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MR2085902
Agashe, Amod(1-TX); Stein, William(1-HRV)
Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank zero. (English. English summary)
With an appendix by J. Cremona and B. Mazur.
Math. Comp. 74 (2005), no. 249, 455--484 (electronic).
11G40 (11G10)
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References: 44 [-] Reference Citations: 3 Review Citations: 0
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[References]

Note: This list, extracted from the PDF form of the original paper, may contain data conversion errors, almost all limited to the mathematical expressions.
  1. A. Agashe, On invisible elements of the Tate-Shafarevich group, C. R. Acad. Sci. Paris S'er. I Math. 328 (1999), no. 5, 369--374. MR1678131 (2000e:11083)
  2. A. Agashe and W. A. Stein, Appendix to Joan-C. Lario and Ren'e Schoof: Some compu- tations with Hecke rings and deformation rings, to appear in J. Exp. Math. cf. MR1959271 (2004b:11072)
  3. A. Agashe and W.A. Stein, Visibility of Shafarevich-Tate Groups of Abelian Varieties, to appear in J. of Number Theory (2002). cf. MR1939144 (2003h:11070)
  4. A. Agashe and W. A. Stein, The Manin constant, congruence primes, and the modular degree, preprint, (2004). http://modular.math.washington.edu/papers/manin-agashe/
  5. W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user lan- guage, J. Symbolic Comput. 24 (1997), no. 3-4, 235--265, Computational algebra and number theory (London, 1993). MR1484478
  6. B.J. Birch, Elliptic curves over Q : A progress report, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), Amer. Math. Soc., Providence, R.I., 1971, pp. 396--400. MR0314845 (47 #3395)
  7. S. Bosch, W. L"utkebohmert, and M. Raynaud, N\'eron models, Springer-Verlag, Berlin, 1990. MR1045822 (91i:14034)
  8. J. W.S. Cassels, " Arithmetic on curves of genus 1 . III. The Tate-\v Safarevi\v c and Selmer groups'', Proc. London Math. Soc. (3) 12 (1962), 259--296. MR0163913 (29 #1212)
  9. J. E. Cremona, Algorithms for modular elliptic curves, second ed., Cambridge University Press, Cambridge, 1997. MR1628193 (99e:11068)
  10. J. E. Cremona and B. Mazur, Visualizing elements in the Shafarevich-Tate group, Ex- periment. Math. 9 (2000), no. 1, 13--28. MR1758797 (2001g:11083)
  11. B. Conrad and W.A. Stein, Component groups of purely toric quotients, Math. Res. Lett. 8 (2001), no. 5--6, 745--766. MR1879817 (2003f:11087)
  12. C. Delaunay, Heuristics on Tate-Shafarevitch groups of elliptic curves defined over Q, Experiment. Math. 10 (2001), no. 2, 191--196. MR1837670 (2003a:11065)
  13. F. Diamond and J. Im, Modular forms and modular curves, Seminar on Fermat's Last Theorem, Providence, RI, 1995, pp. 39--133. MR1357209 (97g:11044)
  14. B. Edixhoven, On the Manin constants of modular elliptic curves, Arithmetic algebraic geometry (Texel, 1989), Birkh"auser Boston, Boston, MA, 1991, pp. 25--39. MR1085254 (92a:11066)
  15. M. Emerton, Optimal quotients of modular Jacobians. Preprint. cf. MR2021024
  16. E.V. Flynn, F. Lepr'evost, E. F. Schaefer, W. A. Stein, M. Stoll, and J. L. Wetherell, Em- pirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comp. 70 (2001), no. 236, 1675--1697. MR1836926 (2002d:11072)
  17. J-M. Fontaine, Groupes finis commutatifs sur les vecteurs de Witt, C. R. Acad. Sci. Paris S'er. A-B 280 (1975), Ai, A1423--A1425. MR0374153 (51 #10353)
  18. B.H. Gross, L -functions at the central critical point, Motives (Seattle, WA, 1991), Amer. Math. Soc., Providence, RI, 1994, pp. 527--535. MR1265543 (95a:11060)
  19. A. Grothendieck, Le groupe de Brauer. III. Exemples et compl\'ements, Dix Expos'es sur la Cohomologie des Sch'emas, North-Holland, Amsterdam, 1968, pp. 88--188. MR0244271 (39 #5586c)
  20. A. Grothendieck, Mod`eles de N\'eron et monodromie in Groupes de monodromie en g\'eom\'etrie alg\'ebrique. I, Springer-Verlag, Berlin, 1972, S'eminaire de G'eom'etrie Alg'ebrique du Bois-Marie 1967--1969 (SGA 7 I), Dirig'e par A. Grothendieck. Vol. 288. MR0354656 (50 #7134)
  21. B. Gross and D. Zagier, Heegner points and derivatives of L -series, Invent. Math. 84 (1986), no. 2, 225--320. MR0833192 (87j:11057)
  22. N.M. Katz, Galois properties of torsion points on abelian varieties, Invent. Math. 62 (1981), no. 3, 481--502. MR0604840 (82d:14025)
  23. V.A. Kolyvagin and D.Y. Logachev, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties, Algebra i Analiz 1 (1989), no. 5, 171--196. MR1036843 (91c:11032)
