CoCalc Public Fileswww / job / SteinProp.tex
Author: William A. Stein
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45
46\begin{document}
47
48
49\normalsize
50\baselineskip=15.9pt
51
52\begin{center}
53{\Large\bf Research Plan}\\
54William A.\ Stein\\
55\today
56\end{center}
57
58\mysection{Introduction}
59My research program reflects the essential interplay between abstract
60theory and explicit machine computation during
61the latter half of the twentieth century;
62it sits at the intersection of recent work of
63B.~Mazur, K.~Ribet, J-P.~Serre, R.~Taylor, and A.~Wiles
64on Galois representations attached to modular abelian varieties
65(see \cite{ribet:modreps, serre:conjectures,
66taylor-wiles:fermat, wiles:fermat})
67with work of J.~Cremona, N.~Elkies, and J.-F.~Mestre
68on explicit computations involving modular forms
69(see \cite{cremona:algs, elkies:ffield}).
70
71In 1969 B.~Birch~\cite{birch:bsd} described computations that
72led to the most fundamental open
73conjecture in the theory of elliptic curves:
74\begin{quote}
75I want to describe some computations undertaken by myself and
76Swinnerton-Dyer on EDSAC
77by which we have calculated the zeta-functions of certain elliptic
78curves. As a result of these computations we have found an analogue
79for an elliptic curve of the Tamagawa number of an algebraic group;
80and conjectures (due to ourselves, due to Tate, and
81due to others) have proliferated.
82\end{quote}
83The rich tapestry of arithmetic conjectures and theory
84we enjoy today would not exist without the ground-breaking
85application of computing by Birch and Swinnerton-Dyer.
86Computations in the 1980s by Mestre were key in convincing
87Serre that his conjectures on modularity of odd
88irreducible  Galois representations
89were worthy of serious consideration (see~\cite{serre:conjectures}).
90These conjectures have inspired much recent work;
91for example, Ribet's proof of the
92$\epsilon$-conjecture, which played an essential role in
93Wiles's proof of Fermat's Last Theorem.
94
95My work on the Birch and Swinnerton-Dyer
96conjecture for modular abelian varieties and search for new
97examples of modular icosahedral Galois representations
98has led me to discover and implement algorithms for
99explicitly computing with modular forms.
100My research, which involves finding ways to compute
101with modular forms and modular abelian varieties,
102is driven by outstanding conjectures in number theory.
103
104\mysection{Invariants of modular abelian varieties}
105Now that the Shimura-Taniyama conjecture has been proved,
106the main outstanding problem in the field
107is the Birch and Swinnerton-Dyer conjecture (BSD conjecture),
108which ties together the arithmetic invariants of an elliptic curve.
109There is no general class of elliptic curves for which
110the full BSD conjecture is known.
111Approaches to the BSD conjecture that rely on congruences
112between modular forms are likely to
113require a deeper understanding of the analogous
114conjecture for higher-dimensional abelian varieties.
115As a first step, I have obtained theorems that make possible explicit
116computation of some of the arithmetic invariants of modular
117abelian varieties.
118
119\mysubsection{The BSD conjecture}
121that every elliptic curve over~$\Q$ is
122a quotient of the curve~$X_0(N)$ whose complex points
123are the isomorphism classes of pairs consisting of a
124(generalized) elliptic curve and a cyclic subgroup of order~$N$.
125Let~$J_0(N)$ denote the Jacobian of $X_0(N)$; this is an abelian
126variety of dimension equal to the genus of~$X_0(N)$ whose points
127correspond to the degree~$0$ divisor classes on~$X_0(N)$.
128
129An {\em optimal quotient} of $J_0(N)$ is a quotient by an abelian subvariety.
130Consider an optimal quotient~$A$ such that $L(A,1)\neq 0$.
131By~\cite{kolyvagin-logachev:totallyreal},~$A(\Q)$ and~$\Sha(A/\Q)$
132are both finite.
