On the Iwasawa invariants of elliptic curves.

*(English)*Zbl 1032.11046Let \(E\) be an elliptic curve over \(\mathbb Q\) and \(p\) be a prime number where \(E\) has good ordinary reduction. If \(E/\mathbb Q\) is assumed to be modular, then the following nonnegative integers are defined. The “algebraic” Iwasawa invariants \(\lambda_E^{{\text{alg}}}\) and \(\mu_E^{{\text{alg}}}\) defined in terms of the structure of the \(p\)-primary subgroup \(\text{Sel}_E(\mathbb Q_{\infty})_p\) of the Selmer group for \(E\) over the cyclotomic \(\mathbb Z_p\)-extension \(\mathbb Q_{\infty}\) of \(\mathbb Q\). The “analytic” Iwasawa invariants \(\lambda_E^{{\text{anal}}}\) and \(\mu_E^{{\text{anal}}}\) are defined in terms of the \(p\)-adic \(L\)-function for \(E\) constructed by B. Mazur and P. Swinnerton-Dyer [Invent. Math. 25, 1-61 (1974; Zbl 0281.14016)]. In the current paper the authors prove that in certain cases \(\mu_E^{{\text{alg}}}=\mu_E^{{\text{anal}}}=0\) and \(\lambda_E^{{\text{alg}}}=\lambda _E^{{\text{anal}}}\). These equalities together with a deep theorem by Kato imply the Main Conjecture for \(E\) over \(\mathbb Q_{\infty}\). We now describe the contents of the paper in more detail. As remarked by the authors the theorems of the paper apply to modular forms and elliptic curves with multiplicative reduction at \(p\) as well.

The algebraic Iwasawa invariants are defined as follows. The Galois group \(\Gamma=\text{Gal}(\mathbb Q_{\infty}/\mathbb Q)\) acts on the Selmer group \(\text{Sel}_E(\mathbb Q_{\infty})\), and its \(p\)-primary component \(\text{Sel}_E(\mathbb Q_{\infty})_p\) can be regarded as a \(\Lambda\)-module, where \(\Lambda=\mathbb Z_p[[\Gamma]]\). Kato proved a conjecture by B. Mazur [Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)] which states that \(\text{Sel}_E(\mathbb Q _{\infty})\) is \(\Lambda\)-cotorsion. Thus its Potrjagin dual \(X_E(\mathbb Q_{\infty})=\text{Sel}_E(\mathbb Q_{\infty})_p^{\wedge}\) is a finitely generated \(\Lambda\)-module, therefore admitting a pseudo-isomorphism \[ X_E(\mathbb Q_{\infty})\sim\left(\bigoplus_{i=1}^n\Lambda/(f_i(T)^{a_i})\right)\oplus\left(\bigoplus_{j=1}^m\Lambda/(p^{\mu_j})\right), \] where we have the identification \(\mathbb Z_p[[T]]\cong\mathbb Z_p[[\Gamma]]\) given by \(T\mapsto\gamma-1\), \(\gamma\) is a fixed topological generator of \(\Gamma\). The \(f_i(T)\)’s are irreducible distinguished polynomials in \(\Lambda\) and the \(a_i\)’s and \(\mu_j\)’s are positive integers. The algebraic Iwasawa invariants are defined by \[ \lambda_E^{{\text{alg}}}= \sum_{i=1}^na_i\deg(f_i(T))\quad\text{and}\quad \mu_E^{{\text{alg}}}=\sum_{j=1}^m\mu_j. \] In order to formulate the Main Conjecture we also need the “characteristic polynomial” of \(X_E(\mathbb Q_{\infty})\), which is defined as \[ f_E^{{\text{alg}}}(T)=p^{\mu_E^{{\text{alg}}}}\prod_{i=1}^nf_i(T)^{a_i}. \] The invariant \(\lambda_E^{{\text{alg}}}\) can also be defined through group theory as \((\text{Sel}_E(\mathbb Q_{\infty})_p)_{\text{div}}\cong (\mathbb Q_p/\mathbb Z_p)^{\lambda_E^{{\text{alg}}}}\).