  24. V.A. Kolyvagin and D.Y. Logachev, Finiteness ofover totally real fields, Math. USSR Izvestiya 39 (1992), no. 1, 829--853. MR1137589 (93d:11063)
  25. D.R. Kohel and W. A. Stein, Component Groups of Quotients of J0( N), Proceedings of the 4th International Symposium (ANTS-IV), Leiden, Netherlands, July 2--7, 2000 (Berlin), Springer, 2000. MR1850621 (2002h:11051)
  26. S. Lang, Number theory. III, Springer-Verlag, Berlin, 1991, Diophantine geometry. MR1112552 (93a:11048)
  27. H. W. Lenstra, Jr. and F. Oort, Abelian varieties having purely additive reduction, J. Pure Appl. Algebra 36 (1985), no. 3, 281--298. MR0790619 (86e:14020)
  28. Joan-C. Lario and Ren'e Schoof, Some computations with Hecke rings and deformation rings, Experiment. Math. 11 (2002), no. 2, 303--311, with an appendix by Amod Agashe and William Stein. MR1959271 (2004b:11072)
  29. B. Mazur, Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183--266. MR0444670 (56 #3020)
  30. B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes'Etudes Sci. Publ. Math. (1977), no. 47, 33--186 (1978). MR0488287 (80c:14015)
  31. B. Mazur, Rational isogenies of prime degree (with an appendix by D. Goldfeld), Invent. Math. 44 (1978), no. 2, 129--162. MR0482230 (80h:14022)
  32. B. Mazur and J. Tate, Points of order 13 on elliptic curves, Invent. Math. 22 (1973/74), 41--49. MR0347826 (50 #327)
  33. J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984), Springer, New York, 1986, pp. 103--150. MR0861974
  34. A. P. Ogg, Rational points on certain elliptic modular curves, Analytic number theory (Proc. Sympos. Pure Math., Vol XXIV, St. Louis Univ., St. Louis, Mo., 1972), Amer. Math. Soc., Providence, R.I., 1973, pp. 221--231. MR0337974 (49 #2743)
  35. B. Poonen and M. Stoll, The Cassels-Tate pairing on polarized abelian varieties, Ann. of Math. (2) 150 (1999), no. 3, 1109--1149. MR1740984 (2000m:11048)
  36. G. Shimura, On the factors of the jacobian variety of a modular function field, J. Math. Soc. Japan 25 (1973), no. 3, 523--544. MR0318162 (47 #6709)
  37. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Prince- ton University Press, Princeton, NJ, 1994, Reprint of the 1971 original, Kan Memorial Lectures, 1. MR1291394 (95e:11048)
  38. G. Stevens, Arithmetic on modular curves, Birkh"auser Boston Inc., Boston, Mass., 1982. MR0670070 (87b:11050)
  39. W. A. Stein, Explicit approaches to modular abelian varieties, Ph.D. thesis, University of California, Berkeley (2000).
  40. W. A. Stein, An introduction to computing modular forms using modular symbols, to appear in an MSRI Proceedings (2002).
  41. W. A. Stein, Shafarevich-Tate groups of nonsquare order, Proceedings of MCAV 2002, Progress of Mathematics (to appear). cf. MR2058655
  42. J. Sturm, On the congruence of modular forms, Number theory (New York, 1984--1985), Springer, Berlin, 1987, pp. 275--280. MR0894516 (88h:11031)
  43. J. Tate, Duality theorems in Galois cohomology over number fields, Proc. Inter- nat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 288--295. MR0175892 (31 #168)
  44. J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog, S'eminaire Bourbaki, Vol. 9, Soc. Math. France, Paris, 1966 (reprinted in 1995), Exp. No. 306, 415--440. MR1610977
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MR2023296
Coleman, Robert F.(1-CA); Stein, William A.(1-HRV)
Approximation of eigenforms of infinite slope by eigenforms of finite slope.
Geometric aspects of Dwork theory. Vol. I, II, 437--449,
Walter de Gruyter GmbH & Co. KG, Berlin, 2004.
11F33
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MR2058655 (2005c:11072)
Stein, William A.(1-HRV)
Shafarevich-Tate groups of nonsquare order. (English. English summary)
Modular curves and abelian varieties, 277--289,
Progr. Math., 224,
Birkhäuser, Basel, 2004.
11G10
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In an unguarded moment, P. Swinnerton-Dyer \ref[in Proc. Conf. Local Fields (Driebergen, 1966), 132--157, Springer, Berlin, 1967; MR0230727 (37 \#6287)] wrote that if the group ${\cyr Sh}(A)$ (of everywhere locally trivial $K$-torsors under an abelian variety $A$ over a number field $K$) is finite---as it is widely conjectured to be---then a theorem of Tate would imply that its order ${\cyr sh}(A)$ is a square, i.e. for every prime $p$, the exponent $v_p({\cyr sh}(A))$ of $p$ in ${\cyr sh}(A)$ is even.

