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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.
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  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
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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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{A review for this item is in process.}


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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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References: 19 [-] Reference Citations: 1 Review Citations: 0

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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Next Item
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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References: 20 [-] Reference Citations: 5 Review Citations: 0

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

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[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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  8. B. Gross and D. Zagier, Heegner points and derivatives of $L$-series, Invent. Math. 84 (1986), no. 2, 225--320. MR0833192 (87j:11057)
  9. B. H. Gross, Kolyvagin's work on modular elliptic curves, $L$-functions and arithmetic (Durham 1989), Cambridge Univ. Press, Cambridge (1991), 235--256. MR1110395 (93c:11039)
  10. V. A. Kolyvagin, On the structure of Shafarevich-Tate groups, Algebraic geometry (Chicago, IL, 1989), Springer, Berlin (1991), 94--121. MR1181210 (94b:11055)
  11. S. Lang and J. Tate, Principal homogeneous spaces over abelian varieties, Amer. J. Math. 80 (1958), 659--684. MR0106226 (21 #4960)
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  13. W. G. McCallum, Kolyvagin's work on Shafarevich-Tate groups, $L$-functions and arithmetic (Durham 1989), Cambridge Univ. Press, Cambridge (1991), 295--316. MR1110398 (92m:11062)
  14. J. S. Milne, Arithmetic duality theorems, Academic Press Inc., Boston, Mass., 1986. MR0881804 (88e:14028)
  15. M. R. Murty and V. K. Murty, Non-vanishing of $L$-functions and applications, Birkhäuser Verlag, Basel 1997. MR1482805 (98h:11106)
  16. C. O'Neil, The Period-Index Obstruction for Elliptic Curves, J. Number Th., to appear. cf. MR1924106 (2003f:11079)
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MR1897901 (2003c:11052)
Buzzard, Kevin(4-LNDIC); Stein, William A.(1-HRV)
A mod five approach to modularity of icosahedral Galois representations. (English. English summary)
Pacific J. Math. 203 (2002), no. 2, 265--282.
11F80
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Let $\rho\colon {\rm Gal}(\overline{\bold Q}/\bold Q)\rightarrow {\rm GL}_2(\bold C)$ be a continuous irreducible two-dimensional complex representation of the absolute Galois group of the field $\bold Q$ of rational numbers. Assume further that $\rho$ is odd, that is, the image of a complex conjugation element in ${\rm Gal}(\overline {\bold Q}/\bold Q)$ has determinant $-1$. A special case of the strong Artin conjecture asserts that there should be a weight one cuspidal newform $f$ whose $L$-function $L(s,f)$ matches the Artin $L$-function $L(s,\rho)$ attached to $\rho$; briefly, $\rho$ should be modular. The conjecture is known to hold when the image of $\rho$ (a finite subgroup of ${\rm GL}_2(\bold C)$) is solvable \ref[R. P. Langlands, Base change for ${\rm GL(2)$}, Ann. of Math. Stud., 96, Princeton Univ. Press, Princeton, N.J., 1980; MR0574808 (82a:10032); J. Tunnell, Bull. Amer. Math. Soc. (N.S.) 5 (1981), no. 2, 173--175; MR0621884 (82j:12015)]. In the remaining cases the projective image of $\rho$ in ${\rm PGL}_2(\bold C)$ is isomorphic to the alternating group $A_5$, the group of rotational symmetries of the icosahedron. For these "icosahedral" Artin representations modularity was (until recently---see below) unknown except in a handful of cases \ref[J. P. Buhler, Icosahedral Galois representations, Lecture Notes in Math., 654, Springer, Berlin, 1978; MR0506171 (58 \#22019); On Artin's conjecture for odd $2$-dimensional representations, Lecture Notes in Math., 1585, Springer, Berlin, 1994; MR1322315 (95i:11001)].

In the paper under review, Buzzard and Stein demonstrate an effective computational approach to proving the modularity of a class of icosahedral Artin representations. They apply this approach to eight representations, thereby demonstrating the modularity of each. The approach is described in detail for the first representation, of conductor $1376=2^5·43$, and the necessary data for carrying out the computations for the remaining seven examples are provided.

