III 2-class groups of CM fields 125
8.4 Proof of the Structure Theorem
haveλ−= 1. Thus,λ=λ−= 1 and the first claim follows
For the second claim recall from Lemma 8.2.2 that λ(Ad−∞) = λ(A−∞). Note that the groups Ac−n ∩A+n are of exponent 2. So if we know that the 2-class group of Ln,d is cyclic and that λ(Ad−∞) = 1, then A+n contains at most 2 elements. As the capitulation kernelAn(L+n,d)→ An(Ln,d) contains at most 2 elements due to [Wash, Theorem 10.3], we see that the 2-class group ofL+n,dis uniformly bounded.
In order to prove our main result (Theorem 8.1.1) we will also need [C-A-Z, Theorem 5] and [C, Theorem 1] which are summarized in the following Theorem.
Theorem 8.3.6. Let n≥1 and assume thatd takes one of the following forms:
a) d=pq, for two distinct primes p and q congruent to 3 (mod 8).
b) d, is a prime congruent to 9 (mod 16).
Then the rank of the 2-class group of Ln,d is2.
8.4 Proof of the Structure Theorem
We will first compute the cardinality of h2(L1,d) and then use this result to prove Theorem 8.1.1.
Lemma 8.4.1. Let dbe an odd positive square-free integer. We have:
a) h2(L1,d) = 2·h2(−d), if d = pq, for two distinct primes p ≡ 5 (mod 8) and q≡3 (mod 8).
b) h2(L1,d) = h2(−2d), if d= pq, for two distinct primes p ≡ q ≡3 (mod 8) or d=p for a primep such that p≡9 (mod 16) and (2p)4= 1.
Proof. The proof is basically a computation in units and class number formulas.
a) Denote by ε2pq the fundamental unit of the quadratic fieldQ(√
2pq). We have ε2pq =x+y√
2pqfor some integers x andy. Sinceε2pq has a positive norm we obtain x2−2pqy2 = 1. Thusx2−1 = 2pqy2. Set y=y1y2 for yi∈Z. Assume which is impossible. Sox±1 is not square inN. From the third and the fourth item of [A-Z-T, Proposition 3.3], we deduce thatq(L1,d) = 4. By Kuroda’s class
number formula (cf. [Wa, p. 201]), we have h2(L1,d) = 1
25q(L1,d)h2(pq)h2(−pq)h2(2pq)h2(−2qp)h2(2)h2(−2)h2(−1)
= 1
25q(L1,d)h2(pq)h2(−pq)h2(2pq)h2(−2qp)
= 1
25 ·4·2·h2(−pq)·2·4
= 2·h2(−pq), which proves the first claim.
b) Suppose now that dis the product of two primes p and q that congruent to 3 mod 8 then we obtain the desired result by [A-C-Z2, Corollary 2]. Ifd=p is a prime congruent to 9 (mod 16) the result follows from the proof of Theorem 1 of [A-C-Z2, p. 7].
Now we have all ingredients to provide a partial proof of our first main Theorem:
Theorem 8.4.2. Let d be of one of the following forms:
d=p be a prime congruent to9 (mod 16) and assume that(2p)4= 1.
d=pq for two primes congruent to3 (mod 8).
Let2r=h2(−2d).Then for n≥1 the2-class group ofLn,d is isomorphic to the group Z/2Z×Z/2n+r−2Z. In the projective limit we obtainZ2×Z/2Z.
Note that Theorem 8.4.2 is point a) of Theorem 8.1.1
Proof. By Theorem 8.3.6 we know that the 2-rank of the 2-class group ofLn,dequals 2 forn≥1. Furthermore, we have λ− = 1 due to Theorem 8.3.2, andh2(L1,d) = 2r by Lemma 8.4.1. By Theorem 8.3.3 the class number ofL+n,dis odd for alln. As there is no capitulation inA−n according to Theorem 7.3.1 andλ−= 1, we see thatA−n has rank one for n large enough (see also Lemma 8.2.1). This implies that the second generator of the 2-class group ofLn,d is a class of a ramified prime in Ln,d/L+n,d. As the class number of L+n,d is odd, these ramified classes have order 2 and we obtain that the 2-class group ofLn,d is isomorphic toZ/2Z×Z/2lnZ.
