Ranks of Abelian varieties over infinite extensions of the rationals
The author studies the rank of the Mordell-Weil groups for some abelian varieties over infinite extensions of \({\mathbb Q}.\) If \(L/{\mathbb Q}\) is finite then the well-known Mordell-Weil Theorem asserts that \(A(L)\) is of finite rank. For infinite extensions the rank of the corresponding Mordell-Weil group might be infinite (cf. [\textit{G. Frey, M. Jarden}, Proc. Lond. Math. Soc. (3) 28, 112--128 (1974; Zbl 0275.14021)] for the case of \(A({\overline{\mathbb Q}})\)). On the other hand the theorem of Kato, Ribet and Rohrlich asserts that if \(E/{\mathbb Q}\) is an elliptic curve, \(\Sigma\) a finite set of rational primes then the group \(E({\mathbb Q}^{ab}_{\Sigma})\) is finitely generated. \({\mathbb Q}^{ab}_{\Sigma}\) denotes here the maximal abelian extension of \({\mathbb Q}\) unramified outside \({\Sigma}\). Let \(S\) be an infinite set of primes of \({\mathbb Z},\) and \({\mathbb Q}^{ab}_{S}\) be the maximal abelian extension of \({\mathbb Q}\) unramified outside \(S\) and contained in \({\overline{\mathbb Q}}.\) For a prime \(p\geq 2\) let \({\mathbb Q}^{(p)}_{S}\) be the compositum of all extensions unramified outside \(S\) of the form \({\mathbb Q}({\mu}_{p}, {\root p \of d})\) for some \(d\in {\mathbb Q}^{\times}.\) Let \(F\subset {\overline{\mathbb Q}}\), \quad \((\sigma) = ({\sigma}_{1},\cdots ,{\sigma}_{n}) \in \text{Gal} ({\overline{\mathbb Q}}/{\mathbb Q})^{n}\) anf \(F^{({\sigma})}\) be the intersection of all fixed fields \(F^{({\sigma}_{i})}.\) The author proves the following: Theorem 1.2: Let \(E/{\mathbb Q}\) be an elliptic curve and let \(S\) be an infinite set of primes. (1) Suppose that \(\text{rank}_{\mathbb Z}(E(\mathbb Q))\) is odd. If the parity conjecture holds for all quadratic twists of \(E\) then the rank of \(E(({\mathbb Q}_{S}^{(2)})^{(\sigma)})\) is infinite for all \(n\geq 0\) and all \((\sigma)\in \text{Gal} ({\overline{\mathbb Q}}/{\mathbb Q})^{n}.\) Hence \(\text{rank}_{\mathbb Z}(E(({\mathbb Q}_{S}^{ab})^{(\sigma)}))\) is infinite as well. (2) Suppose \(E/{\mathbb Q}\) does not have wild ramification at \(2\) and \(3.\) There are infinitely many primes \(p>2\) such that if the parity conjecture holds for \(E\) over extensions of degree \(p\) and we set \({ S^{\prime}} = S \cup \{ p\}\) then the rank of \(E(({\mathbb Q}_{ S^{\prime}}^{ab})^{(\sigma)})\) is infinite for all \(n\geq 0\) and all \((\sigma)\in \text{Gal} ({\overline{\mathbb Q}}/{\mathbb Q})^{n}.\) In particular, if the hypotheses of (1) and (2) are satisfied, then the rank of \(E({\mathbb Q}_{S}^{(\sigma)})\) is infinite. Let \(K\) be a quadratic extension of \(\mathbb Q\). Denote by \((K_{S}^{p-\text{dihe}})\) the maximal abelian \(p\)-elementary extension of \(K\) unramified outside \(S\) and dihedral over \({\mathbb Q}.\) For abelian varieties the author shows the following: Theorem 1.4: Let \((A/{\mathbb Q}, \phi)\) be a polarized abelian variety \(n\geq 0\) and let \((\sigma)) \in \text{Gal} ({\overline{\mathbb Q}}/{\mathbb Q})^{n}\) be fixed. Suppose there is a quadratic extension \(K/{\mathbb Q}\), fixed by \((\sigma)\), such that \( \text{corank}_{{\mathbb Z}_{p}} \text{Sel}_{p^{\infty}}(A/K)\) is odd, for some prime \(p>2\) which splits in \(K\) and such that \(\gcd(p, \deg(\phi))=1\). Let \(S\) be an infinite set of rational primes which does not include any of the primes of bad reduction for \(A/{\mathbb Q}\), and such that \(S\) contains infinitely many primes either inert in \(K\) and congruent to \(-1\bmod p\), or split in \(K\) and congruent to \(1\bmod p.\) Then the corank of \(\text{Sel}_{p^{\infty}}(A/F)\) is unbounded over finite extensions \(F/K\) contained in the field \((K_{S}^{p-\text{dihe}})^{(\sigma)}\).
scientific article; zbMATH DE number 5351817
scientific article