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Diophantine stability. (English) Zbl 1491.14036

Let \(K\) be a number field, \(\overline{ K}\) a separable closure of \(K\), \(V\) an irreducible variety over \(K\) and \(L\) a field containing \(K\). The authors say that \(V\) is {\em diophantine-stable for \(L/K\) } if \(V(L)=V(K)\). If \(\ell\) is a rational prime, they say that \(V\) is {\em \(\ell\)-diophantine stable} if for every positive integer \(n\), and every finite set \(\Sigma\) of places of \(K\), there are infinitely many cyclic extensions \(L/K\) of degree \(\ell^n\), completely split at all places \(v\in\Sigma\), such that \(V\) is diophantine-stable for \(L/K\).
The first main result deals with a simple abelian variety \(A\) over \(K\). Assume that all \({\overline K}\)-endomorphisms of \(A\) are defined over \(K\). Then there is a set \(S\) of rational primes with positive density such that \(A\) is \(\ell\)-diophantine-stable over \(K\) for every \(\ell\in S\).
The second main result, which is a consequence of the first one, deals with an irreducible curve \(X\) over \(K\). Assume that the normalisation and completion \(\tilde X\) of \(X\) has genus \(\ge 1\) and that all \(\overline{K}\)-endomorphisms of the Jacobian of \(\tilde X\) are defined over \(K\); then there is a set \(S\) of rational primes with positive density such that \(X\) is \(\ell\)-diophantine-stable over \(K\) for every \(\ell\in S\).
From the second result the authors deduce the following two statements, the first one by applying repeatedly their result to the modular curve \(X_0(p)\), the second one to an elliptic curve over \({\mathbb Q}\) of positive rank. Let \(p\ge 23\) with \(p\not\in\{37,43,67,163\}\); then there are uncountably many pairwise non-isomorphic subfields \(L\) of \(\overline{\mathbb Q}\) such that no elliptic curve defined over \(L\) possesses an \(L\)-rational subgroup of ordre \(p\). And finally the authors prove that for every prime \(p\), there are uncountably many pairwise non-isomorphic totally real fields \(L\) of algebraic numbers in \({\mathbb Q}_p\) over which the following two statements both hold: (i) There is a diophantine definition of \(\mathbb Z\) in the ring of integers \({\mathcal O}_L\) of \(L\); in particular Hilbert’s Tenth Problem has a negative answer for \({\mathcal O}_L\). (ii) There exists a first-order definition of the ring \(\mathbb Z\) in \(L\); the first-order theory for such fields \(L\) is undecidable.
The appendix by M. Larsen is devoted to the proof of the following result. Let \(A\) be a simple abelian variety defined over \(K\) such that \({\mathcal E}:={\mathrm{End}}_K(A)= {\mathrm{End}}_{\overline K}(A)\). Let \({\mathcal R}\) denote the center of \(\mathcal E\) and \({\mathcal M}={\mathcal R}\otimes{\mathbb Q}\). There is a positive density set \(S\) of rational primes such that for every prime \(\lambda\) of \({\mathcal M}\) lyning above \(S\) we have: (i) there is a \(\tau_0\in G_{K^{\mathrm ab}}\) such that \(A[\lambda]^{(\tau_0)}=0\), and (ii) there is a \(\tau_1\in G_{K^{\mathrm ab}}\) such that \(A[\lambda]/(\tau_1-1)A[\lambda]\) is a simple \({\mathcal E}/\lambda\)-module.

MSC:

14G05 Rational points
11G05 Elliptic curves over global fields
14H25 Arithmetic ground fields for curves
14K05 Algebraic theory of abelian varieties