LetE be the elementary Λ-module associated toA∞. Then according to [Wash, page 282-283]νn,0E= 2νn−1,0E for all n≥2. Indeed, νn,0=νn,n−1νn−1,0. AsE has Z2-rank 1, we know that T acts as 2v on E for some v ∈ Z2. For n ≥ 2 we have νn,n−1 = (T + 1)2n−1−1 + 2. Forn≥2 the term (T + 1)2n−1 −1 = T2n−1 +O(2T) acts as 4v0 on E for some v0 ∈ Z2. Hence, νn,n−1 acts as 2(1 + 2v0) = 2u on E for some unitu∈Z2 and all n≥2.
Consequently,
|E/νn,0E|=|E/2n−1E||E/ν1,0E|= 2n−1+c0
8.4. PROOF OF THE STRUCTURE THEOREM 145 for n ≥ 1 and some constant c0 ≥ 1 independent of n. Note that we can rewrite this as |E/νn,0E| = 2n+c. Since E has only one Z2-generator we can assume that the pseudo-isomorphism φ : A∞ → E is surjective. The maximal finite submodule of A∞ is generated by the classes (cn)n∈N of the ramified primes above 2. Let τ be a generator of Gal(Ld,∞/L0,d). Then τ(cn) = cn as the primes above 2 are totally ramified in L∞,d/L0,d. It follows that T cn = 0. Using that νn,0 is coprime to the characteristic ideal of A∞ for all n, we obtain for every n ≥ 1 that the kernel of φ : A∞/νn,0A∞ → E/νn,0E is isomorphic to the maximal finite submodule in A∞
and contains 2 elements. Let Y be such that A∞/Y ∼= A0. Then An ∼= A∞/νn,0Y (see [Wash, page 281]). We obtain
|An|=|A∞/νn,0Y|=|A∞/νn,0A∞||νn,0A∞/νn,0Y|= 2n+c+1|νn,0A∞/νn,0Y|forn≥1.
As the maximal finite submodule inA∞is annihilated byνn,0, we see that the size of νn,0A∞/νn,0Y is constant independent ofn. Hence, we obtain that the 2-class group of Ln,d is of size 2n+ν for all n≥1. Using that h2(L1,d) = 2r, we obtain ν =r−1.
This yields 2·2ln = 2n+r−1 and we obtain ln = n+r−2. Noting that Ln,d is the n-th step of the cyclotomic Z2-extension of L0,d finishes the proof of the first claim.
As the direct term Z/2Z is norm coherent the second claim is immediate.
Corollary 8.4.3. Let d be of one of the following two forms:
a) d=p a prime congruent to9 (mod 16) and assume that(2p)4= 1, b) d=pq for two distinct primes congruent to 3 (mod 8).
Ifdtakes the first form, setp=u2−2v2 whereu andv are two positive integers such thatu≡1 (mod 8).
Ifdtakes the second form set(pq) = 1and let the integersX, Y, k, l andm be such that 2q=k2X2+ 2lXY + 2mY2 and p=l2−2k2m (see [Ka, p. 356] for their existence).
For all n≥1, we have:
a) Ifdis of the first form, then the 2-class group of Ln,d is isomorphic toZ/2Z× Z/2n+1Z, if and only if (up)4=−1.
Outside of this particular case it is isomorphic to the groupZ/2Z×Z/2n+r−2Z, where r≥4 was defined in Theorem 8.4.2.
b) If d is of the second form then the 2-class group of Ln,d is isomorphic to the groupZ/2Z×Z/2n+1Z if and only if (|X+lY−2 |) =−1.