133The BSD conjecture asserts that
134$$\frac{L(A,1)}{\Omega_A} = 135\frac{\#\Sha(A/\Q)\cdot\prod_{p\mid N} c_p} 136{\# A(\Q)\cdot\#\Adual(\Q)}.$$
137Here the Shafarevich-Tate group $\Sha(A/\Q)$ is a measure of the failure
138of the local-to-global principle; the Tamagawa numbers~$c_p$ are the
139orders of the component  groups of~$A$; the real number~$\Omega_A$ is
140the volume of~$A(\R)$ with respect to a basis of differentials having
141everywhere nonzero good reduction; and~$\Adual$ is the dual of~$A$.
142My goal is to verify the full
143conjecture for many specific abelian varieties on a case-by-case basis.
144This is the first step in a program to verify the above
145conjecture for an infinite family of quotients of~$J_0(N)$.
146
147\mysubsection{The ratio $L(A,1)/\Omega_A$}
148Following Y.~Manin's work on elliptic curves,
149A.~Agash\'e and I proved the following
150theorem in~\cite{stein:vissha}.
151\begin{theorem}\label{thm:ratpart}
152Let~$m$ be the largest square dividing~$N$.
153The ratio $L(A,1)/\Omega_A$ is a rational number that can be
154explicitly computed, up to a unit (conjecturally $1$) in $\Z[1/(2m)]$.
155\end{theorem}
156The proof uses modular symbols combined with an extension of the argument
157used by Mazur in~\cite{mazur:rational} to bound the Manin constant.
158The ratio $L(A,1)/\Omega_A$ is expressed as the lattice index of
159two modules over the Hecke algebra.
160I expect the method to give similar results
161for special values of twists, and of
162$L$-functions attached to eigenforms of higher weight.
163I have computed $L(A,1)/\Omega_A$ for all optimal
164quotients of level $N\leq 1500$; this table continues to be
165of value to number theorists.
166
167\mysubsection{The torsion subgroup}
168I can compute upper and lower bounds on $\#A(\Q)_{\tor}$, but I can not
169determine $\#A(\Q)_{\tor}$ in all cases.  Experimentally, the deviation
170between the upper and lower bound is reflected in congruences with
171forms of lower level; I hope to exploit this in a precise way.
172I also obtained the following
173intriguing corollary that suggests cancellation between
174torsion and~$c_p$; it generalizes to higher weight forms, thus
175suggesting a geometric explanation for reducibility
176of Galois representations.
177\begin{corollary}
178Let~$n$ be the order of the image of
179$(0)-(\infty)$ in $A(\Q)$, and
180let~$m$ be the largest square dividing~$N$.
181Then $\displaystyle n\cdot L(A,1)/\Omega_A$ is an integer,
182up to a unit in  $\Z[1/(2m)]$.
183\end{corollary}
184
185\mysubsection{Tamagawa numbers}
186\begin{theorem}\label{thm:tamagawa}
187When $p^2\nmid N$, the number~$c_p$ can be explicitly computed
188(up to a power of~$2$).
189\end{theorem}
190I prove this in~\cite{stein:compgroup}.  Several related problems
191remain: when $p^2 \mid N$ it may be possible to compute~$c_p$
192using the Drinfeld-Katz-Mazur interpretation
193of~$X_0(N)$; it should also be possible to use my
194methods to treat optimal quotients of $J_1(N)$.
195
196I was surprised to find that systematic computations using this
197formula indicate the following conjectural refinement of a result of
198Mazur~\cite{mazur:eisenstein}.
199\begin{conjecture}
200Suppose~$N$ is prime and~$A$
201is an optimal quotient of $J_0(N)$.  Then $A(\Q)_{\tor}$ is
202generated by the image of $(0)-(\infty)$
203and $c_p = \#A(\Q)_{\tor}$.  Furthermore,
204the product of the~$c_p$ over all optimal factors
205equals the numerator of $(N-1)/12$.
206\end{conjecture}
207I have checked this conjecture for all $N\leq 997$ and,
208up to a power of~$2$, for all $N\leq 2113$.
209The first part is known when~$A$ is an elliptic
210curve (see~\cite{mestre-oesterle:crelle}).
211Upon hearing of this conjecture, Mazur proved it
212when all $q$-Eisenstein quotients'' are simple.
