# zbMATH — the first resource for mathematics

On the $$p$$-adic Birch, Swinnerton-Dyer conjecture for non-semistable reduction. (English) Zbl 1013.11029
Let $$E/{\mathbb{Q}}$$ be an elliptic curve without complex multiplication. The main result of the paper is to prove an analogue of Kato’s result on the $$p$$-adic Birch and Swinnerton-Dyer conjecture (originally done in the case where $$E$$ is ordinary at $$p$$) in the case where $$E$$ is potentially ordinary at $$p$$. The main difficulty is to show that Kato’s Euler systems really map to the $$p$$-adic $$L$$-function.
Let $$F_{\infty}$$ be the $${\mathbb{Z}}_p$$ extension of $${\mathbb{Q}}$$, $$\Gamma=\text{Gal}(F_{\infty}/{\mathbb{Q}})$$, $$\text{Gal}({\mathbb{Q}}(\mu_{p^{\infty}})/{\mathbb{Q}})=\Gamma\times\Delta$$ and $$\text{Sel}(E/F_{\infty})$$ the corresponding Selmer group. It is known [B. Mazur, Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)] that the Pontryagin dual $$\text{Sel}(E/F_{\infty})^{\wedge}$$ of $$\text{Sel}(E/F_{\infty})$$ is a compact, finitely generated module over the Iwasawa algebra $${\mathbb{Z}}_p[[\Gamma]]$$. In the 1970’s Mazur conjectured that $$\text{Sel}(E/F_{\infty})^{\wedge}$$ encoded the formulae predicted by Birch and Swinnerton-Dyer for the Hasse-Weil $$L$$-function $$L(E,s)$$.
In the current paper the author makes the following hypothesis: either $$E$$ is potentially ordinary at $$p\geq 5$$ or $$E$$ is potentially ordinary at $$p=3$$ with semistable reduction over a quadratic extension of $${\mathbb{Q}}_3$$. This hypothesis implies the existence of a character $$\varepsilon$$ of conductor $$p$$ such that $$\text{Frob}_p^{-1}\in G_{{\mathbb{Q}}_p/I_p}$$ has a unique eigenvalue $$\alpha$$ on the subspace $$H^1_{\text{ét}}(E_{\overline{\mathbb{Q}}},\overline{\mathbb{Q}}_{\ell})\otimes\varepsilon^{-1}$$ fixed by the inertia group $$I_p$$ at a prime $$\ell\neq p$$. It was also proved in [D. Delbourgo, Compos. Math. 113, 123-154 (1998; Zbl 0911.11050)] that there exists a unique element $$L_p^{an}\in{\mathbb{Z}}_p[[\Gamma\times\Delta]][\mu_p,p^{-1}]$$ such that for all characters $$\chi\neq\varepsilon$$ of $$p$$-power conductor $\chi(L_p^{an})=\frac{p^m}{\alpha^m\sum_{n=1}^{p^m}\chi^{-1}\varepsilon(n)\exp(2\pi in/p^m)}\times\frac{L(E,\chi^{-1},1)}{\Omega_E^{\text{sign}(\chi)}},$ where $$m=\text{ord}_p(\text{cond}(\chi\varepsilon^{-1}))$$, the right hand side is considered $$p$$-adically via a fixed embedding $$\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_p$$, $$\Omega^+_E$$ (resp. $$\Omega^-_E$$) denotes the real (resp. imaginary) period of a Néron differential associated to a minimal Weierstrass equal of $$E$$ over $${\mathbb{Z}}$$.
The following quantities appear in the nonsemistable $$p$$-adic Birch and Swinnerton-Dyer conjecture. Aside from the Tate-Shafarevich group Ш, the Mordell-Weil group and the Tamagawa factors $$\text{Tam}_{\ell,{\mathbb{Q}}}$$, the $$p$$-adic regulator $$\text{reg}_p(E/{\mathbb{Q}})$$ (expressed in terms of the $$p$$-adic pairing $$\langle , \rangle_{p,{\mathbb{Q}}}$$) and the $$\ell$$-invariant $$\ell_p(E)$$.