Outside of this case it is isomorphic to Z/2Z×Z/2n+r−2Z, where r ≥ 4 was defined in Theorem 8.4.2.
Proof. By Lemma 8.4.1 we know that h2(L1,d) = h2(−2d). Since the 2-rank of Cl(L1,d) equals 2 and |Cl2(Ln,d)| 6= 4 (see [A-C-Z1, Theorem 5.7]) it follows that h2(−2d) is divisible by 8. Thus if dis as in point a) [L-W 1, Theorem 2] shows that h2(−2d) is not divisible by 16 and we obtain r= 3.
If d is in the second case then [Ka, pp. 356-357]) implies that h2(−2d) is not divisible by 16 and againr= 3.
Let us give the following example for Corollary 1.
Example 8.4.4. Set p = 89, u = 17 and v = 10. We have p =u2−2v2 and 2
p
4 = −
u p
4 = 1. So the 2-class group of Ln,p is isomorphic to Z/2Z× Z/2n+1Z for all n≥1.
Let p = 11, q = 19, k = 1, l = 3, m = −1, X = 4 and Y = 1. We have:
p=l2−2k2m and2q=k2X2+ 2lXY + 2mY2. Since −2
|X+lY|
= −27
=−1.
By Corollary 8.4.32-class group of Ln,p is isomorphic to Z/2Z×Z/2n+1Z for alln≥1.
Now we finish the proof of Theorem 8.1.1.
Theorem 8.4.5. Assume that d = pq is the product of two primes p ≡ −q ≡ 5 (mod 8)and2r= 2·h2(−pq). Then forn≥1the 2-class group of Ln,dis isomorphic toZ/2n+r−1Z.
Note that this is point 2 of Theorem 8.1.1.
Proof. We know already from Theorem 8.3.4 that the 2-class group of Ln,d is cyclic, and by Theorem 8.3.2 we know thatλ(L0,d) = 1. In particular, the module A∞ does not contain a finite submodule and is hence isomorphic to its elementary moduleE.
LetY be defined as in the proof of Theorem 8.4.2, then there is no νn,0-torsion and we obtain that the size ofνn,0A∞/νn,0Y is constant independent of n. As before we obtain |An|=|A∞/νn,0A∞||νn,0A∞/νn,0Y|= 2n+d.In particular, Iwasawa’s formula holds for alln≥1. Hence, h2(L1,d) = 2r= 21+ν and ν =r−1. From this the claim follows.
Corollary 8.4.6. Let d = pq be the product of two primes p and q such that p ≡
−q ≡5 (mod 8). Then for all n≥1, we have
a) If (pq) =−1, the 2-class group of Ln,d is isomorphic to Z/2n+1Z.
b) If (pq) = 1 and (qp)4= 1, the 2-class group of Ln,d is isomorphic to Z/2n+2Z. Outside of these two cases, the 2-class group of Ln,d is isomorphic to Z/2n+r−1Z, where r≥4 was defined in Theorem 8.4.5.
Proof. Assume that d is of the first form. By [Co-Hu, 19.6 Corollary] we have h2(−pq) ≡ 2 (mod 4). Hence, r = 2. If d is of the second form we know that Cl2(Q(√
−d)) is cyclic and divisible by 4 [Qi 1, page 1427]. Asp≡5 mod 8 we see that (2p) = −1 and therefore (4p)4 = −1. Then (4qp) = −1 and [Qi 1, Theorem 3.9]
implies thath2(−pq) is not divisible by 8. Hence, h2(−pq) = 4 andr = 3.
For the above corollary we provide the following Example 8.4.7. Let d= 13·19. We have 1319
=−1. So the 2-class group of Ln,p is isomorphic to Z/2n+1Z for all n≥1.
8.5. APPLICATIONS 147
Let d= 5·11. We have 115
= 1 and 115
4 = 1. So the 2-class group of Ln,p
is isomorphic to Z/2n+2Z for all n≥1.
