! p_<<1035>>\setcounter<<1485>>myroman<<1485>><<1486>>5<<1486>>\roman<<1487>>myroman<<1487>><<1035>>^<<809>>2<<809>>;SPMamp; 5^<<810>>4<<810>> \!\cdot\! 149^<<811>>2<<811>> \!\cdot\! p_<<1036>>\setcounter<<1488>>myroman<<1488>><<1489>>4<<1489>>\roman<<1490>>myroman<<1490>><<1036>>^<<813>>2<<813>>;SPMamp; -\\ \hline
125 ;SPMamp; 251 ;SPMamp; 5^<<814>>3<<814>> \!\cdot\! 151^<<815>>2<<815>> \!\cdot\! p_<<1037>>\setcounter<<1491>>myroman<<1491>><<1492>>5<<1492>>\roman<<1493>>myroman<<1493>><<1037>>^<<817>>2<<817>> \!\cdot\! p_<<1038>>\setcounter<<1494>>myroman<<1494>><<1495>>18<<1495>>\roman<<1496>>myroman<<1496>><<1038>>^<<819>>2<<819>>;SPMamp; 5^<<820>>6<<820>> \!\cdot\! 149^<<821>>2<<821>> \!\cdot\! p_<<1039>>\setcounter<<1497>>myroman<<1497>><<1498>>4<<1498>>\roman<<1499>>myroman<<1499>><<1039>>^<<823>>2<<823>> \!\cdot\! p_<<1040>>\setcounter<<1500>>myroman<<1500>><<1501>>5<<1501>>\roman<<1502>>myroman<<1502>><<1040>>^<<825>>2<<825>> \!\cdot\! p_<<1041>>\setcounter<<1503>>myroman<<1503>><<1504>>10<<1504>>\roman<<1505>>myroman<<1505>><<1041>>^<<827>>2<<827>> \!\cdot\! p_<<1042>>\setcounter<<1506>>myroman<<1506>><<1507>>11<<1507>>\roman<<1508>>myroman<<1508>><<1042>>^<<829>>2<<829>>;SPMamp; -\\ \hline
7 ;SPMamp; 197 ;SPMamp; 7 \!\cdot\! 29^<<830>>2<<830>>;SPMamp; 7^<<831>>2<<831>> \!\cdot\! 13^<<832>>4<<832>>;SPMamp; 7^<<833>>3<<833>>\\ \hline
49 ;SPMamp; 197 ;SPMamp; 7^<<834>>2<<834>> \!\cdot\! 29^<<835>>2<<835>> \!\cdot\! p_<<1043>>\setcounter<<1509>>myroman<<1509>><<1510>>10<<1510>>\roman<<1511>>myroman<<1511>><<1043>>^<<837>>2<<837>>;SPMamp; 7^<<838>>4<<838>> \!\cdot\! 13^<<839>>4<<839>> \!\cdot\! p_<<1044>>\setcounter<<1512>>myroman<<1512>><<1513>>9<<1513>>\roman<<1514>>myroman<<1514>><<1044>>^<<841>>2<<841>>;SPMamp; 7^<<842>>6<<842>> \!\cdot\! p_<<1045>>\setcounter<<1515>>myroman<<1515>><<1516>>4<<1516>>\roman<<1517>>myroman<<1517>><<1045>>^<<844>>2<<844>> \!\cdot\! p_<<1046>>\setcounter<<1518>>myroman<<1518>><<1519>>4<<1519>>\roman<<1520>>myroman<<1520>><<1046>>^<<846>>2<<846>> \!\cdot\! p_<<1047>>\setcounter<<1521>>myroman<<1521>><<1522>>5<<1522>>\roman<<1523>>myroman<<1523>><<1047>>^<<848>>2<<848>>\\ \hline
11 ;SPMamp; 89 ;SPMamp; 11 \!\cdot\! 67^<<849>>2<<849>>;SPMamp; 11^<<850>>2<<850>>;SPMamp; 11^<<851>>3<<851>> \!\cdot\! 67^<<852>>2<<852>>\\ \hline
13 ;SPMamp; 53 ;SPMamp; 13;SPMamp; 13^<<853>>2<<853>>;SPMamp; -\\ \hline
17 ;SPMamp; 103 ;SPMamp; 17 \!\cdot\! 613^<<854>>2<<854>>;SPMamp; 17^<<855>>2<<855>> \!\cdot\! 101^<<856>>2<<856>>;SPMamp; 17^<<857>>3<<857>> \!\cdot\! 67^<<858>>2<<858>>\\ \hline
19 ;SPMamp; 191 ;SPMamp; 19 \!\cdot\! 37^<<859>>2<<859>>;SPMamp; 19^<<860>>2<<860>>;SPMamp; 19^<<861>>5<<861>> \!\cdot\! 37^<<862>>2<<862>>\\ \hline
\end<<863>>array<<863>>
$$
\par
The BSD conjecture and this table (and my ``conjecture'' about $\Omega_A$) imply that
for the integers $n$ in the first column of the table, there is an~$A$
such that
$$<<1524>>\mbox<<1525>><<1526>>\fontencoding<<1527>>OT2<<1527>>\fontfamily<<1528>>wncyr<<1528>>\fontseries<<1529>>m<<1529>>\fontshape<<1530>>n<<1530>>\selectfont Sh<<1526>><<1525>><<1524>>(A) \approx (\mathbb<<1531>>Z<<1531>>/n\mathbb<<1532>>Z<<1532>>)\times H$$
with $\gcd(n,\#H)=1$. This is evidence for Conjecture 1, and also gives lots of examples
to show that $\#<<1533>>\mbox<<1534>><<1535>>\fontencoding<<1536>>OT2<<1536>>\fontfamily<<1537>>wncyr<<1537>>\fontseries<<1538>>m<<1538>>\fontshape<<1539>>n<<1539>>\selectfont Sh<<1535>><<1534>><<1533>>(A)$ is neither a square or twice a square in general.
\par
\vspace<<1540>>1ex<<1540>>\noindent<<1541>>\bf Challenge:<<1541>> <<865>>\em Let $E$ be one of the curves considered in the table, let $r$ be its rank,
and notice that in the table $n^r\mid \#<<1542>>\mbox<<1544>><<1546>>\fontencoding<<1547>>OT2<<1547>>\fontfamily<<1548>>wncyr<<1548>>\fontseries<<1549>>m<<1549>>\fontshape<<1550>>n<<1550>>\selectfont Sh<<1546>><<1544>><<1542>>_<<1543>>\mbox<<1545>>\tiny \rm an<<1545>><<1543>>^*$.
The BSD conjecture predicts that this divisibility should always hold. Prove that it
does for infinitely many $\ell$.<<865>>
\par
\end<<866>>document<<866>>
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