Although the invariant \(\lambda_E^{{\text{alg}}}\) can be quite large, it is expected that \(\mu_E^{{\text{alg}}}\) be 0. However, this is not always the case, as B. Mazur (1972, loc. cit.) showed this latter number is positive for certain \(E\) and \(p\). Here are some known results concerning \(\mu_E^{{\text{alg}}}\).

(1) Suppose \(E_1\) and \(E_2\) are elliptic curves defined over \(\mathbb Q\) and \(p>2\) is a prime number where both have good ordinary reduction. If \(E_1[p]\cong E_2[p]\) as Galois modules then \(\text{Sel}_{E_1}(\mathbb Q_{\infty})[p]\) is finite if and only if \(\text{Sel}_{E_2}(\mathbb Q_{\infty})[p]\) is finite. Hence, if \(\text{Sel}_{E_1}(\mathbb Q_{\infty})_p\) is \(\Lambda\)-cotorsion and \(\mu_{E_1} ^{{\text{alg}}}=0\), then \(\text{Sel}_{E_2}(\mathbb Q_{\infty})_p\) is also \(\Lambda\)-cotorsion and \(\mu_{E_2}^{{\text{alg}}}=0\).

(2) Suppose \(E/\mathbb Q\) is an elliptic curve and \(p>2\) is a prime number where \(E\) has good ordinary reduction. If \(E\) admits a cyclic \(\mathbb Q\)-isogeny of degree \(p^f\) with kernel \(\Phi\) and if moreover the action of \(G_{\mathbb Q}\) on \(\Phi\) is ramified at \(p\) and odd, then \(\mu_E^{{\text{alg}}}\geq 1\).

(3) Assume \(E/\mathbb Q\) is an elliptic curve and \(p>2\) is a prime number where \(E\) has good ordinary reduction. If \(E\) admits a \(\mathbb Q\)-isogeny of degree \(p\) with kernel \(\Phi\) and furthermore the action of \(G_{\mathbb Q}\) on \(\Phi\) is either ramified at \(p\) and even or unramified at \(p\) and odd, then \(\mu_E^{{\text{alg}}}=0\).

Let us now pass to the analytic invariants. Suppose \(E\) is a modular elliptic curve over \(\mathbb Q\) and \(p\) is a prime number where \(E\) has good ordinary reduction. For any Dirichlet character \(\rho\), let \(L(E/\mathbb Q,\rho,s)\) be the Hasse-Weil \(L\)-function for \(E\) twisted by \(\rho\). Let \(\Omega_E\) be the real Néron period for \(E\). If \(\rho\) is even, it is known that \(L(E/\mathbb Q,\rho,1)/\Omega_E\in\overline{\mathbb Q}\), fixing an embedding \(\overline{\mathbb Q}\hookrightarrow\mathbb{C}\). Fix also an embedding \(\overline{\mathbb Q}\hookrightarrow\overline{\mathbb Q}_p\). B. Mazur and P. Swinnerton-Dyer (loc. cit.) constructed an element \(\mathcal{L}(E/\mathbb Q,T)\in \Lambda\otimes\mathbb Q_p\) satisfying the following interpolation property. Suppose \(\rho\in\text{Hom}(\Gamma,\mu_{p^{\infty}})\) is a character of finite order. Since \(\gamma\) is a topological generator of \(\Gamma\), \(\rho\) is determined by \(\rho(\gamma) =\zeta\in\mu_{p^{\infty}}\). One can view \(\rho\) as a Dirichlet character of \(p\)-power order and conductor. Assuming that \(\rho\) is nontrivial, its conductor is of the form \(p^m\), and assuming \(p>2\), \(\zeta\) has order \(p^{m-1}\). Then \(\mathcal{L}(E /\mathbb Q,T)\) is characterized by \[ \mathcal{L}(E/\mathbb Q,\zeta-1)=\tau(\rho^{-1})\alpha_p^{-m}\frac{L(E/\mathbb Q,\rho,1)}{\Omega_E}, \] where \(\rho\) runs through all the nontrivial characters of \(\Gamma\), \(\tau(\rho^{-1})\) is the usual Gauss sum and \(\alpha_p\) is the eigenvalue of Frobenius acting on the maximal unramified quotient of the \(p\)-adic Tate module of \(E\). Using the Weierstrass preparation theorem we define the analytic invariants by \[ \mathcal{L}(E/\mathbb Q,T)=p^{\mu_E^{{\text{anal}}}}u(T)f(T), \] where \(f(T)\) is a distinguished polynomial of degree \(\lambda_E^{{\text{anal}}}\) and \(u(T)\) in an invertible power series. The “analytic characteristic polynomial” is defined as \(f_E^{{\text{anal}}}(T)=p^{\mu_E^{{\text{anal}}}}f(T)\). One should have \(\mu_E^{{\text{anal}}}\geq 0\), i.e., \(f_E^{{\text{anal}}}(T)\in\mathbb Z_p[T]\). This is known if \(E[p]\) is irreducible as a Galois module.