213There are three promising approaches to finding
214a complete proof. One involves the explicit
215formula of Theorem~\ref{thm:tamagawa};
216another is based on Ribet's level lowering theorem,
217and a third makes use of a simplicity result of Merel.
218
219Theorem~\ref{thm:tamagawa}
220also suggests a way to compute Tamagawa numbers of
221motives attached to eigenforms of higher weight.
222These numbers appear in the conjectures of Bloch and Kato,
223which generalize the BSD conjecture to motives (see~\cite{bloch-kato}).
224
225
226\mysubsection{Upper bounds on $\#\BigSha$}
227V.~Kolyvagin and K.~Kato~\cite{kolyvagin:structureofsha, scholl:kato}
228obtained upper bounds on~$\#\Sha(A)$.
229To verify the full BSD conjecture for certain abelian
230varieties, it is necessary is to make these bounds explicit.
231Kolyvagin's bounds involve computations with Heegner points,
232and Kato's involve a study of the Galois representations
233associated to~$A$.  I plan to carry out such
234computations in many specific cases.
235
236\mysubsection{Lower bounds on $\#\BigSha$}
237One approach to showing that~$\Sha$ is as large as predicted
238by the BSD conjecture is suggested by Mazur's notion of
239the visible part of~$\Sha$ (see~\cite{cremona-mazur,  mazur:visthree}).
240Let~$\Adual$ be the dual of~$A$.
241The visible part of $\Sha(\Adual/\Q)$ is the
242kernel of $\Sha(\Adual/\Q)\ra \Sha(J_0(N))$.
243Mazur observed that if an element of order~$p$
244in~$\Sha(\Adual/\Q)$ is visible,
245then it is explained by a jump in the rank of Mordell-Weil
246in the sense that there is another abelian subvariety $B\subset J_0(N)$
247such that $p \mid \#(\Adual\intersect B)$ and the rank of~$B$ is positive.
248I think that this observation can be turned around: if there is
249another abelian
250variety~$B$ of positive rank such that $p\mid \#(\Adual\intersect B)$,
251then, under mild hypotheses, there is an element of~$\Sha(\Adual/\Q)$
252of order~$p$.  Thus the theory of congruences between modular
253forms can be used to obtain a lower bound on~$\#\Sha(\Adual/\Q)$.
254I am trying to use the cohomological methods of~\cite{mazur:tower}
255and suggestions of B.~Conrad and Mazur to prove the following conjecture.
256\begin{conjecture}
257Let~$\Adual$ and~$B$ be abelian subvarieties of~$J_0(N)$.
258Suppose that $p\mid \#(\Adual\intersect B)$, that~$p\nmid N$,
259and that~$p$ does not divide the
260order of any of the torsion subgroups
261or component groups of~$A$ or~$B$.  Then
262$(B(\Q) \oplus \Sha(B/\Q))\tensor\Z/p\Z 263\isom (\Adual(\Q)\oplus \Sha(\Adual/\Q) )\tensor\Z/p\Z$.
264\end{conjecture}
265Unfortunately, $\Sha(\Adual/\Q)$ can fail to be
266visible inside~$J_0(N)$.
267For example, I found that the BSD conjecture
268predicts the existence of invisible elements of odd order in~$\Sha$
269for at least~$15$ of the~$37$ optimal quotients of prime
270level $\leq 2113$.
271For every integer~$M$ (Ribet~\cite{ribet:raising} tells us which~$M$
272to choose), we can consider the images of~$\Adual$ in $J_0(NM)$.
273There is not yet enough evidence to conjecture the existence of
274an integer~$M$ such that all of $\Sha(\Adual/\Q)$ is visible in
275$J_0(NM)$.
276I am gathering data to determine whether or not
277to expect the existence of such~$M$.
278
279\mysubsection{Motivation for considering abelian varieties}
280If $A$ is an elliptic curve, then explaining~$\Sha(A/\Q)$ using
281only congruences between elliptic curves is bound to fail.
282This is because pairs of nonisogenous elliptic curves
283with isomorphic $p$-torsion are, according to
284E.~Kani's conjecture, extremely rare.
285It is crucial to understand what happens in all dimensions.