The main theorem states that under the above hypothesis
(A) The Iwasawa module $$\text{Sel}(E/F_{\infty})^{\wedge}$$ is $${\mathbb{Z}}_p[[\Gamma]]$$-torsion.
(B) Denote $$L_p^{\text{alg}}$$ a generator of the characteristic ideal of $$\text{Sel}(E/F_{\infty})^{\wedge}$$, $$\kappa$$ a cyclotomic character at $$p$$ and $$s\in{\mathbb{Z}}_p$$. If $$\text{ Ш}(E/{\mathbb{Q}})_{p^{\infty}}$$ is infinite or $$\langle , \rangle_{p,{\mathbb{Q}}}$$ is degenerate then $$\text{ord}_{s=0}(\kappa^s(L_p^{\text{alg}}))>r_E=\text{rank}(E({\mathbb{Q}}))$$. If $$\text{ Ш}(E/{\mathbb{Q}})_{p^{\infty}}$$ is finite and $$\langle , \rangle_{p,{\mathbb{Q}}}$$ is non-degenerate the previous inequality is indeed an equality and $\frac 1{r!}\left(\frac{d^{r_E}}{ds^{r_E}} \kappa^s(L_p^{\text{alg}})\right)_{s=0} \sim\;\ell_p(E/{\mathbb{Q}}) \times\frac{\text{reg}_p(E/{\mathbb{Q}})\times\#\text{ Ш}(E/{\mathbb{Q}})_{p^{\infty}}\times\prod_{\ell}\text{Tam}_{\ell,{\mathbb{Q}}}}{\#E({\mathbb{Q}})_{\text{tor}}^2} ,$ where $$a\sim b$$ means $$\text{ord}_p(a)=\text{ord}_p(b)$$.
(C) $$L_p^{\text{alg}}$$ divides in $${\mathbb{Z}}_p[[\Gamma]][\mu_p,p^{-1}]$$ the branch of $$L_p^{an}$$ fixed by $$\Lambda$$.
(D) $$\text{ord}_{s=0}(\kappa^s(L_p^{\text{alg}}))\geq r_E$$.
In the potentially supersingular case (A) is false because $$\text{Sel}(E/F_{\infty})^{\wedge}$$ is often pseudoisomorphic to $${\mathbb{Z}}_p [[\Gamma]]$$ which is free. (B) plus (C) will imply (D). Formulae of type in (B) were first proven in the good ordinary case by Perrin-Riou and Schneider and in the multiplicative case by Jones.

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G07 Elliptic curves over local fields
Full Text:
##### References:
 [1] Bloch, S.; Kato, K., L-functions and Tamagawa numbers of motives, The Grothendieck festchrift I, 86, (1990), Birkhäuser Boston, p. 333-400 · Zbl 0768.14001 [2] Coates, J.; Greenberg, R., Kummer theory for abelian varieties over local fields, Invent. math., 124, 129-174, (1996) · Zbl 0858.11032 [3] Coleman, R., Division values in local fields, Invent. math., 53, 91-116, (1979) · Zbl 0429.12010 [4] Colmez, P., Théorie d’Iwasawa des représentations de de Rham, Ann. of math., 148, 485-571, (1998) · Zbl 0928.11045 [5] Delbourgo, D., Iwasawa theory for elliptic curves at unstable primes, Compositio math., 113, 123-154, (1998) · Zbl 0911.11050 [6] de Shalit, E., Iwasawa theory of elliptic curves with complex multiplication, (1987), Academic Press Orlando · Zbl 0674.12004 [7] Fontaine, J.-M., Représentations p-adiques semi-stables, “Périodes p-adiques”, Astérisque, 223, 113-184, (1994) · Zbl 0865.14009 [8] J.