Continuing the previous post, we would like to sketch out how additive combinatorics helps us to solve Hilbert’s tenth problem for rings of integers of general number fields. As explained previously, We are reduced to showing existence of a certain elliptic curve with whose group of rational points has positive rank but doesn’t grow under a fixed finite extensions of number fields. In fact, we can further relax it as follows:
(MRS24, Theorem 3.1 and 4.8) If for every quadratic extension of number fields, there exists an abelian variety such that , then Hilbert’s tenth problem has a negative solution for the ring of integers of every number field.
This strengthening is needed for the proof in [ABHS] but not for the one in [KP]. We will present the former proof since it is shorter, but for the latter, there are great expositions by the authurs in the recorded talks L1 and L2.
WLOG by taking Weil restriction we can assume contains a primitive -th root of unity for some odd prime to be chosen. The main player is the hyperelliptic curve of genus . For , let denotes the twist . Note that for any , and are isomorphic over . There is a natural isomorphism of on , hence on its Jacobian .
For a quadratic extension , the -quadratic twist of is the curve , which is isomorphic to over , similarly . We also have -twists for , as is isomorphic to over .
Since we have an injection , we can view as a self-isogeny of of degree and study its Selmer group (analogous to ususal Selmer group). In particular, from the long exact sequence induced from the short exact sequence we have that .
The key realization is that we can find a large set of primes of such that the local conditions they impose on the -Selmer group is vacuous, in the sense that . (Recall that where and is the image of the boundary map .) This set of primes consists of those that are coprime to and remain inert or ramify in , by the following lemma.
Suppose is inert or ramified in . Then .
Proof: Since contains , we see that the galois module is isomorphic to its own Cartier dual. Hence, local Tate duality gives , and the local Euler characteristic formula reads . The last equality follows because .
The starting point is a result by Yu that enables us to use -twist to produce curves with zero Selmer group (and thus the group of rational points is of rank zero):
There exists such that . Moreover, we can choose such that for all primes that ramify in .
The next result shows that if we further twist by -units where is essentially (up to finitely many primes) the set of silent primes, the -Selmer rank stays unchanged.
There is a subset of primes which is the union of the set of silent primes with a finite set of primes such that if and for all , then .
The idea is that both Selmer groups and can be viewed as living inside the common ambient space since . It remains to show that their corresponding local conditions are equal inside for all primes . If , this is because the two curves are isomorphic over . Otherwise, if is inert or ramified in , then by the above lemma.
It remains to find enough twists that has positive rank. Define The following proposition holds:
The set is infinite, and for all but fnintely many , the Jacobian has positive rank.
Then we can deduce the main theorem as follows. Recall the goal is to construct an abelian variety with the property that . Let be an odd prime not dividing the discriminant of . Then and the -th cyclotomic field are linearly disjoint inside a common algebraic closure. We may assume and is unramified at proimes above . Let and choose as above. Then choose as in the proposition, so that . By another lemma, we also have has rank 0. Hence so that over is the sought-after abelian variety.