In Chapter 2, we study the infinite families of quadratic and cubic curves of the elliptic curve A=X0(27), such that they have complex multiplication with the integer ring of Q(√ . −3). In Chapter 7, we briefly discuss the main conjectures attached to E/H, how they relate to the main conjecture for H∞/H and the p part of the Birch–Swinnerton–Dyer conjecture.
The p-part of the Birch–Swinnerton-Dyer Conjecture
The primes for bad reduction for E(D2) are again 3 and the primes divisor D, since the discriminant of E(D2) is -27D4.
Quadratic Twists
The sign will not matter, since we are most interested in the case where D is an integer. Further, x=X under the change of coordinates which gives the minimal Weierstrass form, and thus we have
Cubic Twists
In the case of q= 7, we can take E to be any quadratic twist of the elliptic curve A=X0(49) with Eq. From now on, assume that E is a quadratic twist of A(q) with a quadratic extension of H of the form H(√ .. λ), where λ is some nonzero element of K and the discriminant of H(√.
Descent theory over H
This is obtained by the restriction of the Serre–Tate character of the abelian variety B/K [18, Theorem 10], which is the restriction of the scalars E from H to K. Thus, Eb is not a Lubin–Tate group by itself, but Eb together with the homomorphism λbE(v ) : Eb → Ebσv relative Lubin–Tate group studied by de Shalit in [8, I §1]. Let u be any place in H where E has a good reduction, and let w be its restriction on K .
Then for any n >1, the extension Hv(Ewn)/Hv is completely branched and its Galois group is isomorphic to (O/wn)×. Now, E has good ordinary reduction in primes of H over p. Let v be any site of F that does not lie over p, and let Uv be the units of the ring of integers complementing F to v. Let also T =P ∪B where B denotes the set of primes of H where E is ill-reduced, and similarly defined.
By Tate local duality, the dual of the discrete group H1(Hv, E) is E(Hv), and this induces the duality between H1(Hv, E)πn and E(Hv)/π∗nE(Hv) for any positive integer n. Then π∗n is an automorphism of the formal group of E atv, and reduction modulo v induces an isomorphism.
Descent theory over extensions of H
Note that Pn lies above p and p ̸= p∗, so π∗m is an automorphism of the group E1(Fn,Pn) of Fn,Pn-rational points in the core of reduction modulo Pn. We will study this in more detail in Chapter 7 to formulate the main conjecture for E/H. Before doing this, in the next chapter we will construct the p-adic L-functions that appear in the statement of the main conjectures. Thanks to the unique scaling factor cE(a) in our definition of the rational functions, it can be shown that this constant is equal to 1.
We fix a generator f of the ideal f, and define Q = Φ(Ω∞/f, L) such that Q is a primitive f-subpoint on E. The statement of the lemma now follows immediately, after noting that cE( a) is a unit at v. Let I be the ring of integers of the completion of the maximum unbranched extension Kpur of Kp.
Considering z as a parameter of the formal additive group, exp(z, Lσ) is an exponential mapping of Ebσ. Then follows the uniqueness of exponential mappings for formal groups which.
Construction of the p-adic L-function for H ∞ /H
Finally, we note that our methods easily give an analogue of Theorem 4.1.11 for the p-adic interpolation of L(ψkE/H, k) when k varies over any fixed residue class modulus. This will create the p-adic function L for H∞/H for all p, and as we will see in Chapter 7, will be an essential ingredient for the key conjecture for E/H for p= 2. We claim that we we can remove this factor and obtain a pseudo-measure which is independent of D.
In fact, we have χkp(θD) = ck(D)h, so it remains to show that θD|Γ generates the augmentation ideal of I[[Γ]]. There exists a unique elementνp belonging to the quotient field ΛI(G) such that we have for all integers k >1 with k ≡0 mod #(∆). Recall from section 4.1 that we can decompose νp as a sum of elements ineχΛI(Γ) if we additionally assume that (p, h) = 1.
In Section 4.1 we showed that the p-adic L-function µE ∈ΛI(G) interpolates the values of L(ψkE/H, k) when k is odd, and ofLS(ψkE/H, k) when k is even, where S is the set of primes of H that divide f. Then Ψp interpolates the values of L(ψkE/H, k) fork spanning all the residue classes modulo #(∆) in the following way.
Elliptic Units
However, we can choose any prime qdivision f, then q̸=p and we can apply the same argument by writing Pn⊕Qas a point sum of the q-power division and a point W ∈Eb with (b,q) = 1. Let UHn denote the set of semilocal units of Hn⊗KKp =⊕P|pHn,P which are congruent to 1 modulo primes over p. We denote by UH∞ the projective limit of the groups UHn in relation to the rate maps.
We define the group of elliptic units CHn as the group generated by RσD(Pnσ) for all σ ∈G, where Pnσ is a primitive pn subpoint onEσ, asD runs over the index setI. Similarly, let UFn denote the group of semi-local units of Fn⊗K Kp = ⊕P|pFn,P that are congruent to 1 modulo the primes abovep, and denote by UF∞ the projective limit of the groups UFn with respect to the standard cards. Let CFn denote the group generated by:=Rσa(Pnσ) for all σ ∈G, ¯CFn the termination of CFn in UFn, and write.