If \(p>2\) is a prime number, one identifies \(\Gamma\cong\text{Gal}(\mathbb Q(\mu_{p^{\infty}})/\mathbb Q(\mu_p))\). Let \(\chi\) be the cyclotomic character which gives the action of \(\text{Gal}(\mathbb Q(\mu_{p^{\infty}})/\mathbb Q)\) on \(\mu_{p^{\infty}}\). Letting \(\kappa=\chi_{|\Gamma}\), this induces an isomorphism \(\Gamma\cong 1+p\mathbb Z_p\). The \(p\)-adic \(L\)-function \(L_p(E/\mathbb Q,s)\) is defined by \[ L_p(E/\mathbb Q,s)=\mathcal{L}(E/\mathbb Q,\kappa(\gamma)^{1-s}-1). \] Although \(\mathcal{L}(E/\mathbb Q,T)\) depends on the choice of \(\gamma\), the function \(L_p(E/\mathbb Q,s)\) is independent of this choice. Also, \(L_p(E/\mathbb Q,1)=\mathcal{L}(E/\mathbb Q,0)\) and if furthermore \(E\) has good ordinary reduction at \(p\), then \[ L_p(E/\mathbb Q,1)=\mathcal{L}(E/\mathbb Q,0)=(1-\alpha_p^{-1})\frac{L(E/\mathbb Q,1)}{\Omega_E}. \] The Main Conjecture (Mazur) is stated as \(f_E^{{\text{alg}}}(T)=f_E^{{\text{anal}}}(T)\). This clearly implies \(\mu_E^{{\text{alg}}}=\mu_E^{{\text{anal}}}\) and \(\lambda_E^{{\text{alg}}}=\lambda_E^{{\text{anal}}}\). In fact, Kato proved a weaker statement, i.e., \(f_E^{{\text{alg}}}(T)\) divides \(f_E^{{\text{anal}}}(T)\) in \(\mathbb Q_p[T]\) [cf. A. Scholl, An introduction to Kato’s Euler systems, Lond. Math. Soc. Lect. Notes Ser. 254, 379-460 (1998; Zbl 0952.11015) and K. Rubin, Euler systems and modular elliptic curves, Lond. Math. Soc. Lect. Notes Ser. 254, 351-367 (1998; Zbl 0952.11016)]. Thus, if \(\lambda_E^{{\text{alg}}}=\lambda_E^{{\text{anal}}}\) then \(f_E^{{\text{alg}}}(T)\) and \(f_E^{{\text{anal}}}(T)\) differ by multiplication by a power of \(p\). If moreover \(\mu_E^{{\text{alg}}}=\mu_E^{{\text{anal}}}\), then both equalities imply the Main Conjecture.

The main results of the current paper are the two following theorems. First, assume that \(E\) is a modular elliptic curve and that \(p>2\) is a prime number where \(E\) has good ordinary reduction. Assume also that \(E\) admits a \(\mathbb Q\)-rational isogeny of degree \(p\) and kernel \(\Phi\). Furthermore, suppose that the action of \(G_{\mathbb Q}\) on \(\Phi\) is either ramified at \(p\) and even or unramified at \(p\) and odd. Then \(\lambda_E^{{\text{alg}}}=\lambda_E^{{\text{anal}}}\) and \(\mu_E^{{\text{alg}}}=\mu_E^{{\text{anal}}} =0\). Second, assume \(E_1\) and \(E_2\) are modular elliptic curves defined over \(\mathbb Q\), \(p>2\) is a prime number where both \(E_1\) and \(E_2\) have good ordinary reduction and \(E_1[p]\cong E_2[p]\) as Galois modules and these are irreducible. If the equalities \(\mu_{E_1}^{{\text{alg}}}=\mu_{E_1}^{{\text{anal}}}=0\) and \(\lambda_{E_1}^{{\text{alg}}}=\lambda_{E_1}^{{\text{anal}}}\) hold, then so do the equalities \(\mu_{E_2}^{{\text{alg}}}=\mu_{E_2}^{{\text{anal}}}=0\) and \(\lambda_{E_2}^{{\text{alg}}}=\lambda_{E_2}^{{\text{anal}}}\). The range of prime numbers for which the result apply is limited. For instance, for the first one, only for \(p=3,5,7,13\) or \(37\). However, for the first four of these primes, there are infinitely many distinct \(j\)-invariants \(j_E\) which can occur.