What the results of J. Tate \ref[in Proc. Internat. Congr. Mathematicians (Stockholm, 1962), 288--295, Inst. Mittag-Leffler, Djursholm, 1963; MR0175892 (31 \#168)] and M. Flach \ref[J. Reine Angew. Math. 412 (1990), 113--127; MR1079004 (92b:11037)] do imply is that $v_p({\cyr sh}(A))$ is even, if $A$ admits a suitable polarisation (cf. Theorem 1.2). Admitting a principal polarisation is sufficient for the odd part of ${\cyr sh}(A)$ to have square order.

And indeed, B. Poonen and M. Stoll \ref[Ann. of Math. (2) 150 (1999), no. 3, 1109--1149; MR1740984 (2000m:11048)] came up with an explicit Jacobian surface $A$ over $\bold Q$ such that ${\cyr sh}(A)=2$; they also gave a criterion for the Jacobian variety $A$ of a (smooth, projective, absolutely connected) curve $X$ of genus $g\ge2$ over $K$ to have odd $v_2({\cyr sh}(A))$: such is the case if the (finite) number of places of $K$ where $X$ fails to have a $0$-cycle of degree $g-1$ is odd. Numerous further examples have been found by B. W. Jordan and R. A. Livné \ref[Bull. London Math. Soc. 31 (1999), no. 6, 681--685; MR1711026 (2000j:11090)] and by S. Baba \ref[J. Number Theory 87 (2001), no. 1, 96--108; MR1816038 (2002b:11085)].

The author gives the first examples of odd $v_p({\cyr sh}(A))$ for an odd prime $p$. His main result implies that for every $p<25000$ (with $p\neq37$), there is a twist $A$ of the power $E^{p-1}$ of the abelian curve $E\colon y^2+y=x^3-x$ (the curve 37A) such that $v_p({\cyr sh}(A))$ is odd (Theorem 3.1). To get an example where $v_{37}({\cyr sh}(A))$ is odd, use the curve 43A instead.

The restriction $p<25000$ (cf. Proposition 2.3) comes from the fact that for these primes his tireless computer has been able to find a certain auxiliary prime $l$ (cf. Conjecture 2.1) needed for constructing $A$. A sample of his instructions to the computer is included.

