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Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes. (English) Zbl 1431.11123

Let \(E\) be an elliptic curve defined over \(\mathbb Q\) with supersingular reduction at a prime \(p \geq 5\). Let \(K_{\infty}/K\) be the anticyclotomic \(\mathbb Z_p\)-extension of an imaginary quadratic field \(K\) and denote the Iwasawa algebra of \(\Gamma := \mathrm{Gal}(K_{\infty}/K) \simeq \mathbb Z_p\) by \(\Lambda := \mathbb Z_p[[\Gamma]]\).
In this article, the authors show that (under certain hypotheses) the \(\Lambda\)-corank of certain plus/minus Selmer groups of \(E\) over \(K_{\infty}\) is \(1\). Moreover, they study the asymptotic behaviour of \(p\)-primary Selmer groups of \(E\) along the anticyclotomic extension. For this, they adapt the \(\Lambda\)-adic Euler system method of M. Bertolini [Compos. Math. 99, No. 2, 153–182 (1995; Zbl 0862.11043)] to the supersingular case. This is interesting in its own right. Moreover, it is shown in a short appendix that the two main results can alternatively be deduced from results of J. Nekovář [Lond. Math. Soc. Lect. Note Ser. 320, 471–547 (2007; Zbl 1152.11023)], V. Vatsal [Duke Math. J. 116, No. 2, 219–261 (2003; Zbl 1065.11048)], and of A. Iovita and R. Pollack [J. Reine Angew. Math. 598, 71–103 (2006; Zbl 1114.11053)].
We now describe the two main results in a little more detail. Let \(N>3\) be the conductor of \(E\). Write \(N = MD\), where \(D\) is a squarefree product of an even number of primes, and such that \(D\) and \(M\) are coprime. Assume that \(K\) satisfies a modified Heegner hypothesis, i.e., all primes dividing \(pM\) split in \(K\) and all primes dividing \(D\) are inert in \(K\). Moreover, assume that the two primes of \(K\) above \(p\) are totally ramified in \(K_{\infty}\). Finally, assume that the attached Galois representation \[ \rho_{E,p}: G_{\mathbb Q} \rightarrow \mathrm{Aut}(T_p(E)) \] is surjective. Note that there are infinitely many supersingular primes for which the latter condition holds.
Under these hypotheses, the authors introduce plus/minus Selmer groups à la Kobayashi and show that their \(\Lambda\)-corank is \(1\). If in addition \(D=1\), the \(\mathbb Z_p\)-corank of the Selmer groups \(\mathrm{Sel}_{p^{\infty}}(E/K_m)\) is \(p^m + O(1)\), where \(K_m\) denotes the \(m\)-th layer of \(K_{\infty}/K\) as usual. This might be seen as the supersingular analogue of the well-known formula for ordinary primes.

MSC:

11R23 Iwasawa theory
11G05 Elliptic curves over global fields
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References:

[1] Agboola, A., Howard, B.: Anticyclotomic Iwasawa theory of CM elliptic curves II. Math. Res. Lett. 12(5), 611-621 (2005) · Zbl 1130.11058 · doi:10.4310/MRL.2005.v12.n5.a1
[2] Balister, P.N., Howson, S.: Note on Nakayama’s lemma for compact \[\Lambda\] Λ-modules. Asian J. Math. 1(2), 224-229 (1997) · Zbl 0904.16019 · doi:10.4310/AJM.1997.v1.n2.a2
[3] Bertolini, M.: Selmer groups and Heegner points in anticyclotomic \[{\bf Z}_p\] Zp-extensions. Compos. Math. 99(2), 153-182 (1995) · Zbl 0862.11043
[4] Bertolini, M.: Iwasawa theory for elliptic curves over imaginary quadratic fields. J. Théor. Nombres Bordx. 13(1), 1-25 (2001) · Zbl 1061.11058 · doi:10.5802/jtnb.300
[5] Bertolini, M., Darmon, H.: Kolyvagin’s descent and Mordell-Weil groups over ring class fields. J. Reine Angew. Math. 412, 63-74 (1990) · Zbl 0712.14008
[6] Bertolini, M., Darmon, H.: Heegner points on Mumford-Tate curves. Invent. Math. 126(3), 413-456 (1996) · Zbl 0882.11034 · doi:10.1007/s002220050105
[7] Brown, K.S.: Cohomology of Groups. Graduate Texts in Mathematics. Springer, New York (1982) · Zbl 0584.20036
[8] Çiperiani, M.: Tate-Shafarevich groups in anticyclotomic \[{\mathbb{Z}}_p\] Zp-extensions at supersingular primes. Compos. Math. 145(2), 293-308 (2009) · Zbl 1257.11055 · doi:10.1112/S0010437X08003874
[9] Çiperiani, M., Wiles, A.: Solvable points on genus one curves. Duke Math. J. 142(3), 381-464 (2008) · Zbl 1168.14014 · doi:10.1215/00127094-2008-010
[10] Cornut, C.: Mazur’s conjecture on higher Heegner points. Invent. Math. 148(3), 495-523 (2002) · Zbl 1111.11029 · doi:10.1007/s002220100199
[11] Cornut, C., Vatsal, V.: Nontriviality of Rankin-Selberg \[L\] L-Functions and CM Points, \[L\] L-Functions and Galois Representations. London Mathematical Society Lecture Note Series, vol. 320, pp 121-186. Cambridge University Press, Cambridge (2007) · Zbl 1153.11025
[12] Darmon, H., Iovita, A.: The anticyclotomic main conjecture for elliptic curves at supersingular primes. J. Inst. Math. Jussieu 7(2), 291-325 (2008) · Zbl 1146.11057 · doi:10.1017/S1474748008000042
[13] Elkies, N.D.: The existence of infinitely many supersingular primes for every elliptic curve over \[{\bf Q}\] Q. Invent. Math. 89(3), 561-567 (1987) · Zbl 0631.14024 · doi:10.1007/BF01388985
[14] Greenberg, R.: On the Birch and Swinnerton-Dyer conjecture. Invent. Math. 72(2), 241-265 (1983) · Zbl 0546.14015 · doi:10.1007/BF01389322
[15] Greenberg, R., Introduction to Iwasawa theory for elliptic curves. In: Conrad, B., Rubin, K. (eds.) Arithmetic Algebraic Geometry (Park City, UT, 1999), IAS/Park City Mathematics Series, vol. 9, pp. 407-464. American Mathematical Society, Providence (2001) · Zbl 1002.11048
[16] Gross, B.H.: Kolyvagin’s work on modular elliptic curves. In: Coates, J., Taylor, M.J. (eds.) \[L\] L-Functions and Arithmetic (Durham, 1989). London Mathematical Society Lecture Note Series, vol. 153, pp. 235-256. Cambridge University Press, Cambridge (1991) · Zbl 0743.14021
[17] Howard, B.: The Heegner point Kolyvagin system. Compos. Math. 140(6), 1439-1472 (2004) · Zbl 1139.11316 · doi:10.1112/S0010437X04000569
[18] Iovita, A., Pollack, R.: Iwasawa theory of elliptic curves at supersingular primes over \[{\mathbb{Z}}_p\] Zp-extensions of number fields. J. Reine Angew. Math. 598, 71-103 (2006) · Zbl 1114.11053
[19] Kim, B.D.: The parity conjecture for elliptic curves at supersingular reduction primes. Compos. Math. 143(1), 47-72 (2007) · Zbl 1169.11022 · doi:10.1112/S0010437X06002569
[20] Kobayashi, S.: Iwasawa theory for elliptic curves at supersingular primes. Invent. Math. 152(1), 1-36 (2003) · Zbl 1047.11105 · doi:10.1007/s00222-002-0265-4
[21] Longo, M., Rotger, V., Vigni, S.: Special values of \[L\] L-functions and the arithmetic of Darmon points. J. Reine Angew. Math. 684, 199-244 (2013) · Zbl 1312.11051
[22] Longo, M., Vigni, S.: Quaternion algebras, Heegner points and the arithmetic of Hida families. Manuscr. Math. 135(3-4), 273-328 (2011) · Zbl 1320.11054 · doi:10.1007/s00229-010-0409-6
[23] Longo, M., Vigni, S.: A refined Beilinson-Bloch conjecture for motives of modular forms. Trans. Am. Math. Soc. 369(10), 7301-7342 (2017) · Zbl 1432.14008 · doi:10.1090/tran/6947
[24] Mazur, B.: Rational points of abelian varieties with values in towers of number fields. Invent. Math. 18(3-4), 183-266 (1972) · Zbl 0245.14015 · doi:10.1007/BF01389815
[25] Mazur, B.: Modular curves and arithmetic. In: Proceedings of the International Congress of Mathematicians, vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, pp. 185-211 (1984) · Zbl 0597.14023
[26] Mazur, B., Rubin, K.: Kolyvagin systems. Mem. Am. Math. Soc. 168(799), viii+96 (2004) · Zbl 1055.11041
[27] Milne, J.S.: Arithmetic Duality Theorems, 2nd edn. BookSurge, LLC, Charleston (2006) · Zbl 1127.14001
[28] Nekovář, J.: Kolyvagin’s method for Chow groups of Kuga-Sato varieties. Invent. Math. 107(1), 99-125 (1992) · Zbl 0729.14004 · doi:10.1007/BF01231883
[29] Nekovář, J.: The Euler system method for CM points on Shimura curves. In: Burns, D., Buzzard, K., Nekovár, J. (eds.) \[L\] L-Functions and Galois Representations. London Mathematical Society Lecture Note Series, vol. 320, pp. 471-547. Cambridge University Press, Cambridge (2007) · Zbl 1152.11023
[30] Neukirch, J., Schmidt, A., Wingberg, K.: Cohomology of Number Fields. Grundlehren der Mathematischen Wissenschaften, vol. 323. Springer, Berlin (2000) · Zbl 0948.11001
[31] Perrin-Riou, B.: Fonctions \[L\] L \[p\] p-adiques, théorie d’Iwasawa et points de Heegner. Bull. Soc. Math. Fr. 115(4), 399-456 (1987) · Zbl 0664.12010 · doi:10.24033/bsmf.2085
[32] Pollack, R., Weston, T.: On anticyclotomic \[\mu\] μ-invariants of modular forms. Compos. Math. 147(5), 1353-1381 (2011) · Zbl 1259.11101 · doi:10.1112/S0010437X11005318
[33] Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(4), 259-331 (1972) · Zbl 0235.14012 · doi:10.1007/BF01405086
[34] Serre, J.-P.: Linear Representations of Finite Groups. Graduate Texts in Mathematics, vol. 42. Springer, New York (1977) · Zbl 0355.20006
[35] Silverman, J.H.: The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics, vol. 106. Springer, New York (1986) · Zbl 0585.14026
[36] Vatsal, V.: Special values of anticyclotomic \[L\] L-functions. Duke Math. J. 116(2), 219-261 (2003) · Zbl 1065.11048 · doi:10.1215/S0012-7094-03-11622-1
[37] Zhang, S.-W.: Heights of Heegner points on Shimura curves. Ann. Math. (2) 153(1), 27-147 (2001) · Zbl 1036.11029 · doi:10.2307/2661372
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