Let nowX0,Y0 andZ be three positive integers satisfying the Legendre equation
pX02+qY02−Z2= 0 (8.1)
such that
(X0, Y0) = (Y0, Z) = (Z0, X0) = (p, Y0Z) = (q, X0Z) = 1, (8.2) and
X0 odd, Y0 even andZ ≡1 (mod 4). (8.3) (see [L-W2] for more details)
Corollary 8.4.8. Let d = pq be the product of two primes p and q satisfying p ≡
−q ≡ 5 (mod 8), (pq) = 1 and (−qp )4 = 1. Let X0, Y0 and Z be three positive integers satisfying equation(8.1)and the conditions(8.2)and (8.3). If (Zp)4 6= (2XZ0), the 2-class group of Ln,d is isomorphic to Z/2n+3Z. Otherwise, it is isomorphic to Z/2n+r−1Z, for r ≥5 defined as in Theorem 8.4.5.
Proof. It is immediate that the assumptions of Corollary 8.4.6 are not satisfied.
Hence, we have r ≥ 4. By [L-W2, Theorem 2] h2(−pq) is divisible by 16 if and only if
Z p
4 =
2X0 Z
. Hence if (Zp)46= (2XZ0) we get 4≤r <5. Otherwise we obtain that 2·16|2r andr ≥5.
Now we close this section with some numerical examples illustrating the above corollary
Example 8.4.9. Let p = 5, q = 19 and d =−pq. Then X0 = 1, Y0 = 2 and Z = 9 are solutions of equation (8.1) satisfying the condition (8.2) and (8.3).
Furthermore,(95)4=−(29) =−1. Thus, the 2-class group of Ln,d is isomorphic toZ/2n+3Z.
Let p = 37, q = 11 and d = −pq = −407. Then X0 = 1, Y0 = 56518 and Z = 187449 are solutions of equation (8.1) satisfying the condition (8.2) and (8.3). Furthermore, (18744937 )4 = (1874492 ) = 1. Thus, the 2-class group of Ln,d is isomorphic toZ/2n+r−1Z for somer ≥5. Indeed with these settingsr = 5 (see [L-W2, p. 230]).
8.5 Applications
Note that Theorem 7.3.1 and Theorem 7.2.5 only hold for CM fields containing the 4-th root of unityi. Therefore, we cannot compute the 2-class groups of layers of the cyclotomic Z2-extension of imaginary quadratic fields using the same techniques as in the proof of Theorem 8.1.1. But we can still use Theorem 8.4.2 to deduce results on the class field tower of imaginary quadratic fields.
Ln,d
Kn L+n,d Kn,d
K+n
Figure 8.1: Subfields ofLn,d/K+n.
Theorem 8.5.1. Letdbe a positive square-free integer andr such that2r=h2(−2d).
Let K0,d=Q(√
−d) and denote by Kn,d the n-th step of the cyclotomic Z2-extension of K0,d. Suppose that dtakes one of the following forms:
a) d=pq, for two distinct primes p and q congruent to 3 (mod 8).
b) d=p, for a prime p≡9 (mod 16) such that that (2p)4 = 1.
Then for alln≥1 the 2-class group of Kn,d is isomorphic to Z/2Z×Z/2n+r−1Z. In the projective limit we obtainZ2×Z/2Z.
Note that this is point 1 of Theorem 8.1.2.
Proof. LetKn=Q(ζ2n+2) andKn,d=Q(ζ2n+2+ζ2−1n+2,√
−d) =K+n(√
−d) (see Figure 8.1). By the class number formula (cf. [Le 1, Proposition 3 and equation (1)]) we have:
h2(Ln,d) = QLn,d
QKnQKn,d · µLn,d
µKnµKn,d ·h2(Kn)h2(Kn,d)h2(L+n,d) h2(K+n)2 .