The main result (Theorem 2.14) establishes an exact sequence $$0\rightarrow E(\bold Q)/pE(\bold Q)\rightarrow {}_{p^\infty}{\cyr Sh}(A)\rightarrow {}_{p^\infty}{\cyr Sh}(E_L)\rightarrow {}_{p^\infty}{\cyr Sh}(E)\rightarrow 0 $$ for an abelian curve $E$ over $\bold Q$ and an odd prime $p$ which does not divide any of the Tamagawa numbers of $E$ and for which $\rho_{E,p}\colon {\rm Gal}(\overline{\bold Q}|\bold Q)\rightarrow{\rm Aut}({}_p E(\overline{\bold Q}))$ is surjective. The auxiliary prime $l$ should be $\equiv1\pmod p$, it should not divide the conductor of $E$, the function $L(E,\chi,s)$ should not vanish at $s=1$ for some---and hence for all $p-1$---character(s) $\chi\colon (\bold Z/l\bold Z)^\times\rightarrow{}_p\bold C^\times$ of level $l$ and order $p$, and, finally, $p$ should not divide ${\rm Card}\, E(\bold F_l)$. The degree-$p$ cyclic extension $L$ is contained in the field of $l$th roots of $1$, and $A$ is the kernel of the trace map $\roman{Res}_{L|\bold Q}E_L\rightarrow E$; it turns out to be a twist of $E^{p-1}$ (Proposition 2.4).

If ${}_{p^\infty}{\cyr Sh}(E)$ is finite, then so are the other two ${\cyr Sh}$ by a deep theorem of Kazuya Kato, applicable by the choice of $l$. In that case ${\rm rk}\,E(\bold Q)$ and $v_p({\cyr sh}(A))$ have the same parity, in view of the surjectivity of $\rho_{E,p}$ and the fact that the last two groups in the displayed exact sequence are of square order. The author gets the desired examples of odd $v_p({\cyr sh}(A))$ by choosing an $E$ for which ${\rm rk}\,E(\bold Q)$ is odd---such as the rank-1 curve 37A. For this curve he also verifies, for good measure, that ${\cyr Sh}=\{0\}$, using the results of Kolyvagin and the programmes of Cremona.

However, $v_q({\cyr sh}(A))$ is even for every prime $q\neq p$, if ${}_{q^\infty}{\cyr Sh}(E)$ is finite (Proposition 2.16).

REVISED (January, 2005)

Current version of review. Go to earlier version.

Reviewed by Chandan Singh Dalawat

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MR2052021 (2005c:11070)
Stein, William(1-HRV); Watkins, Mark(1-PAS)
Modular parametrizations of Neumann-Setzer elliptic curves.
Int. Math. Res. Not. 2004, no. 27, 1395--1405.
11G05 (11G18)
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Let $E/\bold Q$ be an elliptic curve of conductor $N$. G. Stevens \ref[Invent. Math. 98 (1989), no. 1, 75--106; MR1010156 (90m:11089)] conjectured that the optimal quotient of $X_1(N)$ in the isogeny class of $E$ is the curve in this isogeny class with minimal Faltings height. In this paper the authors verify Stevens' conjecture in the case where $N$ is prime. To do so, first recall that in \ref[J.-F. Mestre and J. Oesterlé, J. Reine Angew. Math. 400 (1989), 173--184; MR1013729 (90g:11078)] the isogeny class of an elliptic curve $E/\bold Q$ of prime conductor $p>37$ contains exactly one curve, unless $p=u^2+64$ and $E$ is one of the two Neumann-Setzer curves \ref[O. Neumann, Math. Nachr. 49 (1971), 107--123; MR0337999 (49 \#2767a); B. Setzer, J. London Math. Soc. (2) 10 (1975), 367--378; MR0371904 (51 \#8121)]: $$ E_0\colon y^2 + xy = x^3 - \frac{u+1}{4} x^2 + 4x - u,$$ $$ E_1\colon y^2 + xy = x^3 - \frac{u+1}{4} x^2 - u. $$ To study Stevens' conjecture it then suffices to consider the second case. The Faltings height of $E_1$ is smaller than that of $E_0$; this follows by exhibiting an isogeny $ E_1 \rightarrow E_0 $ that extends to an étale morphism of the respective Néron models. Analyze the kernel of this isogeny and of the natural map from the Jacobian of $X_0(p)$ to that of $X_1(p)$, coupled with the fact that $E_0$ is $X_0(p)$-optimal \ref[J.-F. Mestre and J. Oesterlé, op. cit.], and it follows that $E_1$ is $ X_1(p)$-optimal.