A summary of the approach: Suppose that $\rho$ is an icosahedral representation which is unramified at $5$, and for which the eigenvalues of a Frobenius element at $5$ have distinct reduction modulo $5$. By the main theorem of \ref[K. Buzzard and R. L. Taylor, Ann. of Math. (2) 149 (1999), no. 3, 905--919; MR1709306 (2000j:11062)], it suffices to establish that the ${\rm mod}\,5$ reduction $\overline{\rho}$ of $\rho$ is modular; that is, that there is some ${\rm mod}\,5$ cuspidal eigenform $f$ such that for all but finitely many primes $p$ the eigenvalue of the Hecke operator $T_p$ on $f$ is equal to the trace of $\overline\rho$ applied to a Frobenius element at $p$. By computing the space of ${\rm mod}\,5$ modular forms of weight $5$ and appropriate level, the authors identify a ${\rm mod}\,5$ modular form $f$ whose first few Hecke eigenvalues match the corresponding traces of Frobenius of $\overline\rho$; this form is then almost certainly the one required. They then compute enough information about the icosahedral extension of $\bold Q$ cut out by the ${\rm mod}\,5$ representation $\overline\rho_f$ associated to $f$ to identify it uniquely as an element of Table 1 of \ref[ On Artin's conjecture for odd $2$-dimensional representations, Lecture Notes in Math., 1585, Springer, Berlin, 1994; MR1322315 (95i:11001)], which lists icosahedral extensions of $\bold Q$ of small discriminant, and hence match $\overline\rho_f$ with $\overline\rho$.

The paper also contains a result which makes it practical to determine computationally when two normalized cuspidal eigenforms of the same level $N>4$, weight $k$ and character, over a field of characteristic not dividing $N$, are equal: essentially, the number of coefficients of the $q$-expansions of the eigenforms that have to be checked to guarantee equality is at worst linear in $N$ (for fixed $k$). For computational purposes, this improves considerably on similar results of J. Sturm \ref[in Number theory (New York, 1984--1985), 275--280, Lecture Notes in Math., 1240, Springer, Berlin, 1987; MR0894516 (88h:11031)] which require checking on the order of $N^3$ coefficients.

After the first draft of this paper was written, two more relevant articles appeared \ref[K. Buzzard et al., Duke Math. J. 109 (2001), no. 2, 283--318; MR1845181 (2002k:11078); R. Taylor, "On icosahedral Artin representations. II", to appear]. Each of these establishes the modularity of a general icosahedral Artin representation, subject to various local conditions. However, none of the eight examples in this paper is covered by the first of these articles, and only three of them by the second.

Reviewed by Mark Edward Tristan Dickinson

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[References]

  1. E. Artin, \"Uber eine neue Art von $L$- reihen, Abh. Math. Sem. in Univ. Hamburg, 3(1) (1923/1924), 89--108.
  2. W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp., 24(3-4) (1997), 235--265, CMP 1 484 478, Zbl 0898.68039, http://www.maths.usyd.edu. au:8000/u/magma/. MR1484478
  3. J.P. Buhler, Icosahedral Galois representations, Springer-Verlag, Berlin, 1978, Lecture Notes in Mathematics, Vol. 654, , Zbl 0374.12002. MR0506171 (58 #22019)
  4. K. Buzzard, M. Dickinson, N. Shepherd-Barron and R. Taylor, On icosahedral Artin representations, Duke Math. J., 109(2) (2001), 283--318, CMP 1 845 181. MR1845181 (2002k:11078)
  5. K. Buzzard and R. Taylor, Companion forms and weight one forms, Ann. of Math. (2), 149(3) (1999), 905--919, , Zbl 0965.11019. MR1709306 (2000j:11062)
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  17. R. Taylor, On icosahedral Artin representations II, in preparation.
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MR1901355 (2003m:11074)
Stein, William A.(1-HRV); Verrill, Helena A.(D-HANN-IM)
Cuspidal modular symbols are transportable. (English. English summary)
LMS J. Comput. Math. 4 (2001), 170--181 (electronic).
11F67
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Summary: "Modular symbols of weight 2 for a congruence subgroup $\Gamma$ satisfy the identity $\{\alpha,\gamma(\alpha)\}=\{\beta,\gamma(\beta)\}$ for all $\alpha,\ \beta$ in the extended upper half plane and $\gamma\in\Gamma$. The analogue of this identity is false for modular symbols of weight greater than 2. This paper provides a definition of transportable modular symbols, which are symbols for which an analogue of the above identity holds, and proves that every cuspidal symbol can be written as a transportable symbol. As a corollary, an algorithm is obtained for computing periods of cusp forms."
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MR1879817 (2003f:11087)
Conrad, Brian(1-MI); Stein, William A.(1-HRV)
Component groups of purely toric quotients. (English. English summary)
Math. Res. Lett. 8 (2001), no. 5-6, 745--766.
11G18 (11G10 11G40 14K15)
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References: 18 [-] Reference Citations: 2 Review Citations: 0