The algebraic Iwasawa invariants are defined as follows. The Galois group \(\Gamma=\text{Gal}(\mathbb Q_{\infty}/\mathbb Q)\) acts on the Selmer group \(\text{Sel}_E(\mathbb Q_{\infty})\), and its \(p\)-primary component \(\text{Sel}_E(\mathbb Q_{\infty})_p\) can be regarded as a \(\Lambda\)-module, where \(\Lambda=\mathbb Z_p[[\Gamma]]\). Kato proved a conjecture by B. Mazur [Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)] which states that \(\text{Sel}_E(\mathbb Q _{\infty})\) is \(\Lambda\)-cotorsion. Thus its Potrjagin dual \(X_E(\mathbb Q_{\infty})=\text{Sel}_E(\mathbb Q_{\infty})_p^{\wedge}\) is a finitely generated \(\Lambda\)-module, therefore admitting a pseudo-isomorphism \[ X_E(\mathbb Q_{\infty})\sim\left(\bigoplus_{i=1}^n\Lambda/(f_i(T)^{a_i})\right)\oplus\left(\bigoplus_{j=1}^m\Lambda/(p^{\mu_j})\right), \] where we have the identification \(\mathbb Z_p[[T]]\cong\mathbb Z_p[[\Gamma]]\) given by \(T\mapsto\gamma-1\), \(\gamma\) is a fixed topological generator of \(\Gamma\). The \(f_i(T)\)’s are irreducible distinguished polynomials in \(\Lambda\) and the \(a_i\)’s and \(\mu_j\)’s are positive integers. The algebraic Iwasawa invariants are defined by \[ \lambda_E^{{\text{alg}}}= \sum_{i=1}^na_i\deg(f_i(T))\quad\text{and}\quad \mu_E^{{\text{alg}}}=\sum_{j=1}^m\mu_j. \] In order to formulate the Main Conjecture we also need the “characteristic polynomial” of \(X_E(\mathbb Q_{\infty})\), which is defined as \[ f_E^{{\text{alg}}}(T)=p^{\mu_E^{{\text{alg}}}}\prod_{i=1}^nf_i(T)^{a_i}. \] The invariant \(\lambda_E^{{\text{alg}}}\) can also be defined through group theory as \((\text{Sel}_E(\mathbb Q_{\infty})_p)_{\text{div}}\cong (\mathbb Q_p/\mathbb Z_p)^{\lambda_E^{{\text{alg}}}}\).

Although the invariant \(\lambda_E^{{\text{alg}}}\) can be quite large, it is expected that \(\mu_E^{{\text{alg}}}\) be 0. However, this is not always the case, as B. Mazur (1972, loc. cit.) showed this latter number is positive for certain \(E\) and \(p\). Here are some known results concerning \(\mu_E^{{\text{alg}}}\).

(1) Suppose \(E_1\) and \(E_2\) are elliptic curves defined over \(\mathbb Q\) and \(p>2\) is a prime number where both have good ordinary reduction. If \(E_1[p]\cong E_2[p]\) as Galois modules then \(\text{Sel}_{E_1}(\mathbb Q_{\infty})[p]\) is finite if and only if \(\text{Sel}_{E_2}(\mathbb Q_{\infty})[p]\) is finite. Hence, if \(\text{Sel}_{E_1}(\mathbb Q_{\infty})_p\) is \(\Lambda\)-cotorsion and \(\mu_{E_1} ^{{\text{alg}}}=0\), then \(\text{Sel}_{E_2}(\mathbb Q_{\infty})_p\) is also \(\Lambda\)-cotorsion and \(\mu_{E_2}^{{\text{alg}}}=0\).

(2) Suppose \(E/\mathbb Q\) is an elliptic curve and \(p>2\) is a prime number where \(E\) has good ordinary reduction. If \(E\) admits a cyclic \(\mathbb Q\)-isogeny of degree \(p^f\) with kernel \(\Phi\) and if moreover the action of \(G_{\mathbb Q}\) on \(\Phi\) is ramified at \(p\) and odd, then \(\mu_E^{{\text{alg}}}\geq 1\).

(3) Assume \(E/\mathbb Q\) is an elliptic curve and \(p>2\) is a prime number where \(E\) has good ordinary reduction. If \(E\) admits a \(\mathbb Q\)-isogeny of degree \(p\) with kernel \(\Phi\) and furthermore the action of \(G_{\mathbb Q}\) on \(\Phi\) is either ramified at \(p\) and even or unramified at \(p\) and odd, then \(\mu_E^{{\text{alg}}}=0\).

Let us now pass to the analytic invariants. Suppose \(E\) is a modular elliptic curve over \(\mathbb Q\) and \(p\) is a prime number where \(E\) has good ordinary reduction. For any Dirichlet character \(\rho\), let \(L(E/\mathbb Q,\rho,s)\) be the Hasse-Weil \(L\)-function for \(E\) twisted by \(\rho\). Let \(\Omega_E\) be the real Néron period for \(E\). If \(\rho\) is even, it is known that \(L(E/\mathbb Q,\rho,1)/\Omega_E\in\overline{\mathbb Q}\), fixing an embedding \(\overline{\mathbb Q}\hookrightarrow\mathbb{C}\). Fix also an embedding \(\overline{\mathbb Q}\hookrightarrow\overline{\mathbb Q}_p\). B. Mazur and P. Swinnerton-Dyer (loc. cit.) constructed an element \(\mathcal{L}(E/\mathbb Q,T)\in \Lambda\otimes\mathbb Q_p\) satisfying the following interpolation property. Suppose \(\rho\in\text{Hom}(\Gamma,\mu_{p^{\infty}})\) is a character of finite order. Since \(\gamma\) is a topological generator of \(\Gamma\), \(\rho\) is determined by \(\rho(\gamma) =\zeta\in\mu_{p^{\infty}}\). One can view \(\rho\) as a Dirichlet character of \(p\)-power order and conductor. Assuming that \(\rho\) is nontrivial, its conductor is of the form \(p^m\), and assuming \(p>2\), \(\zeta\) has order \(p^{m-1}\). Then \(\mathcal{L}(E /\mathbb Q,T)\) is characterized by \[ \mathcal{L}(E/\mathbb Q,\zeta-1)=\tau(\rho^{-1})\alpha_p^{-m}\frac{L(E/\mathbb Q,\rho,1)}{\Omega_E}, \] where \(\rho\) runs through all the nontrivial characters of \(\Gamma\), \(\tau(\rho^{-1})\) is the usual Gauss sum and \(\alpha_p\) is the eigenvalue of Frobenius acting on the maximal unramified quotient of the \(p\)-adic Tate module of \(E\). Using the Weierstrass preparation theorem we define the analytic invariants by \[ \mathcal{L}(E/\mathbb Q,T)=p^{\mu_E^{{\text{anal}}}}u(T)f(T), \] where \(f(T)\) is a distinguished polynomial of degree \(\lambda_E^{{\text{anal}}}\) and \(u(T)\) in an invertible power series. The “analytic characteristic polynomial” is defined as \(f_E^{{\text{anal}}}(T)=p^{\mu_E^{{\text{anal}}}}f(T)\). One should have \(\mu_E^{{\text{anal}}}\geq 0\), i.e., \(f_E^{{\text{anal}}}(T)\in\mathbb Z_p[T]\). This is known if \(E[p]\) is irreducible as a Galois module.