By an intricate analysis of the Eisenstein ideals \ref[B. Mazur, Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 33--186 (1978); MR0488287 (80c:14015)], the authors also show that the modular degree of $E_0$ is odd if and only if $ u\equiv 3 \pmod 8 $, and they post various conjectures concerning the parity of the modular degree of elliptic curves over $\bold Q$ (sample Conjecture 4.2: there are infinitely many elliptic curves over $\bold Q$ with odd modular degree). The paper ends with numerical data for the frequency of nontrivial $p$-{\cyr Sh} (presumably computed under the Birch-Swinnerton-Dyer conjecture) for the Neumann-Setzer curves.

Reviewed by Siman Wong

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[References]

  1. A. Abbes and E. Ullmo, \`A propos de la conjecture de Manin pour les courbes elliptiques modulaires [ The Manin conjecture for modular elliptic curves], Compositio Math. 103 (1996), no. 3, 269--286 (French). MR1414591 (97f:11038)
  2. A. Brumer and O. McGuinness, The behavior of the Mordell-Weil group of elliptic curves, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 375--382, http://www.oisinmc.com/math/310716/. MR1044170 (91b:11076)
  3. F. Calegari and W. Stein, Conjectures about discriminants of Hecke algebras of prime level, to appear in ANTS VI proceedings, Springer-Verlag Lecture Notes in Computer Science Series, http://web.ew.usna.edu/$\sim$ants/.
  4. J. E. Cremona, Elliptic curves of conductor $\leq 20000$, http://www.maths.nott.ac.uk/personal/jec/ftp/data.
  5. C. Delaunay, Heuristics on Tate-Shafarevitch groups of elliptic curves defined over $\Bbb{Q}$ Experiment. Math. 10 (2001), no. 2, 191--196. MR1837670 (2003a:11065)
  6. P. Deligne, Preuve des conjectures de Tate et de Shafarevitch (d'apr\`es G. Faltings) [ Proof of the Tate and Shafarevich conjectures (after G. Faltings)], Astérisque (1985), no. 121--122, 25--41 (French), Seminaire Bourbaki, Vol. 1983/84. MR0768952 (87c:11026)
  7. F. Diamond and J. Im, Modular forms and modular curves, Seminar on Fermat's Last Theorem (Toronto, Ontario, 1993--1994), CMS Conf. Proc., vol. 17, American Mathematical Society, Rhode Island, 1995, pp. 39--133. MR1357209 (97g:11044)
  8. M. Emerton, Optimal quotients of modular Jacobians, preprint, 2001. cf. MR2021024
  9. G. Frey, Links between solutions of ${\rm A} - {\rm B} = {\rm C}$ and elliptic curves, Number Theory (Ulm, 1987) (H. P. Schlickewei and E. Wirsing, eds.), Lecture Notes in Math., vol. 1380, Springer, New York, 1989, pp. 31--62. MR1009792 (90g:11069)
  10. R. K. Guy, Unsolved Problems in Number Theory, Problem Books in Mathematics, Springer-Verlag, New York, 1994. MR1299330 (96e:11002)
  11. G. H. Hardy and J. E. Littlewood, Some problems of "Partitio numerorum": III. On the expression of a number as a sum of primes, Acta Math. 44 (1922), 1--70.
  12. S. Ling and J. Oesterlé, The Shimura subgroup of ${\rm J}_0 ({\rm N})$, Astérisque (1991), no. 196--197, 171--203 (1992). MR1141458 (93b:14038)
  13. B. Mazur, Three lectures about the arithmetic of elliptic curves, http://swc.math.arizona.edu/notes/files/98MazurLN.pdf.
  14. B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33--186. MR0488287 (80c:14015)
  15. L. Merel, L'accouplement de Weil entre le sous-groupe de Shimura et le sous-groupe cuspidal de ${\rm J}_0 ({\rm p})$ [ Weil pairing of the Shimura subgroup and the cuspidal subgroup of ${\rm J}_0 ({\rm p})$], J. reine angew. Math. 477 (1996), 71--115 (French). MR1405312 (97f:11045)
  16. J.-F. Mestre and J. Oesterlé, Courbes de Weil semi-stables de discriminant une puissance ${\rm m}$- i\`eme [ Semistable Weil curves with discriminant an ${\rm m}$ th power], J. reine angew. Math. 400 (1989), 173--184 (French). MR1013729 (90g:11078)
  17. D. Mumford, Abelian Varieties, Tata Institute of Fundamental Research Studies in Mathematics, no. 5, Oxford University Press, London, 1970. MR0282985 (44 #219)
  18. O. Neumann, Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I, Math. Nachr. 49 (1971), 107--123 (German). MR0337999 (49 #2767a)
  19. O. Neumann, Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. II, Math. Nachr. 56 (1973), 269--280 (German). MR0338000 (49 #2767b)
  20. A. P. Ogg, Hyperelliptic modular curves, Bull. Soc. Math. France 102 (1974), 449--462. MR0364259 (51 #514)
  21. K. A. Ribet and W. A. Stein, Lectures on Serre's conjectures, Arithmetic Algebraic Geometry (Park City, Utah, 1999), IAS/Park City Math. Ser., vol. 9, American Mathematical Society, Rhode Island, 2001, pp. 143--232. MR1860042 (2002h:11047)
  22. B. Setzer, Elliptic curves of prime conductor, J. London Math. Soc. (2) 10 (1975), 367--378. MR0371904 (51 #8121)
  23. W. A. Stein and M. Watkins, A database of elliptic curves---first report, Algorithmic Number Theory (Sydney 2002) (C. Fieker and D. Kohel, eds.), Lecture Notes in Comput. Sci., vol. 2369, Springer, Berlin, 2002, pp. 267--275. MR2041090
  24. G. Stevens, Stickelberger elements and modular parametrizations of elliptic curves, Invent. Math. 98 (1989), no. 1, 75--106. MR1010156 (90m:11089)
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  26. V. Vatsal, Multiplicative subgroups of ${\rm J}_0 ({\rm N})$ and applications to elliptic curves, preprint, 2003, http://www.math.ubc.ca/$\sim$vatsal/page.html.
  27. M. Watkins, Computing the modular degree of an elliptic curve, Experiment. Math. 11 (2002), no. 4, 487--502. MR1969641 (2004c:11091)
  28. A. Wiles, Modular elliptic curves and Fermat's last theorem, Ann. of Math. (2) 141 (1995), no. 3, 443--551. MR1333035 (96d:11071)
Next Item
MR2053457
Dummigan, Neil(4-SHEF-PM); Stein, William(1-HRV); Watkins, Mark(1-PAS)
Constructing elements in Shafarevich-Tate groups of modular motives. (English. English summary)
Number theory and algebraic geometry, 91--118,
London Math. Soc. Lecture Note Ser., 303,
Cambridge Univ. Press, Cambridge, 2003.
11F33 (11F67 11F80 11G18)
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MR2029169 (2004k:11094)
Conrad, Brian(1-MI); Edixhoven, Bas(NL-LEID-MI); Stein, William(1-HRV)
$J\sb 1(p)$ has connected fibers. (English. English summary)
Doc. Math. 8 (2003), 331--408 (electronic).
11G18 (11F11 14H40)
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Let $p$ be a prime number and $J_1(p)$ the Jacobian of the moduli curve $X_1(p)$ over $Q$ that parametrizes pairs $(E,P)$ where $E$ is an elliptic curve and $P$ is a point of $E$ of order $p$. One of the main results of the paper is that $ J_1(p)$ has trivial component group at $p$.