Let $R$ be a discrete valuation ring, $K$ its field of fractions and $k$ its residue field. Suppose $J$ is an abelian variety over $K$, endowed with a symmetric principal polarization and let $\pi\colon J\rightarrow A$ be an optimal quotient of $J$ meaning that the kernel of $\pi$ is an abelian variety.

The principal polarization on $J$ induces a polarization $\theta_A$ on $A$, whose degree is the square of a positive integer $m_A$.

Assume that the special fibre of the Néron model of $A$ is the extension of a finite group $\Phi_A$ by a torus. Let $X_A$ denote the group of $\overline k$-characters of this torus.

Similarly, let $X_J$ denote the group of $\overline k$-characters of the toric part of the special fibre of the Néron model of $J$.

A. Grothendieck defined in \ref[ Groupes de monodromie en g\'eom\'etrie alg\'ebrique. I, Lecture Notes in Math., 288, Springer, Berlin, 1972; MR0354656 (50 \#7134)] a monodromy pairing between $X_A$ and $X_{A^\vee}$, inducing an exact sequence $$ 0 \rightarrow X_{A^\vee} \rightarrow {\rm Hom}(X_A,Z) \rightarrow \Phi_A \rightarrow 0 $$ and similarly for $J$, the symmetric principal polarization on $J$ allowing one to write the pairing as $$ 0\rightarrow X_J \rightarrow {\rm Hom}(X_J,Z) \rightarrow \Phi_J \rightarrow 0. $$

By functoriality of Néron models and characters, $\pi\colon J\rightarrow A$ induces a map $\pi^*\colon X_A\rightarrow X_J$, the saturation of whose image is denoted by $\scr L$. One deduces from the monodromy pairing a map $\alpha\colon X_J \rightarrow {\rm Hom}(\scr L,Z)$; let $\Phi_X$ be its cokernel. Moreover, let $m_X$ be the order of the finite group $\alpha(X_J)/\alpha(\scr L)$.

The main result of this paper (Theorem 6.1) implies the equality $$ {\#\Phi_A \over m_A} = {\#\Phi_X \over m_X}. $$

This situation is quite common in the context of modular forms, where $J$ is the Jacobian of a modular curve and $A$ arises from a newform. Using modular symbols, one can then compute $m_A$ explicitly. Moreover, using the method of graphs or the ideal theory of quaternion algebras, one can compute $m_X$ and $\Phi_X$. The main theorem of this paper can thus be used to compute $\#\Phi_A$.

Two tables of computations are given.

It should finally be noted that this paper also offers proofs of some more or less well-known facts concerning group schemes, but for which adequate references are missing. They certainly will be of independent interest.

Reviewed by Antoine Chambert-Loir

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[References]

  1. A. Agashe and W. A. Stein, Visibility of Shafarevich-Tate groups of abelian varieties: Evidence for the Birch and Swinnerton-Dyer conjecture, (2001). MR1939144 (2003h:11070)
  2. S. Bosch, W. Lütkebohmert, and M. Raynaud, N\'eron models. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21. Springer-Verlag, Berlin, 1990. MR1045822 (91i:14034)
  3. C. Chevalley, Une d\'emonstration d'un th\'eor\`eme sur les groupes alg\'ebriques, J. Math. Pures Appl. (9) 39 1960, 307--317. MR0126447 (23 #A3743)
  4. B. Conrad, A modern proof of Chevalley's theorem on algebraic groups, http://www-math.mit.edu/$\sim$dejong/papers/chev.dvi
  5. B. Edixhoven, L'action de l'alg\`ebre de Hecke sur les groupes de composantes des jacobiennes des courbes modulaires est "Eisenstein". Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). Astérisque No. 196-197, (1991), 7--8, 159--170 (1992). MR1141457 (92k:11059)
  6. M. Emerton, Optimal quotients of modular Jacobians, (2001), preprint. cf. MR2021024
  7. E. V. Flynn, F. Leprévost, E. F. Schaefer, W. A. Stein, M. Stoll, and J. L. Wetherell, Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves, Math. Comp. 70 (2001), no. 236, 1675--1697 (electronic). MR1836926 (2002d:11072)
  8. A. Grothendieck, \'El\'ements de g\'eom\'etrie alg\'ebrique, Publications Mathématiques IHES, 4,8,11,17,20,24,28,32, 1960--7. MR0217083 (36 #177a)
  9. A. Grothendieck, Groupes de monodromie en g\'eom\'etrie alg\'ebrique, Lecture Notes in Math 288, Springer-Verlag, Heidelberg (1972). MR0354656 (50 #7134)
  10. N. Katz, B. Mazur, Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, 108. Princeton University Press, Princeton, NJ, 1985. MR0772569 (86i:11024)
  11. D. R. Kohel, Hecke module structure of quaternions. Class field theory---its centenary and prospect (Tokyo, 1998), 177--195, Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo, 2001. MR1846458 (2002i:11059)
  12. D. R. Kohel and W. A. Stein, Component Groups of Quotients of $J_0(N)$, Proceedings of the 4th International Symposium (ANTS-IV), Leiden, Netherlands, July 2--7, 2000 (Berlin), Springer, 2000. MR1850621 (2002h:11051)
  13. B. Mazur, Modular curves and the Eisenstein ideal. Inst. Hautes Études Sci. Publ. Math. No. 47 (1977), 33--186 (1978). MR0488287 (80c:14015)
  14. J.-F. Mestre, La m\'ethode des graphes. Exemples et applications. Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986), 217--242, Nagoya Univ., Nagoya, 1986. MR0891898 (88e:11025)
  15. J.-F. Mestre and J. Oesterlé, Courbes de Weil semi-stables de discriminant une puissance $m$- i\`eme, J. Reine Angew. Math. 400 (1989), 173--184. MR1013729 (90g:11078)
  16. D. Mumford, Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London 1970. MR0282985 (44 #219)
  17. K. A. Ribet, Letter about component groups of elliptic curves, arXiv:math.AG/0105124v1 (2001).
  18. J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil. Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 33--52. Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975. MR0393039 (52 #13850)

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MR1860042 (2002h:11047)
Ribet, Kenneth A.(1-CA); Stein, William A.(1-HRV)
Lectures on Serre's conjectures.
Arithmetic algebraic geometry (Park City, UT, 1999), 143--232,
IAS/Park City Math. Ser., 9,
Amer. Math. Soc., Providence, RI, 2001.
11F80 (11F66 11G05)
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This is a nicely written survey article on the conjectures in the title of the paper. The conjectures of Serre in question are about the modularity of mod\,$p$, 2-dimensional, continuous, odd, absolutely irreducible representations of the absolute Galois group $G_Q$ of $Q$. There is a more refined version which also predicts certain minimal modular invariants from which these Galois representations arise. While the conjectures in their qualitative form are still wide open there has been considerable progress in proving that the qualitative form of the conjecture implies the refined form. It is this implication, which is a consequence of deep work of many mathematicians, that this paper surveys in the main. The paper also has useful exercises that will be of help to someone wishing to learn about this area, and two appendices by K. Buzzard and B. Conrad on mod $ l$ multiplicity one principles and constructions of Galois representations attached to weight 2 newforms.