It is known thath2(Kn) = 1 and by Theorems 8.3.3 and 8.4.2 we have h2(L+n,d) = 1 andh2(Ln,d) = 2n+r−1, respectively. Therefore
2·2n+r−1= QLn,d
QKnQKn,d ·h2(Kn,d). (8.4) It is also known thatQKn = 1. Let k=Q(i,√
d). As the natural norm NL1,d/k :WL1,d/WL2
1,d →Wk/Wk2
is surjective, we obtain that QL1,d divides Qk (cf. [Le 1, Proposition 1]). Since Qk = 1 (cf. [Az, p. 19] and the proof of [A-C-Z2, Lemma 4]), we have QL1 = 1.
SinceNLn,d/Ln−1,d :WLn,d/W2
Ln →WLn−1/W2
Ln−1 is onto, it follows that QLn,d divides QLn−1,d. Thus, by inductionQLn,d = 1.
Note that the extensions Kn,d are essentially ramified (cf. [Le 1, p. 349]) for all n ≥ 1. Since µKn,d = 2 we obtain QKn,d = 1 by [Le 1, Theorem 1] . Hence, h2(Kn,d) = 2n+rfor alln≥1. Since the rank of the 2-class group ofK1,dequals 2 (cf.
[M-C-R, Proposition 4]) and the 2-class group of Kn,d is of type (2,2•) for n large enough (cf [Miz, p. 119]), we achieve the result.
8.5. APPLICATIONS 149 Now using Theorem 8.4.5 we can finish the proof of Theorem 8.1.2.
Theorem 8.5.2. Assume that d = pq is the product of two primes p ≡ −q ≡ 5 (mod 8)and 2r= 2·h2(−pq). LetK0,d =Q(√
−d) and denote for n≥3 by Kn,d the n-th step of the cyclotomic Z2-extension of K0,d. Then forn≥1the 2-class group of Kn,d is isomorphic to Z/2n+r−1Z.
Note that this is point 2 of Theorem 8.1.2
Proof. We keep similar notations and proceed as in the proof of Theorem 8.5.1. Note that by [A-C-Z2, Proposition 3], we haveh2(L+n,d) = 2. So as above we obtain
h2(Ln,d) = QLn,d
QKnQKn,d · µLn,d
µKnµKn,d ·h2(Kn)h2(Kn,d)h2(L+n,d) h2(K+n)2 · Thus,
2n+r−1 = 1
1·1· 2n
2n·2 ·1·h2(Kn,d)·2
1 ·
It follows that h2(Kn,d) = 2n+r−1 for all n. Since Ln,d/Kn,d is ramified, it is obvi-ous that 2-rank(Cl(Kn,d)) ≤ 2-rank(Cl(Ln,d)) = 1 (Theorem 8.4.5). In particular, Cl(Kn,d) is cyclic.
Bibliography
[Ax] Ax, J.,(1965)On the units of an algebraic number field Illinois Journal of Math-ematics, 9, pp. 584-589.
[Az] Azizi, A., (1999) Unit´es de certains corps de nombres imaginaires et ab´eliens sur Q,Ann. Sci. Math. Qu´ebec, 23, 15-21.
[A-C-Z1] Azizi, A., Chems-Eddin, M.M. and Zekhnini, A., On the rank of the 2-class group of some imaginary triquadratic number fields. arXiv:1905.01225v3. To appear in Rendiconti del Circolo Matematico di Palermo series 2.
[A-C-Z2] Azizi, A., Chems-Eddin, M. M. and Zekhnini, A., On some imaginary tri-quadratic number fields k with cl2(k) = (2,4) or (2,2,2) arXiv:2002.03094. To appear in Commentationes Mathematicae Universitatis Carolinae.
[A-Z-T] Azizi, A., Zekhnini, A. and Taous, M. (2016) On the strongly ambiguous classes of some biquadratic number fields.Math. Bohem.,141 , 363–384.
[Ba-Sa] Barman, R. and Saikia A. (2010)A note on Iwasawaµ-invariants of ellipitc curvesBull Baz. Math Soc. New Series 41(3), pp. 399-407.
