An analogue of Kida’s formula for the Selmer groups of elliptic curves.

*(English)*Zbl 1081.11508Summary: A main purpose of this paper is to describe the behavior of the Selmer groups of elliptic curves in a \(p\)-extension of the cyclotomic \(\mathbb Z_p\)-extension of a number field, in particular to give an analogue of Kida’s formula. Let \(p\) be a prime number and \(E\) an elliptic curve over a number field \(K\) with good ordinary reduction at all the primes above \(p\). Mazur investigated the behavior of the Selmer groups of \(E\) in the cyclotomic \(\mathbb Z_p\)-extenion \(K_\infty/K\). Let \({\mathcal X}(E/K_\infty)\) be the Pontryagin dual of the Selmer group of \(E/K_\infty\) (the precise definition is given in the paper). By the action of \(\text{Gal}(K_\infty/K)\), we may regard \({\mathcal X}(E/K_\infty)\) as a module over \(\Lambda= \mathbb Z_p[[T]]\). It is known that this module is finitely generated over \(\Lambda\). Furthermore, Mazur conjectured that this is \(\Lambda\)-torsion. Under this conjecture, we may consider the \(\lambda\)- and \(\mu\)-invariants of \({\mathcal X}(E/K_\infty)\). The \(\lambda\)-invariant is just equal to the \(\mathbb Z_p\)-corank of the maximal divisible subgroup of the Selmer group.

The main theorem is as follows: We assume \(p\geq 5\) in this introduction for simplicity.

Theorem. Let \(L/K\) be a Galois extension of degree a power of \(p\). Assume that \({\mathcal X}(E/K_\infty)\) is \(\Lambda\)-torsion, and its \(\mu\)-invariant is zero. Then \({\mathcal X}(E/L_\infty)\) is also \(\Lambda\)-torsion, and its \(\mu\)-invariant is zero. Furthermore, the \(\lambda\)-invariants of these, \(\lambda_E(K)\) and \(\lambda_E(L)\), satisfy the following formula: \[ \lambda_E(L)= [L_\infty:K_\infty] \lambda_E(K)+ \sum_{w\in P_1} (e_{L_\infty/K_\infty}(w)-1)+ 2 \sum_{w\in P_2} (e_{L_\infty/K_\infty}(w)-1). \] Here \(P_1\) and \(P_2\) are the sets of the primes of \(L_\infty\) defined as

\(P_1= \{w\mid w\nmid p\), \(E\): split multiplicative redution at \(w\}\),

\(P_2= \{w\mid w\nmid p\), \(E\): good reduction at \(w\), \(E(L_{\infty,w})\) has a point of order \(p\}\),

and \(e_{L_\infty/K_\infty}(w)\) denotes the ramification index.

If we assume that \(E\) has multiplicative reduction instead of good ordinary reduction at all the primes above \(p\), it is still conjectured that \({\mathcal X}(E,K_\infty)\) is \(\Lambda\)-torsion. We can also give a formula similar to the above in this case (see Theorem 8.1).

We reduce the proof of the above theorem to the calculation of the Herbrand quotient of the Selmer group following Iwasawa’s method, which gives another proof of Kida’s formula. For the calculation, we use Cassels-Tate-Poitou’s duality and the Hochschild-Serre spectral sequence.

J. Coates pointed out that the main theorem may have useful applicaitons to his \(\text{GL}_2\)-Iwasawa theory.

The main theorem is as follows: We assume \(p\geq 5\) in this introduction for simplicity.

Theorem. Let \(L/K\) be a Galois extension of degree a power of \(p\). Assume that \({\mathcal X}(E/K_\infty)\) is \(\Lambda\)-torsion, and its \(\mu\)-invariant is zero. Then \({\mathcal X}(E/L_\infty)\) is also \(\Lambda\)-torsion, and its \(\mu\)-invariant is zero. Furthermore, the \(\lambda\)-invariants of these, \(\lambda_E(K)\) and \(\lambda_E(L)\), satisfy the following formula: \[ \lambda_E(L)= [L_\infty:K_\infty] \lambda_E(K)+ \sum_{w\in P_1} (e_{L_\infty/K_\infty}(w)-1)+ 2 \sum_{w\in P_2} (e_{L_\infty/K_\infty}(w)-1). \] Here \(P_1\) and \(P_2\) are the sets of the primes of \(L_\infty\) defined as

\(P_1= \{w\mid w\nmid p\), \(E\): split multiplicative redution at \(w\}\),

\(P_2= \{w\mid w\nmid p\), \(E\): good reduction at \(w\), \(E(L_{\infty,w})\) has a point of order \(p\}\),

and \(e_{L_\infty/K_\infty}(w)\) denotes the ramification index.

If we assume that \(E\) has multiplicative reduction instead of good ordinary reduction at all the primes above \(p\), it is still conjectured that \({\mathcal X}(E,K_\infty)\) is \(\Lambda\)-torsion. We can also give a formula similar to the above in this case (see Theorem 8.1).

We reduce the proof of the above theorem to the calculation of the Herbrand quotient of the Selmer group following Iwasawa’s method, which gives another proof of Kida’s formula. For the calculation, we use Cassels-Tate-Poitou’s duality and the Hochschild-Serre spectral sequence.

J. Coates pointed out that the main theorem may have useful applicaitons to his \(\text{GL}_2\)-Iwasawa theory.