-M. Fontaine, B. Mazur, Geometric Galois representations, inProceedings of a Conference on Elliptic Curves and Modular Forms, pp. 47-78, December 18-21 1993, International Press, Hong Kong, 1995. [9] Imai, H., A remark on the rational points of abelian varieties with values in cyclotomic $$Z$$_{l}-extensions, Proc. Japan acad. ser. A math. sci., 51, 12-16, (1975) · Zbl 0323.14010 [10] Jones, J., Iwasawa L-functions of multiplicative abelian varieties, Duke math. J., 59, 399-420, (1989) · Zbl 0716.14008 [11] Jones, J., p-adic heights for semi-stable abelian varieties, Compositio math., 73, 31-56, (1990) · Zbl 0743.14031 [12] Jones, J., Iwasawa L-functions of elliptic curves with additive reduction, J. number theory, 51, 103-117, (1995) · Zbl 0844.11042 [13] K. Kato, Generalized explicit reciprocity laws, preprint. · Zbl 1024.11029 [14] K. Kato, Euler systems, Iwasawa theory and Selmer groups, preprint. · Zbl 0993.11033 [15] K. Kato, p-adic Hodge theory and special values of zeta functions of elliptic cusp forms, preprint. [16] K. Kato, M. Kurihara, T. Tsuji, in preparation. [17] Kolyvagin, V., Euler systems, The Grothendieck festchrift II, 87, 435-483, (1990) · Zbl 0742.14017 [18] Mazur, B., Rational points on abelian varieties with values in towers of number fields, Invent. math., 18, 183-266, (1972) · Zbl 0245.14015 [19] McCallum, W., Duality theorems for Néron models, Duke math. J., 53, 1093-1124, (1986) · Zbl 0623.14023 [20] Milne, J.S., Arithmetic duality theorems, Perspectives in mathematics, 1, (1986), Academic Press Boston · Zbl 0613.14019 [21] Nasybullin, A., Elliptic Tate curves over local γ-extensions, Math. notes USSR, 13, 322-327, (1973) · Zbl 0268.14011 [22] Perrin-Riou, B., Théorie d’Iwasawa des représentations p-adiques sur un corps local, Invent. math., 115, 81-149, (1994) · Zbl 0838.11071 [23] Perrin-Riou, B., Systèmes d’Euler p-adiques et théorie d’Iwasawa, Ann. inst., 48, 1231-1307, (1998) · Zbl 0930.11078 [24] Rohrlich, D., On L-functions of elliptic curves and cyclotomic towers, Invent. math., 75, 409-423, (1984) · Zbl 0565.14006 [25] K. Rubin, Euler systems and modular elliptic curves, inProceedings of a Conference on Galois Representations in Arithmetic Algebraic Geometry, Durham, 1996, Cambridge Univ. Press, pp. 351-367, 1998, Cambridge, UK. [26] Rubin, K., Euler systems, annals of math. studies, (2000), Princeton Univ. Press New Jersey [27] Schneider, P., p-adic height pairings I, Invent. math., 69, 401-409, (1982) · Zbl 0509.14048 [28] Schneider, P., Iwasawa L-functions of varieties over algebraic number fields. A first approach, Invent. math., 71, 251-293, (1983) · Zbl 0511.14010 [29] Schneider, P., p-adic height pairings II, Invent. math., 79, 329-374, (1985) · Zbl 0571.14021 [30] A. J. Scholl, An introduction to Kato’s Euler systems, inProceedings of a Conference on Galois Representations in Arithmetic Algebraic Geometry, Durham, pp. 379-460, 1996, Cambridge Univ. Press, Cambridge, UK, 1998. [31] Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. math., 15, 259-331, (1972) · Zbl 0235.14012 [32] Barré-Sireix, K.; Diaz, G.; Gramain, F.; Philibert, G., Une preuve de la conjecture Mahler-Manin, Invent. math., 124, 1-9, (1996) · Zbl 0853.11059 [33] Vishik, M.M., Non-Archimedean measures associated with Dirichlet series, Mat. sbornik, 99, 248-260, (1976) · Zbl 0358.14014
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.