286
287Within the range accessible by computer,
288abelian varieties exhibit more richly textured
289structure than elliptic curves.
290For example, I discovered a visible element of prime order $83341$
291in the Shafarevich-Tate group of an abelian variety of
292prime conductor~$2333$; in contrast, over all optimal elliptic
293curves of conductor up to $5500$, it appears that
294the largest order of an element of a Shafarevich-Tate
295group is~$7$.
296
297
298\mysection{Conjectures of Artin, Merel, and Serre}
299
300\mysubsection{Icosahedral Galois representations}
301E.~Artin conjectured in~\cite{artin:conjecture}
302that the $L$-series associated to any continuous irreducible
303representation $\rho:\GQ\ra \GL_n(\C)$, with $n>1$, is entire.
304Recent exciting work of Taylor and others suggests that a complete proof
305of Artin's conjecture, in the case when $n=2$ and~$\rho$ is odd,
306is on the horizon.
307This case of Artin's conjecture is known when the image
308of~$\rho$ in $\PGL_2(\C)$ is solvable
309(see~\cite{tunnell:artin}), and in infinitely
310many cases when the image of~$\rho$ is not solvable (see~\cite{bdsbt}).
311
312In 1998, K.~Buzzard suggested a way to combine the
313main theorem of~\cite{buzzard-taylor}, along with
314a computer computation, to deduce modularity of certain
315icosahedral Galois representations.  Buzzard and I recently obtained
316the following theorem.
317\begin{theorem}
318The icosahedral Artin representations of conductor~$1376=2^5\cdot 43$
319are modular.
320\end{theorem}
321We expect our method to yield several more examples.
322These ongoing computations are laying a small part of the technical
323foundations necessary for a full proof of the Artin conjecture
324for odd two dimensional~$\rho$, as well as stimulating
325the development of new algorithms for computing with modular forms
326using modular symbols in characteristic~$\ell$.
327
328\mysubsection{Cyclotomic points on modular curves}
329If~$E$ is an elliptic curve over~$\Q$
330and~$p$ is an odd prime, then the $p$-torsion
331on~$E$ can not all lie in~$\Q$; because of
332the Weil pairing the $p$-torsion generates a field
333that contains~$\Q(\mu_p)$.
334For which primes~$p$ does
335there exist an elliptic curve~$E$ over $\Q(\mu_p)$
336with all of its $p$-torsion rational over $\Q(\mu_p)$?
337When $p=2,3,5$ the corresponding moduli space has genus zero
338and infinitely many examples exist.  Recent work of L.~Merel,
339combined with computations he enlisted me to do,
340suggest  that these are the only primes~$p$ for
341which such elliptic curves exist.
342In~\cite{merel:cyclo}, Merel exploits
343cyclotomic analogues of the techniques used
344in his proof of the uniform boundedness conjecture
345to obtain an explicit criterion that can
346be used to answer the above question for many primes~$p$,
347on a case-by-case basis.
348Theoretical work of Merel, combined with my computations of
349twisted $L$-values and character groups of tori,
350give the following result (see~\cite[\S3.2]{merel:cyclo}):
351\begin{theorem}
352Let $p \equiv 3\pmod{4}$ be a prime satisfying
353$7 \leq p < 1000$. There are no elliptic curves
354over $\Q(\mu_p)$ all of whose $p$-torsion is rational
355over $\Q(\mu_p)$.
356\end{theorem}
357The case in which~$p$ is congruent to~$1$ modulo~$4$ presents
358additional difficulties that involve showing that~$Y(p)$ has no
359$\Q(\sqrt{p})$-rational points.  We hope to tackle these in
360the near future.
361
362
363\mysubsection{Genus one curves}
364The index of an algebraic curve~$C$ over~$\Q$ is the
365order of the cokernel of the degree map $\Div_\Q(C)\ra\Z$;
366rationality of the canonical divisor implies that the index
367divides $2g-2$, where~$g$ is the genus of~$C$.
368When $g=1$ this is no condition at all; Artin conjectured, and
369Lang and Tate~\cite{lang-tate} proved, that for every integer~$m$
370there is a genus one curve of index~$m$ over some number field.