Seeing this, the given element in the annihilator is a finite sum, so we can eliminate terms involving elements of Gn by subtracting terms of the form σ −Nσ. Thus, with the help of the Čebotarev density theorem, we can choose another primeq' of Fn, which is mapped to 1 +N and corrects µM.
Statement of the Main Conjecture for H ∞ /H
So if we denote by Dn the subgroup of X(Hn) generated by the factorization group of the primes of Hn above p, we have an exact series. Furthermore, the class field theory identifies A(H∞) with Gal(L(H∞)/H∞), where A(H∞) denotes the inductive limit of A(Hn) taken with respect to the natural maps resulting from the recording of fields. In this section we treat the Iwasawa modules appearing in the fundamental exact sequence 4.4.2 as Λ(G) =Zp[[G]] modules.
Given a finitely generated torsion Zp[[G]]-moduleM, write charΛ(M) for the characteristic ideal M given by the structure theorem for finitely generated torsional Λ(G)χ ≃ Zp[[Γ]]-modules, and charΛ( M)χ for the characteristic ideal Mχ as a Λ(G)χ-module. The purpose of this chapter is to define and study the Euler system of elliptic units ¯CH∞ defined in Chapter 4 for the tower H∞/H. Furthermore, HnK(q)/Hn is completely branched on primes above q and unbranched everywhere else, so the statements of the lemma follow.
Then the submodule Zp ¯EF of UF generated by the image of EF into UF has rank Zp r1+r2−1−vF for an integer vF >0. For ease of notation, given a Λ(G)-module, let Y(n) denote the quotient Y /I(Hn)Y and let πY denote the map Y(n)→YΓn induced by the map of projection.
An Application of the Čebotarev Density Theorem
Recall that τq is a fixed generator of the cyclic group Gq, and let x1−τq denote the image of x1−τq modQ inside (OHn/Q)×/((OHn/Q)×)pℓ. Let b be the element of peGal(Hn′(v1/pℓ)/Hn′) corresponding to β via the Kummer map such that. Furthermore, I(G) annihilates Gal(Ln∩Hn′/Hn), since Hn′ is abelian over H, so we can consider c as an element of peGal(Ln/L′n).
By the Čebotar density theorem, there are infinitely many primitives Hn of degree one, unbranched in Hn′(v1/pℓ)/K, whose Frobenius in Gal(LnHn′(v1/pℓ)/Hn) is equal to ρ. First, the fact that Q has first degree and ρ fixes L′n implies that q splits completely into Hn′ and thus q∈ Iℓ.
The Inductive Argument
Given a finitely generated torsion ΛI(G)-module M, we recall from Section 4.4 that char(M)⊂ΛI(G) denotes a characteristic ideal of M. This is justified since we are only interested in char(X)χ and the characteristic ideals of Γ -modules behave well when expanding scalars. The invariants of Iwasawa Λ(G)-modules are defined similarly, and if M = X⊗bZpI is obtained from a Λ(G)-module X by scalar expansion on I, the invariants of M coincide with those of X.
Recall from the proof of Theorem 5.1.5 that X(H∞)/I(Hn)X(H∞) equals Gal(M(Hn)/H∞), where M(Hn) is the largest abelian p-extension of Hn , which is unbranched outside the prime numbers over p. Let UHn,P denote the group of units in the complement of Hn at P that are consistent with 1 modulo P, and let >0 such that p−tOP ⊂logUHn,P for every P∈P. Further, let ∆Hn/K denote the discriminant of Hn/K and choose the generator ∆n of the ideal ∆Hn/KOp.
Let M(Hn) be the maximal abelian p-extension of Hn which is unbranched outside the primes in P. Thus we can identify Gal(L(Hn)/Hn) with A(Hn), the p-prime part of the ideal class, the group of Hn.
The Iwasawa Invariants of the p-adic L-function
As discussed at the beginning of Section 6.1, this together with the divisibility relation obtained in Theorem 5.3.4 completes the proof of the main conjecture for H∞/H. In this section we briefly discuss the main assumption for E/H and its connection with the main assumption for H∞/H that we proved in Chapter 6. Then a generator of the characteristic ideal of Y∞ is given by fY∞(χp(γ) )(1 +T)−1), where γ is the fixed topological generator of Γ.
A rational prime p is a special partitioned prime if and only if it divides into K, andψ(p)≡ ±1 against 4for both primespofK above p. Now let p be any prime that divides into K, and let p be one of the prime ideals of K above p. Then the Frobenius automorphism of K acts on E[4] by multiplication by ψ(p), thanks to the main complex multiplication theorem.
Recall from the theory of complex multiplication that for a prime ideal p of K prime to 3, we have ψE/K(p) =π, where π is the unique generator of p, namely 1 mod 3OK. On the "2-part" of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication.