×

zbMATH — the first resource for mathematics

A finiteness theorem for the symmetric square of an elliptic curve. (English) Zbl 0781.14022
Let \(E\) be an elliptic curve and \(M\) be symmetric square of the Tate module \(T_ p(E)\). Bloch and Kato have given a conjectural formula for the value of the motivic \(L\)-function attached to \(M\) at the point \(s=2\). It includes an order of a generalization of the Tate-Shafarevich group for \(M\). If we assume that \(E\) is a modular elliptic curve then the modular parametrisation of \(E\) gives an explicit expression for this value through the degree \(\deg\pi\) of the parametrisation and other invariants of \(E\). In this case it is known that the \(L\)-function is an entire function on the whole \(s\)-plane. The author proves that \(\deg \pi\) annihilates the Tate-Shafarevich group if \(p>3\), \(E\) has good reduction in \(p\) and the image of representation of \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\) on \(E_ p\) equals \(\text{GL}_ 2(\mathbb{F}_ p)\). Thus the author proves the finiteness of this group. The main tool of the proof is an extension of Kolyvagin’s technique for the usual Tate-Shafarevich group.
This paper is becoming widely known because in the end of June A. Wiles has announced the proof of the Taniyama-Weil conjecture for stable elliptic curves. The key ingredient of his proof is a generalization of the Flach theorem giving an exact value for the Tate-Shafarevich group.

MSC:
14H52 Elliptic curves
14G40 Arithmetic varieties and schemes; Arakelov theory; heights
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Bloch, S., Ogus, A.: Gersten’s Conjecture and the homology of schemes. Ann. Sci. ?c. Norm. Sup?r4, 181-202 (1974) · Zbl 0307.14008
[2] Bloch, S.: A note on Gersten’s conjecture in the mixed characteristic case. In: Bloch, S. et al. (eds.) Applications of Algebraic K-theory to Number theory and Algebraic Geometry. Boulder 1983. (Contemp. Math. vol. 55) Providence, RI: Am. Math. Soc. 1985
[3] Bloch, S., Kato, K.: L-functions and Tamagawa numbers of motives. In Cartier, P. et al (eds.) The Grothendieck Festschrift, vol. 1. Boston Basel Stuttgart: Birkh?user 1990 · Zbl 0768.14001
[4] Brown, K. S.: Cohomology of Groups. (Grad. Texts Math., vol. 87) Berlin Heidelberg New York: Springer 1982 · Zbl 0584.20036
[5] Coates, J., Schmidt, C.G.: Iwasawa theory for the symmetric square of an elliptic curve. J. Reine Angew. Math.375/376, 104-156 (1987) · Zbl 0609.14013
[6] Deligne, P., Rapoport, M.: Les sch?mas de modules de courbes elliptiques. in: Deligne, P., Kuyk, W. (eds.) Modular Functions of One Variable II. (Lect. Notes Math., vol. 349) Berlin Heidelberg New York: Springer 1973 · Zbl 0281.14010
[7] Edixhoven, B.: On the Manin constant of modular elliptic curves. In: van der Geer, G., Oort, F., Steenbrink, J. (eds.) Arithmetic Algebraic Geometry. (Prog. Math., vol. 89) Boston Basel Stuttgart: Birkh?user 1991 · Zbl 0749.14025
[8] Faltings, G.: Crystalline Cohomology and p-adic Galois representations. In: Igusa, Jun-Ichi (ed.) Algebraic Analysis, Geometry and Number Theory. Baltimore: The John Hopkins University Press 1991
[9] Flach, M.: Selmer groups for the symmetric square of an elliptic curve. Ph.D. dissertation, University of Cambridge (1990)
[10] Flach, M.: A generalisation of the Cassels-Tate pairing. J. Reine Angew. Math.412, 113-127 (1990) · Zbl 0711.14001
[11] Gross, B.H.: Kolyvagin’s work on modular elliptic curves. In: Coates, J., Taylor, M. (eds.) L-functions and Arithmetic. (Lond. Math. Soc. Lect. Note Ser., vol. 153) Cambridge London: Cambridge University Press 1991 · Zbl 0743.14021
[12] Grothendieck, A.: Le groupe de Brauer III. In: Dix exposes sur la cohomologie des schemas pp. 88-188. Amsterdam: North Holland 1968
[13] Hida, H.: Congruences of Cusp Forms and Special Values of Their Zeta Functions. Invent. Math.63, 225-261 (1981) · Zbl 0459.10018
[14] Hochschild, G., Serre, J.P.: Cohomology of Group Extensions. Trans. Am. Math. Soc.74, 110-134 (1953) · Zbl 0050.02104
[15] Jannsen, U.: Continuous ?tale cohomology. Math. Ann.280, 207-245 (1988) · Zbl 0649.14011
[16] Jannsen, U.: Mixed Motives and Algebraic K-theory. (Lect. Notes Math., vol. 1400) Berlin Heidelberg New York: Springer 1990 · Zbl 0691.14001
[17] Katz, N., Mazur, B.: Arithmetic moduli of elliptic curves. Princeton: Princeton University Press 1985 · Zbl 0576.14026
[18] Mazur, B.: Deformations of Galois representations. In: Ihara, Y. et al. (eds.) Galois Groups over ?. (Publ. Math. Sci. Res. Inst. vol. 16) New York: Springer 1989 · Zbl 0739.11021
[19] Mildenhall, S.: Cycles in a product of elliptic curves, and a group analogous to the class group. (Preprint 1991) · Zbl 0788.14004
[20] Milne, J.S.: Arithmetic Duality Theorems. (Perspect. Math., vol. 1) Academic Press 1986 · Zbl 0613.14019
[21] Quillen, D.: Higher algebraic K-Theory I. In: Bass, H. (ed.) Algebraic K-Theory I. (Lect. Notes Math. vol. 341, pp. 85-147) Berlin Heidelberg New York: Springer 1973 · Zbl 0292.18004
[22] Raskind, W.: Torsion algebraic cycles on varieties over local fields. In: Jardine, J.F., Snaith, V.P. (eds.) Algebraic K-Theory: Connections with Geometry and Topology, pp. 343-388. Dordrecht: Kluver Academic Publishers 1989 · Zbl 0709.14005
[23] Scholl, A.J.: On Modular Units. Math. Ann.285, 503-510 (1989) · Zbl 0663.10025
[24] Serre, J.P.: Propri?t?s galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math.15, 259-331 (1972) · Zbl 0235.14012
[25] Sherman, C.: Some theorems on the K-theory of coherent sheaves. Commun. Algebra714, 1489-1508 (1979) · Zbl 0429.18017
[26] Shimura, G.: The special values of zeta functions associated with cusp forms. Commun. Pure Appl. Math.29, 783-804 (1976) · Zbl 0348.10015
[27] Zagier, D.B.: Modular Parametrizations of elliptic curves. Canad. Math. Bull.28 (3), 372-384 (1985) · Zbl 0579.14027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.