[B-G-S] Bernardi, D., Goldstein, C., Stephens, N. (1984). Notes p-adiques sur les courbes elliptiques: J. Reine Angew. Math. 351, pp. 129-170.
[Bl] Bley, W. (2006) Equivariant tamagawa conjecture for abelian extensions of a auadratic imaginary field: Dokumenta Mathematica 11, pp 73-118
[Br] Brumer, A. (1967)On the units of algebraic number fields Mathematika 14(2), pp. 121-124.
[C] Chems-Eddin, M. M., The rank of the 2-class group of some number fields with large degree. arXiv:2001.00865v2.
[C-A-Z] Chems-Eddin, M. M, Azizi, A. and Zekhnini, A., On the 2-class group of some number fields with large degree. arXiv:1911.11198. To appear in Algebra colloquium
[C-K-L] Choi, J., Kezuka, Y., Li, Y. (2019). Analogues of Iwasawa’s µ= 0 conjec-ture and weak Leopoldt theorem for certain non-cyclotomic Z2-extensions: Asian Journal of Mathematics, Vol. 23, No. 3, pp 383-400
151
[Co] Coates, J. (1991).Elliptic curves with complex multiplication and Iwasawa the-ory: Bull. London Math. Soc. 23, pp. 321-350.
[Co-Go] Coates, J., Goldstein, C. (1983). Some remarks on the main conjecture for elliptic curves with complex multiplication: American J. of Mathematics 105, pp.
337-366.
[Co-Wi 1] Coates, J., Wiles, A. (1977). Kummer’s criterion for Hurwitz numbers:
Algebraic number theory, Kyoto 1976, Japan Society for the Promotion of Science, pp. 9-23.
[Co-Wi 2] Coates, J., Wiles, A. (1977). On the conjecture of Birch and Swinnerton-Dyer: Invent. Math., 39, pp. 223-251.
[Co-Wi 3] Coates, J., Wiles, A. (1978). On p-adic L-functions and elliptic units: J.
Australian Math, Soc. 26, pp. 1-25.
[Col] Coleman, R. (1979). Division values in local fields: Invent. Math., 53, pp. 91-116.
[Co-Hu] Conner, P. E. and Hurrelbrink, J., (1988)Class number parity Ser. Pure Math., vol.8, World Scientific, Singapore .
[Cr] Cri¸san, V. (2019) The split prime µ conjecture and further topics in Iwasawa theoryPhD Thesis University G¨ottingen.
[Cr-M] Cri¸san, V., M¨uller, K. (2020)The Vanishing of theµ-Invariant for Split Prime Zp-extensions over Imaginary Quadratic Fields The Asian Journal of Mathemat-ics, 24(2), 267-302.
[dS] de Shalit, E. (1987) The Iwasawa theory of elliptic curves with complex multi-plication: Perspect. Math. Vol.3.
[Fe] Federer, L. (1982)p-adic L-functions, regulators and Iwasawa modulesPhd The-sis, Princton.
[Fe-Gr] Federer, L., Gross, B.H: (1981) Regulators and Iwasawa modules(1981) In-vent. Mathematicae 62, pp. 443-457.
[Fe-Wa] Ferrero, B., Washington, L.C. (1979). The Iwasawa invariant µp vanishes for abelian number fields: Ann. of Math. (2) 109, no. 2, pp. 377-395.
[Fr] Fr¨ohlich, A., (1983) Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields, Contemporary Mathematics vol. 24American Mathematicla Society, Providence .
[Gil 1] Gillard, R. (1985). Fonctions L p-adiques des des corps quadratiques imagi-naires et de leurs extensions ab`eliennes: J. Reine Angew. Math. 358, pp. 76-91.
BIBLIOGRAPHY 153 [Gil 2] Gillard, R. (1987). Transformation de Mellin-Leopoldt des fonctions
ellip-tiques: J. Number Theory 25, no. 3, pp. 379-393.