{For the entire collection see MR1860012 (2002d:11003).}

Reviewed by Chandrashekhar Khare

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MR1857596 (2003d:11082)
Merel, Loïc(F-PARIS6-MI); Stein, William A.(1-HRV)
The field generated by the points of small prime order on an elliptic curve.
Internat. Math. Res. Notices 2001, no. 20, 1075--1082.
11G05
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References: 4 [-] Reference Citations: 2 Review Citations: 2

Let $p$ be a prime number, and let $Q(\mu_p)$ denote the $p$th cyclotomic field. The authors prove the following theorem: If there exists an elliptic curve over $Q(\mu_p)$ such that the points of order $p$ on $E$ are all $Q(\mu_p)$-rational, then $p=2,3,5,13$, or $p > 1000$. (In addition, the case $p=13$ has recently been ruled out by M. Rebolledo.) This result generalizes previous results of L. Merel \ref[Duke Math. J. 110 (2001), no. 1, 81--119; MR1861089 (2002k:11080)], and the techniques used in the two papers are similar. The main new ingredient is the (quite nontrivial) verification of a technical hypothesis on $p$ involving the non-vanishing of certain $L$-functions. The reduction (for each fixed $p$) of the main theorem to the verification of this hypothesis is discussed in Sections 1 and 2 of the paper. The computational methods used for verifying the hypothesis are described in detail in Section 3.

Reviewed by Matthew H. Baker

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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), 369--374. MR1678131 (2000e:11083)
  2. J. Cremona, Algorithms for Modular Elliptic Curves, 2d ed., Cambridge Univ. Press, Cambridge, 1997. MR1628193 (99e:11068)
  3. L. Merel, Sur la nature non-cyclotomique des points d'ordre fini des courbes elliptiques, Duke Math. J. 110, 81--119. MR1861089 (2002k:11080)
  4. J.-F. Mestre, "La méthode des graphes. Exemples et applications" in Proceedings of the International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields (Katata, 1986), Nagoya University, Nagoya, Japan, 1986, 217--242. MR0891898 (88e:11025)

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MR1836926 (2002d:11072)
Flynn, E. Victor(4-LVRP); Leprévost, Franck(F-GREN-F); Schaefer, Edward F.(1-STCL-CS); Stein, William A.(1-HRV); Stoll, Michael(D-DSLD-MI); Wetherell, Joseph L.(1-SCA)
Empirical evidence for the Birch and Swinnerton-Dyer conjectures for modular Jacobians of genus 2 curves. (English. English summary)
Math. Comp. 70 (2001), no. 236, 1675--1697 (electronic).
11G40 (11G10 11G30)
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References: 46 [-] Reference Citations: 4 Review Citations: 0

For an abelian variety $A$ over a number field $K$, the conjectures of B. J. Birch and H. P. F. Swinnerton-Dyer \ref[J. Reine Angew. Math. 218 (1965), 79--108; MR0179168 (31 \#3419)] and of J. T. Tate \ref[in S\'eminaire Bourbaki, Vol.\ 9, Exp. No. 306, 415--440, Soc. Math. France, Paris, 1995; see MR1610880 (99f:00041) MR1610977 ] relate arithmetic properties of $A$ to the analytic behaviour of its $L$-function $L(A,s)$. The first conjecture states that the rank of the (finitely generated commutative) group $A(K)$ of $K$-rational points on $A$ is equal to the order of vanishing of the function $L(A,s)$ at $s=1$. The second conjecture expresses the leading coefficient $L^*(A,1)$ in the Taylor expansion of $L(A,s)$ at $s=1$ in terms of certain arithmetic invariants of $A$, among them the order of the group ${\cyr Sh}(A,K)$ of those principal homogeneous $A$-spaces over $K$ which become isomorphic to $A$ over every completion of $K$.

Each of these conjectures requires an act of faith even for its statement. For the first one, the analytic continuation of the function $L(A,s)$ to a domain containing the point $s=1$ needs to be assumed; for the second, the finiteness of the group ${\cyr Sh}(A,K)$ needs to be assumed as well. As of now, neither of these two requirements is known to hold in general.

