| Abstract | Za eliptičku krivulju E/Q i za svaki prost broj p najprije određujemo sve moguće torzijske grupe E(Q∞,p)tors , gdje je Q∞,p jedinstveno Zp-proširenje od Q, tj.\ jedinstveno Galoisovo proširenje od Q takvo da je Gal (Q∞,p/Q)≃Zp. Za eliptičku krivulju E/Q i prost broj p vrijedi: Ako je p≥5, onda je ... More E(Q∞,p)tors=E(Q)tors. Ako je p=3, onda je grupa E(Q∞,3)tors izomorfna nekoj od grupa iz Mazurovog teorema ili nekoj od grupa Z/21Z i Z/27Z. Ako je p=2, onda je grupa E(Q∞,2)tors izomorfna nekoj od grupa iz Mazurovog teorema. Treba biti oprezan, u slučajevima p=2 i p=3 ne vrijedi nužno da je E(Q∞,p)tors=E(Q)tors. Na ovo pitanje također dajemo detaljan odgovor te nalazimo primjere za sve moguće slučajeva rasta torzije Q→Q∞,p, gdje je p∈{2,3}. Promatramo također i torziju nad kompozitumom svih Zp-proširenja od Q. Neka je K≥5=∏p≥5 prostQ∞,pteK=∏p prostQ∞,p. Dokazali smo da za eliptičku krivulju E/Q vrijedi da je E(K≥5)tors=E(Q)tors te da je E(K)tors izomorfno nekoj od grupa iz Mazurovog teorema ili nekoj od grupa Z/13Z, Z/21Z i Z/27Z. Na kraju navodimo neke rezultate o ponašanju torzije eliptičke krivulje E/Q nad poljima Q(μp∞)=∞⋃k=1Q(μpk),gdje jeμn={ω∈C : ωn=1}. Preciznije, dokazan je sljedeći rezultat za eliptičke krivulje E/Q: E(Q(μ2∞))tors=E(Q(μ24))tors,E(Q(μ3∞))tors=E(Q(μ33))tors teE(Q(μp∞))tors=E(Q(μp))tors,za svaki prost broj p≥5. Less |
| Abstract (english) | We determine, for an elliptic curve E/Q and for all prime numbers p, all the possible torsion groups E(Q∞,p)tors, where Q∞,p is the Zp-extension of Q. For a prime number p, denote by Q∞,p the unique Zp-extension of Q and for a positive integer n, denote by Qn,p the nth layer of ... More Q∞,p, i.e.\ the unique subfield of Q∞,p such that Gal(Qn,p/Q)≃Z/pnZ. Let, as always, μn={ω∈C : ωn=1} be the set of all nth roots of unity. We also define μp∞=⋃k∈Nμpk. Note that Q(μpk)=Q(ζpk), where ζn is nth primitive root of unity. Recall that the Zp-extension of Q is the unique Galois extension Q∞,p of Q such that Gal(Q∞,p/Q)≃Zp, where Zp is the additive group of the p-adic integers and is constructed as follows: Let G=Gal(Q(μp∞)/Q)=lim←nGal(Q(μpn+1)/Q)∼→lim←n(Z/pn+1Z)×=Z×p. Here we know that G=Δ×Γ, where Γ≃Zp and Δ≃Z/(p−1)Z for p≥3 and Δ≃Z/2Z (generated by complex conjugation) for p=2, so we define Q∞,p:=Q(μp∞)Δ. We also see that every layer is uniquely determined by (for p≥3) Qn,p=Q(μpn+1)∩Q∞,p, so for p≥3 it is the unique subfield of Q(μpn+1) of degree pn over Q. More details and proofs of these facts about Zp-extensions and Iwasawa theory can be found in [56, Chapter 13]. Iwasawa theory for elliptic curves (see [19]) studies elliptic curves in Zp-extensions, in particular the growth of the rank and n-Selmer groups in the layers of the Zp-extensions. In this paper we completely solve the problem of determining how the torsion of an elliptic curve defined over Q grows in the Zp-extensions of Q. These results, interesting in their own right, might also find applications in other problems in Iwasawa theory for elliptic curves and in general. For example, to show that elliptic curves over Q∞,p are modular for all p, Thorne [55] needed to show that E(Q∞,p)tors=E(Q)tors for two particular elliptic curves. In this work we did that thing in general case. Our results are the following: Let E/Q be an elliptic curve. Let p≥5 be a prime number. Then E(Q∞,p)tors=E(Q)tors. Group E(Q∞,2)tors is isomorphic to exactly one of the following groups: Z/NZ,1≤N≤10, or N=12,Z/2Z⊕Z/2NZ,1≤N≤4, and for each group G from the list above there exists an E/Q such that E(Q∞,2)tors≃G. Group E(Q∞,3)tors is isomorphic to exactly one of the following groups: Z/NZ,1≤N≤10, or N=12,21 or 27,Z/2Z⊕Z/2NZ,1≤N≤4. and for each group G from the list above there exists an E/Q such that E(Q∞,3)tors≃G. By Mazur's theorem we see that {E(Q∞,2)tors:E/Q elliptic curve}={E(Q)tors:E/Q elliptic curve},{E(Q∞,3)tors:E/Q elliptic curve}={E(Q)tors:E/Q elliptic curve}∪{Z/21Z,Z/27Z}. However, given a specific E/Q it is not necessarily the case that E(Q∞,p)tors=E(Q)tors. Indeed there are many elliptic curves for which torsion grows from Q to Q∞,p, and we investigate this question further in Section 3.6. Specifically, for each prime p we find for which groups G there exists infinitely many j-invariants j such that there exists an elliptic curve E/Q with j(E)=j and such that E(Q)tors⊊E(Q∞,p)tors≃G. Furthermore, after we understood the behaviour of the torsion of elliptic curve E/Q over the field Q∞,p, we tried to find out what will happen if we look at the compositum of all of those fields. We answered that question completely too. Let K≥5=∏p≥5 primeQ∞,p and let K=∏p primeQ∞,p. We proved that for an elliptic curve E/Q it holds that E(K≥5)tors=E(Q)tors and also that E(K)tors is isomorphic to one of the following groups Z/nZ,1≤n≤10 or n∈{12,13,21,27},Z/2Z⊕Z/2nZ,1≤n≤4. For each group G from the list above there exists an E/Q such that E(Q∞,3)tors≃G. At the end, in chapter 5 we state some results about the behaviour of the torsion of elliptic curve E/Q over the fields Q(μp,∞)=∞⋃k=1Q(μpk). More precisely, we prove the following result Let E/Q be an elliptic curve, then for a prime number p≥5 it holds that E(Q(μp∞))tors=E(Q(μp))tors. Furthermore E(Q(μ3∞))tors=E(Q(μ33))torsandE(Q(μ2∞))tors=E(Q()μ24)tors. In chapter 6 we exhibit all magma [2] codes that we used for computations. Less |