[Go-Sch] Goldstein, C., Schappacher, N. (1981). S´eries d’Eisenstein et fonctions L de courbes elliptiques `a multiplication complexe: Journal f¨ur die Reine und Ange-wandte Mathematik 327, pp. 184-218 .
[Gre 1] Greenberg, R. (1971) On some questions concerning the Iwasawa invariants Princton University Thesis.
[Gre 2] Greenberg, R. (1973) On a certain l-adic representation Inventiones math.
21 pp. 117-124.
[Gre 3] Greenberg, R. (1976)On the Iwasawa invariants of totally real number fields American Journal of Mathematics, Vol.98, No1. pp. 263-284.
[Gre 4] Greenberg, R. (1978). On the structure of certain Galois groups: Invent.
Math., 47, pp. 85-99.
[Gre 5] Greenberg, R. (2003)Galois theory of the Selmer group of an abelian variety Compositio Mathematicae (136) pp. 255-297.
[Gre 6] Greenberg, R.Topics in Iwasawa theoryhttps://sites.math.washington.
edu/~greenber/book.pdf
[Gre-Vat] Greenberg, R. and Vatsal, V. (2000)On the Iwasawa inariants of elliptic curvesInvent. Math. 142 , pp 17-63.
[Gr] Greither, C. (1992).Class groups of abelian fields, and the main conjecture An-nales de L’Institut Fourier 42, no 3, pp 449-499.
[Gro 1] Gross, B.H. (1981) p-adic L-series at s = 0 J. Math Sci. University Tokyo Sec. Math 28(3), pp. 979-994.
[Gro 02] Gross, B. (1980) Arithmetic on elliptic curves with complex multiplication:
Lecture Notes in Mathematics 766, Springer-Verlag. Berlin Heidelberg.
[Gu] Guillott, P. (2018) A Gentle Course in Local Class Field Theory Cambridge University Press, Cambridge.
[Ho-Kl] Hofer, M. and Kleine, S.(2020)On the Gross Order of Vanishing Conjecture for Large Vanishing Orders, preprint arXiv:2102.01573.
[Ha-Le] Hatley, J. and Lei, A. (2019)Arithmetic Properties of Signed Selmer Groups at Non-Ordinary Primes Annales de l’Institut Fourier, Tome 69 no 3, pp. 1259-1294.
[Ha-Le-Vi] Hatley J. Lei, A. and Vigni S. (2020) On Λ-Submodules of Finite Index of Anticyclotomic Plus an Minus Selmer Groupspreprint arXiv:2003.10301
[Iw 1] Iwasawa, K., (1959) On Γ-extensions of algebraic number fields, Bull. Amer.
Math. Soc.,65 , 183–226.
[Iw 2] Iwasasawa, K. (1973) On Zl-extensions of algebraic number fields Annales of Mathematics, 2nd series, 98 (2) pp. 246-326.
[Iw 3] Iwasawa, K. (1986)Local class field theoryOxford mathematical Monographs, Oxford University Press, New York.
[Ja] Janusz, G. J. (1973), Algebraic number fields Pure and Applied Mathematics Volume 55, !st edition, Academic Press, New York.
[Jau 1] Jaulent, J.-F. (2002)Classes logarithmiques des corps totalement r´eels Acta Arith., 103(1), pp. 1-7.
[Jau 2] Jaulent, J.-F. (2017) Normes cyclotomiques na¨ıves et unit´es logarithmiques Acta Math., 108(6) pp. 545-554.
[Ka] Kaplan, P., (1976)Sur le 2-groupe des classes d’id´eaux des corps quadratiques, J. Reine Angew. Math., 283-284, 313–363.
[Kat] Kato, K. (2004) P-adic hodge theory and values of Zeta-functions of modular forms Ast´erisque, tome 295, pp.117-290.