The proof involves the study of the component groups at $p$ of Jacobians of intermediate curves between $X_1(p)$ and $X_0(p)$. (The case of $X_0(p)$ was treated by Mazur-Rapoport.) More precisely, for any subgroup $H$ of $(Z/pZ)^\times/\{±1\}$ the authors consider the curve $X_H(p)=X_1(p)/H$ and its Jacobian $J_H(p)$. They prove that the natural surjective map $J_H(p)\to J_0(p)$ induces an injection $\Phi(J_H(p))\to \Phi(J_0(p))$ between the component groups of mod $p$ fibers and that $\Phi(J_H(p)) $ is cyclic of order $|H|/\gcd(|H|,6)$ over $\overlineF_p$. Furthermore, viewing $\Phi(J_0(p))$ as a quotient of $(Z/pZ)^\times/\{±1\}$, the image of $\Phi(J_H(p))$ coincides with the image of $H$. In particular, $\Phi(J_H(p))$ is always Eisenstein in the sense of Mazur and Ribet and $\Phi(J_1(p))$ is trivial. In order to reach these results they compute a regular proper model of $X_H(p)$ over $ Z_{(p)}$, adapting the classical Jung-Hirzebruch method for complex sufaces. This method enables them to resolve tame cyclic quotient singularities on curves over a discrete valuation ring.