If \(p>2\) is a prime number, one identifies \(\Gamma\cong\text{Gal}(\mathbb Q(\mu_{p^{\infty}})/\mathbb Q(\mu_p))\). Let \(\chi\) be the cyclotomic character which gives the action of \(\text{Gal}(\mathbb Q(\mu_{p^{\infty}})/\mathbb Q)\) on \(\mu_{p^{\infty}}\). Letting \(\kappa=\chi_{|\Gamma}\), this induces an isomorphism \(\Gamma\cong 1+p\mathbb Z_p\). The \(p\)-adic \(L\)-function \(L_p(E/\mathbb Q,s)\) is defined by \[ L_p(E/\mathbb Q,s)=\mathcal{L}(E/\mathbb Q,\kappa(\gamma)^{1-s}-1). \] Although \(\mathcal{L}(E/\mathbb Q,T)\) depends on the choice of \(\gamma\), the function \(L_p(E/\mathbb Q,s)\) is independent of this choice. Also, \(L_p(E/\mathbb Q,1)=\mathcal{L}(E/\mathbb Q,0)\) and if furthermore \(E\) has good ordinary reduction at \(p\), then \[ L_p(E/\mathbb Q,1)=\mathcal{L}(E/\mathbb Q,0)=(1-\alpha_p^{-1})\frac{L(E/\mathbb Q,1)}{\Omega_E}. \] The Main Conjecture (Mazur) is stated as \(f_E^{{\text{alg}}}(T)=f_E^{{\text{anal}}}(T)\). This clearly implies \(\mu_E^{{\text{alg}}}=\mu_E^{{\text{anal}}}\) and \(\lambda_E^{{\text{alg}}}=\lambda_E^{{\text{anal}}}\). In fact, Kato proved a weaker statement, i.e., \(f_E^{{\text{alg}}}(T)\) divides \(f_E^{{\text{anal}}}(T)\) in \(\mathbb Q_p[T]\) [cf. A. Scholl, An introduction to Kato’s Euler systems, Lond. Math. Soc. Lect. Notes Ser. 254, 379-460 (1998; Zbl 0952.11015) and K. Rubin, Euler systems and modular elliptic curves, Lond. Math. Soc. Lect. Notes Ser. 254, 351-367 (1998; Zbl 0952.11016)]. Thus, if \(\lambda_E^{{\text{alg}}}=\lambda_E^{{\text{anal}}}\) then \(f_E^{{\text{alg}}}(T)\) and \(f_E^{{\text{anal}}}(T)\) differ by multiplication by a power of \(p\). If moreover \(\mu_E^{{\text{alg}}}=\mu_E^{{\text{anal}}}\), then both equalities imply the Main Conjecture.

The main results of the current paper are the two following theorems. First, assume that \(E\) is a modular elliptic curve and that \(p>2\) is a prime number where \(E\) has good ordinary reduction. Assume also that \(E\) admits a \(\mathbb Q\)-rational isogeny of degree \(p\) and kernel \(\Phi\). Furthermore, suppose that the action of \(G_{\mathbb Q}\) on \(\Phi\) is either ramified at \(p\) and even or unramified at \(p\) and odd. Then \(\lambda_E^{{\text{alg}}}=\lambda_E^{{\text{anal}}}\) and \(\mu_E^{{\text{alg}}}=\mu_E^{{\text{anal}}} =0\). Second, assume \(E_1\) and \(E_2\) are modular elliptic curves defined over \(\mathbb Q\), \(p>2\) is a prime number where both \(E_1\) and \(E_2\) have good ordinary reduction and \(E_1[p]\cong E_2[p]\) as Galois modules and these are irreducible. If the equalities \(\mu_{E_1}^{{\text{alg}}}=\mu_{E_1}^{{\text{anal}}}=0\) and \(\lambda_{E_1}^{{\text{alg}}}=\lambda_{E_1}^{{\text{anal}}}\) hold, then so do the equalities \(\mu_{E_2}^{{\text{alg}}}=\mu_{E_2}^{{\text{anal}}}=0\) and \(\lambda_{E_2}^{{\text{alg}}}=\lambda_{E_2}^{{\text{anal}}}\). The range of prime numbers for which the result apply is limited. For instance, for the first one, only for \(p=3,5,7,13\) or \(37\). However, for the first four of these primes, there are infinitely many distinct \(j\)-invariants \(j_E\) which can occur.

Reviewer: Amílcar Pacheco (Rio de Janeiro)