371Their construction yields genus one curves over~$\Q$ only for a few
372values of~$m$, and they ask whether one can find genus one curves
373over~$\Q$ of every index.  I have answered
374this question for odd~$m$.
375\begin{theorem}
376Let~$K$ be any number field.  There are genus one curves over~$K$
377of every odd index.
378\end{theorem}
379The proof involves showing that enough cohomology classes in
380Kolyvagin's Euler system of Heegner points do not vanish
381combined with explicit Heegner point computations.
382I hope to show that curves  of every index occur, and
383to determine the consequences of my
384nonvanishing result for Selmer groups.  This can be viewed
385as a contribution to the problem of understanding $H^1(\Q,E)$.
386
387\mysubsection{Serre's conjecture modulo $pq$}
388Let~$p$ and~$q$ be primes, and consider a continuous
389representation $\rho:\GQ\ra \GL(2,\Z/pq\Z)$
390that is irreducible in the sense that its reductions
391modulo~$p$ and modulo~$q$ are both irreducible.
392Call~$\rho$ {\em modular} if there is a modular
393form~$f$ such that a mod~$p$ representation attached to~$f$
394is the mod~$p$ reduction of~$\rho$, and ditto for~$q$.
395I have carried out specific computations suggested by Mazur
396in hopes of determining when one should expect
397that such mod~$pq$ representations are modular; the computation
398suggests that the right conjectures are elusive.
399Ribet's theorem (see~\cite{ribet:raising})
400produces infinitely many levels $pq\ell$ at which there is
401a form giving rise to $\rho$~mod~$p$ and another
402giving rise to $\rho$~mod~$q$; we hope to determine if for some~$\ell$
403there is a single form  giving rise to both reductions.
404
405\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
406\newcommand{\closer}{\vspace{-1.5ex}}
407\begin{thebibliography}{10}
408
409\bibitem{agashe}
410A.~Agash\'{e}, \emph{On invisible elements of the {T}ate-{S}hafarevich group},
411  C. R. Acad. Sci. Paris S\'er. I Math. \textbf{328} (1999), no.~5, 369--374.
412\closer
413
414\bibitem{stein:vissha}
415A.~Agash\'{e} and W.\thinspace{}A. Stein, \emph{Visibility of
416  {S}hafarevich-{T}ate groups of modular abelian varieties}, in preparation
417  (1999).
418\closer
419
420\bibitem{artin:conjecture}
421E.~Artin, \emph{{\"U}ber eine neue {A}rt von {L}-{R}eihen}, Abh. Math. Sem. in
422  Univ. Hamburg \textbf{3} (1923), 89--108.
423\closer
424
425\bibitem{birch:bsd}
426B.\thinspace{}J. Birch, \emph{Elliptic curves over \protect{${\mathbf{Q}}$:
427  {A}} progress report}, 1969 Number Theory Institute (Proc. Sympos. Pure
428  Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math.
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430\closer
431
432\bibitem{bloch-kato}
433S.~Bloch and K.~Kato, \emph{\protect{${L}$}-functions and \protect{T}amagawa
434  numbers of motives}, The Grothendieck Festschrift, Vol. \protect{I},
435  Birkh\"auser Boston, Boston, MA, 1990, pp.~333--400.
436\closer
437
439C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the modularity of
440  elliptic curves over~\protect{$\Q$}},
441  in preparation.
442\closer
443
444\bibitem{bdsbt}
445K.~Buzzard, M.~Dickinson, N.~Shepherd-Barron, and R.~Taylor, \emph{On
446  icosahedral {A}rtin representations},
447  available at {\tt http://www.math.harvard.edu/\~{}rtaylor/}.
448\closer
449
450\bibitem{buzzard-taylor}
451K.~Buzzard and R.~Taylor, \emph{Companion forms and weight one
452  forms},  Annals of Math. (1999).
453\closer\vspace{-3ex}
454
455\bibitem{cremona:algs}
456J.\thinspace{}E. Cremona, \emph{Algorithms for modular elliptic curves}, second
457  ed., Cambridge University Press, Cambridge, 1997.