The last part of the paper is devoted to computer computations concerning the arithmetic of $J_1(p)$. The authors give a conjectural formula for the order of the torsion subgroup of $J_1(p)(Q)$.

Reviewed by Alessandra Bertapelle

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[References]

  1. A. Agashé, On invisible elements of the Tate-Shafarevich group, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 5, 369--374. MR1678131 (2000e:11083)
  2. A. Agashe and W. Stein, Visibility of Shafarevich-Tate groups of abelian varieties, J. Number Theory 97 (2002), no. 1, 171--185. MR1939144 (2003h:11070)
  3. A. Agashe and W. Stein, Visible evidence for the Birch and Swinnerton-Dyer conjecture for modular abelian varieties of analytic rank 0, to appear in Math. Comp. cf. MR2085902
  4. A. Agashe and W. Stein, The Manin constant, congruence primes, and the modular degree, in progress.
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MR2041090
Stein, William A.(1-HRV); Watkins, Mark(1-PAS)
A database of elliptic curves---first report.
Algorithmic number theory (Sydney, 2002), 267--275,
Lecture Notes in Comput. Sci., 2369,
Springer, Berlin, 2002.
11G05 (11Yxx)
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References: 0 Reference Citations: 2 Review Citations: 0
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MR1959271 (2004b:11072)
Lario, Joan-C.(E-UPBMS); Schoof, René(I-ROME2)
Some computations with Hecke rings and deformation rings. (English. English summary)
With an appendix by Amod Agashe and William Stein.
Experiment. Math. 11 (2002), no. 2, 303--311.
11F80 (11F11 11F25)
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Let $E$ be the elliptic curve over $Q$ of conductor 142, having Weierstrass equation $Y^{2}+XY=X^{3}-X^{2}-X-3$. The representation $\overline\rho\colon {\rm Gal}(\overlineQ/Q)\to {\rm GL}_{2}(F_{3})$ provided by the $3$-torsion points is unramified outside $3$ and $71$. For $N=71, 142$ and $284$ the authors determine explicitly the structure of the local Hecke algebra $ T_{N}$ generated over $Z_{3}$ by the Hecke operators acting on the weight 2 and level $N$ cusp forms whose associated mod 3 representation is isomorphic to $\overline\rho$. More precisely, they show that $T_{N}\simeq Z_{3}[[X,Y]]/I_{N}$, where generators of the ideals $I_{N}$ are explicitly computed. By the results of A. J. Wiles \ref[Ann. of Math. (2) 141 (1995), no. 3, 443--551; MR1333035 (96d:11071)] and R. L. Taylor and Wiles \ref[Ann. of Math. (2) 141 (1995), no. 3, 553--572; MR1333036 (96d:11072)], in the case $N=71$ (resp. $N=284$) the algebra $T_{N}$ is the universal deformation ring of $\overline\rho$ for a deformation problem which is minimal (resp. non-minimal) at 71; it is a complete intersection, as we can directly see from the description given in this paper. For the case $N=142$ two natural Hecke algebras are considered, corresponding to the eigenvalues $±1$ for the Hecke operator $T_{2}$. Both algebras turn out to be complete intersections. The main tool of the construction is the determination, in the appendix, of a bound (depending on the level $N$) on the greatest index $n$ such that the Hecke operators $T_{r}$ with $r\leq n$ generate the whole Hecke algebra. This allows the authors to do computations by dealing with a finite number of vectors with entries in $Z_{3}$.

Reviewed by Lea Terracini

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[References]

  1. G. Cornell, J.H. Silverman and G. Stevens, Eds. Modular forms and Fermat's Last Theorem, Springer-Verlag, New York 1997. MR1638473 (99k:11004)
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MR1939144 (2003h:11070)
Agashe, Amod(1-TX); Stein, William(1-HRV)
Visibility of Shafarevich-Tate groups of abelian varieties. (English. English summary)
J. Number Theory 97 (2002), no. 1, 171--185.
11G40 (11G10 14K15)
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To a short exact sequence $0\to A\to J\to Q\to 0$ of abelian varieties over a field $K$ corresponds a long exact sequence $$ 0\to A(K)\to J(K)\to Q(K)\to H^1(K,A)\to H^1(K,J)\to\cdots $$ of cohomology groups. B. C. Mazur says that a class $c\in H^1(K,A)$ is visible in $J$ if it gets killed in $H^1(K,J)$. The authors show that every class $c$ is visible in some $J$ (Proposition 1.3)---indeed, one can take $J$ to have dimension less than $dn^{2d}$, where $d$ is the dimension of $A$ and $n$ is the order of $c$ in $H^1(K,A)$ (Proposition 2.3).