However, for modular abelian varieties $A$ over $\bold Q$, the analytic continuation of $L(A,s)$ to the whole of $\bold C$ is known. For such varieties, V. A. Kolyvagin and others \ref[V. A. Kolyvagin and D. Yu. Logachëv, Algebra i Analiz 1 (1989), no. 5, 171--196; MR1036843 (91c:11032)] have shown that if the $L$-function $L(A,s)$ has at most a simple zero at $s=1$, then the order of vanishing equals the rank of the group $A(Q)$ (as predicted by the first conjecture) and the group ${\cyr Sh}(A,Q)$ is finite (so the statement of the second conjecture is meaningful).

One of the triumphs of recent years has been to show that all 1-dimensional abelian varieties over $\bold Q$ are modular \ref[C. Breuil et al., J. Amer. Math. Soc. 14 (2001), no. 4, 843--939 (electronic); MR1839918 (2002d:11058)]. Extensive calculations, beginning with Birch and Swinnerton-Dyer in the early 1960s on one of the first electronic computers at Cambridge, have lent support to the conjectures in this $1$-dimensional case \ref[J. E. Cremona, Algorithms for modular elliptic curves, Second edition, Cambridge Univ. Press, Cambridge, 1997; MR1628193 (99e:11068)].

The authors extend these calculations to some 2-dimensional cases. They consider thirty-two curves $C$ of genus 2 over $\bold Q$ whose Jacobians $J$ are modular abelian surfaces. For each $J$ they compute, with a high degree of precision, the leading coefficient $L^*(J,1)$ and the arithmetic invariants $t$ (the order of the torsion subgroup of $J(Q)$), $c$ (the product of the local Tamagawa numbers at the finite places), $R$ (the regulator) and $\Omega$ (the period). Within the accuracy of their computations, the number $L^*(J,1)t^2/cR\Omega$---conjecturally the order of the group ${\cyr Sh}(J,Q)$---does turn out to be an integer. In all thirty-two cases, this integer happens to be equal to the order of the $2$-torsion subgroup of ${\cyr Sh}(J,Q)$. So the second conjecture has been reduced for them to the statement that the number $L^*(J,1)t^2/cR\Omega$ is an integer and the group ${\cyr Sh}(J, Q)$ is annihilated by $2$. As an example, for the last curve on their list, namely $$y^2+(x^3+x+1)y+(x^3-x^2-x)=0,$$ the Jacobian $J$ satisfies the conjecture if the number $L^*(J,1)t^2/cR\Omega$, which agrees with 1 to 44 decimal places, is indeed equal to 1 and if the group ${\cyr Sh}(J,Q)$ is trivial.

Reviewed by Chandan Singh Dalawat

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[References]

  1. A. Agashé and W.A. Stein, Some abelian varieties with visible Shafarevich-Tate groups, preprint, 2000. cf. MR1772005 (2001g:17019)
  2. A. Agashé, and W.A. Stein, The generalized Manin constant, congruence primes, and the modular degree, in preparation, 2000.
  3. B. Birch and H.P.F. Swinnerton-Dyer, Notes on elliptic curves. II, J. Reine Angew. Math., 218 (1965), 79--108. MR0179168 (31 #3419)
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MR1850621 (2002h:11051)
Kohel, David R.(5-SYD); Stein, William A.(1-CA)
Component groups of quotients of $J\sb 0(N)$. (English. English summary)
Algorithmic number theory (Leiden, 2000), 405--412,
Lecture Notes in Comput. Sci., 1838,
Springer, Berlin, 2000.
11G18 (11F11 11G10 11G40 14G35)
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Let $A$ be an abelian variety over $Q$ and let $p$ be a prime number. An important arithmetic invariant attached to $A$ and $p$ is the order of the group $\Phi_{A,p}$ of connected components of the reduction modulo $p$ of the N{é}ron model of $A$ over $ Z$.

To each modular newform $f$ of weight 2 for the congruence subgroup $\Gamma_0(N)$ ($N\ge1$), Shimura has associated an abelian variety $A_f$ defined over $Q$; it is a certain quotient of $J_0(N)$ and has good reduction at primes which do not divide $N$.

The authors give an algorithm for computing the order of $\Phi_{A_f,p}$ when the prime $p$ divides $N$ but $p^2$ does not divide $N$. They include a table listing these orders when $N\le127$.

{For the entire collection see MR1850596 (2002d:11002).}

Reviewed by Chandan Singh Dalawat

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