458\closer
459
460\bibitem{cremona-mazur}
461J.\thinspace{}E. Cremona and B.~Mazur, \emph{Visualizing elements in the
462  \protect{Shafarevich-Tate} group}, Proceedings of the Arizona Winter School
463  (1998).
464\closer
465
466\bibitem{elkies:ffield}
467N.\thinspace{}D. Elkies, \emph{Elliptic and modular curves over finite fields
468  and related computational issues}, Computational perspectives on number
469  theory (Chicago, IL, 1995), Amer. Math. Soc., Providence, RI, 1998,
470  pp.~21--76.
471\closer
472
473\bibitem{kolyvagin:structureofsha}
474V.\thinspace{}A. Kolyvagin, \emph{On the structure of {S}hafarevich-{T}ate
475  groups}, Algebraic geometry (Chicago, IL, 1989), Springer, Berlin, 1991,
476  pp.~94--121.
477\closer
478
479\bibitem{kolyvagin-logachev:totallyreal}
480V.\thinspace{}A. Kolyvagin and D.\thinspace{}Y. Logachev, \emph{Finiteness of
481  \protect{$\Sha$} over totally real fields}, Math. USSR Izvestiya \textbf{39}
482  (1992), no.~1, 829--853.
483\closer
484
485\bibitem{lang-tate}
486S.~Lang and J.~Tate, \emph{Principal homogeneous spaces over abelian
487  varieties}, Amer. J. Math. \textbf{80} (1958), 659--684.
488\closer
489
490\bibitem{mazur:tower}
491B.~Mazur, \emph{Rational points of abelian varieties with values in towers of
492  number fields}, Invent. Math. \textbf{18} (1972), 183--266.
493\closer
494
495\bibitem{mazur:eisenstein}
496\bysame, \emph{Modular curves and the \protect{Eisenstein} ideal}, Inst. Hautes
497  \'Etudes Sci. Publ. Math. (1977), no.~47, 33--186 (1978).
498\closer
499
500\bibitem{mazur:rational}
501\bysame, \emph{Rational isogenies of prime degree (with an appendix by {D}.
502  {G}oldfeld)}, Invent. Math. \textbf{44} (1978), no.~2, 129--162.
503\closer
504
505\bibitem{mazur:visthree}
506\bysame, \emph{Visualizing elements of order three in the {S}hafarevich-{T}ate
507  group}, preprint (1999).
508\closer
509
510\bibitem{merel:cyclo}
511L.~Merel, \emph{Sur la nature non-cyclotomique des points d'ordre fini des
512  courbes elliptiques}, preprint (1999).
513\closer
514
515%\bibitem{mestre:graphs}
516%J.-F. Mestre, \emph{La m\'ethode des graphes. \protect{Exemples} et
517%  applications}, Proceedings of the international conference on class numbers
518%  and fundamental units of algebraic number fields (Katata) (1986), 217--242.
519%\closer
520
521\bibitem{mestre-oesterle:crelle}
522J.-F. Mestre and J.~Oesterl{\'e}, \emph{Courbes de {W}eil semi-stables de
523  discriminant une puissance \protect{$m$}-i\eme}, J. Reine Angew. Math.
524  \textbf{400} (1989), 173--184.
525\closer
526
527\bibitem{ribet:modreps}
528K.\thinspace{}A. Ribet, \emph{On modular representations of \protect{${\rm 529 {G}al}(\overline{\bf{Q}}/{\bf {Q}})$} arising from modular forms}, Invent.
530  Math. \textbf{100} (1990), no.~2, 431--476.
531\closer
532
533\bibitem{ribet:raising}
534\bysame, \emph{Raising the levels of modular representations}, S\'eminaire de
535  Th\'eorie des Nombres, Paris 1987--88, Birkh\"auser Boston, Boston, MA, 1990,
536  pp.~259--271.
537\closer
538
539\bibitem{scholl:kato}
540A.\thinspace{}J. Scholl, \emph{An introduction to {K}ato's {E}uler systems},
541  Galois Representations in Arithmetic Algebraic Geometry, Cambridge University
542  Press, 1998, pp.~379--460.
543\closer
544
545\bibitem{serre:conjectures}
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573
574
575
576\end{document}
577
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