When $K$ is a number field, the notion of visibility in $J$ applies to elements of the subgroup ${\cyr X}(A)\subset H^1(K,A)$ of those classes whose restriction to every completion of $K$ is trivial. If $d=1$, the upper bound $dn^{2d}=n^2$ can be improved to $n$ for elements of ${\cyr X}(A)$ (Proposition 2.4).

The main theorem (Theorem 3.1) provides a method for constructing elements of the kernel of ${\cyr X}(A)\to{\cyr X}(J)$, which is the $J$-visible subgroup of ${\cyr X}(A)$. Namely, if one can find an abelian subvariety $B\subset J$ and an integer $n$ satisfying a certain number of properties which are too technical to reproduce here, then there is a natural map $\varphi$ from $B(K)/nB(K)$ to the $J$-visible subgroup of ${\cyr X}(A)$; the order of the kernel of $\varphi$ is at most $n^r$, where $r$ is the rank of $A(K)$.

As an application, the authors give an example (Proposition 4.1) of a 20-dimensional abelian subvariety $A$ of $J_0(389)$ and an elliptic curve $B\subset J_0(389)$ such that by taking $J=A+B$ and $n=5$ in the main theorem, one concludes that $\varphi$ embeds $(\bold Z/5\bold Z)^2$ into the subgroup of $J$-visible elements of ${\cyr X}(A)$, thus providing evidence for the Birch and Swinnerton-Dyer conjecture in this case.

As another application (Proposition 4.2), the authors treat the elliptic curve $E$ of conductor $5389$ considered by J. E. Cremona and B. C. Mazur \ref[Experiment. Math. 9 (2000), no. 1, 13--28; MR1758797 (2001g:11083)] for which the conjectural order of ${\cyr X}(E)$ is $9$ but no element of order 3 is visible in $J_0(5389)$. The authors produce 9 elements of ${\cyr X}(E)$ and show that they are all visible at the higher level of $J_0(7·5389)$.

Reviewed by Chandan Singh Dalawat

Previous Item
[References]

  1. A. Agashe and W.A. Stein, Visible Evidence for the Birch and Swinnerton-Dyer Conjecture for Rank 0 Modular Abelian Varieties, preprint. cf. MR2085902
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  3. W. Bosma, J. Cannon, and C. Playoust, The magma algebra system. I. The user language, J. Symbolic Comput. 24, No. 3-4 (1997), 235--265; Computational Algebra and Number Theory, London, 1993. MR1484478
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  7. J.E. Cremona and B. Mazur, Visualizing elements in the Shafarevich-Tate group, Exp. Math. 9, No. 1 (2000), 13--28. MR1758797 (2001g:11083)
  8. B. Edixhoven, The weight in Serre's conjectures on modular forms, Invent. Math. 109, No. 3 (1992), 563--594. MR1176206 (93h:11124)
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Next Item
MR1900139 (2003c:11059)
Stein, William A.(1-HRV)
There are genus one curves over $\Bbb Q$ of every odd index. (English. English summary)
J. Reine Angew. Math. 547 (2002), 139--147.
11G05 (11G18)
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References: 21 [-] Reference Citations: 1 Review Citations: 0
For a genus $1$ curve $X$ over a field $K$, let $r$ be the smallest degree of an extension $L \vert K$ such that $X(L)$ is non-empty, called the index of $X \vert K$. The author shows, for each $r$ not divisible by $8$, that there are infinitely many genus $1$ curves over $K$ of index $r$, partially answering a question of S. Lang and J. Tate \ref[Amer. J. Math. 80 (1958), 659--684; MR0106226 (21 \#4960)]. The paper starts by giving a cohomological definition of the index $r$ of $X \vert K$ and then some background on Heegner points and Kolyvagin's Euler system. The author proves an intermediate result for $K = \bold Q$ using Kolyvagin's Euler system. Using some additional computations, the author then deduces the main result by considering twists of $E = X_0(17)$.

Reviewed by Imin Chen

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