[Ke 1] Kezuka, Y. (2017) On the Main Conjecture of Iwasawa theory for certain elliptic curves with complex multiplication, PhD-Thesis, University of Cambridge, Cambridge.
[Ke 2] Kezuka, Y. (2019)On the Main conjecture of Iwasawa theory for certain non-cyclotomic Zp-Extension J. London Math. Soc., Vol. 100, pp. 107-136.
[Ki] Kida,Y. Cyclotomic Z2-extensions of J-fields,(1982) J. Number Theory, 14 , 340-352.
[Kim] Kim, B.D. (2009)The Iwasawa invariants of Plus/Minus Selmer groupsAsian Journal of Mathematics, 132 pp.181-190.
[Kl 1] Kleine, S. (2015) A new approach to the investigation of Iwasawa invariants PhD-Thesis University G¨ottingen.
[Kl 2] Kleine, S. (2019)T-ranks of Iwasawa modules Journal of Number Theory, Vol 196, pp. 61-86.
[Ku-La] Kubert, D.S., Lang, S. (1981).Modular Units, Grundelehren der mathema-tischen Wissenschaften 244: Springer.
[Kum] Kumakawa M. (2016)Greenberg’s Conjecture for the cyclotomicZ2-extension of Certain Number Fields of Degree 4 Tokyo J. Math Vol 39 Nr 1.
[Ku 1] Kuz’min, L.v. (1972)The Tate module for algebraic number fields Izv. Akad.
Nauk SSSR Ser. Mat. 36(2), pp. 267-327.
BIBLIOGRAPHY 155 [Ku 2] Kuz’min, L.v. (2018) Local and global universal norms in the cyclotomic Zp
extension of an algebraic number field Izvestiya: Mathematics, Volume 82, Issue 3, pp. 532-548.
[La] Lang, S. (1990) Cyclotomic Fields I and II. With an Appendix by Karl Rubin 2nd edition, New York, Springer.
[Le 1] F. Lemmermeyer,(1995)Ideal class groups of cyclotomic number fields I,Acta Arith., 72 , 347–359..
[Le 2] Lemmermeyer, F. (2013)The ambigous class number formula revistedJournal of the Ramanujan Math. Soc. Vol. 28 (4), pp. 415-421.
[L-W 1] Leonard, P. A. and Williams,K. S. (1982)On the divisibility of the class numbers of Q(√
−d) andQ(√
−2d) by 16, Can. Math. Bull., 25, 200-206.
[L-W2] Leonard, P. A. and Williams,K. S. (1983)On the divisibility of the class num-ber of Q(√
−pq) by 16, Proc. Edinburgh Math. Soc.,26 , 221-231.
[Li-Mu] Lim, M.F. and Murty, V.K. (2016)The growth of fine Selmer GroupsJournal of the Ramanujan Math Soc. Voulme 31, Issue 1, pp. 79-94.
[L-Y-X-Z] Li,J., Ouyang Y., Xu, Y.and Zhang,S.l-class groups of fields in Kummer towers, arXiv:1905.04966.
[Lu] Lubin, J. (1964). One parameter formal Lie groups over p-adic integer rings:
Annals of Mathematics, Second Series, Vol. 80, No. 3, pp. 464-484.
[Ma] Matar, A. (2020)On the Lambda-Cotorsion subgroup of the Selmer groupAsian Journal of Mathematics, Volume 24, no 3, pp. 437-456.
[M-C-R] McCall, T. M, Parry,C. J., Ranalli,R. R. (1995) Imaginary bicyclic bi-quadratic fields with cyclic 2-class group, J. Number Theory, 53 , 88-99.
[Mil] Milne, J.S. (2006) Arithmetic Duality TheoremsBookSurge, LLC, Charleston, SC.
[Miz] Mizusawa, Y. (2010)On the maximal unramified pro-2-extension over the cy-clotomic Z2-extension of an imaginary quadratic field, J. de Theor. des Nr. de Bordeaux, 22, 115-138.
[Mu] K. M¨uller. (2019)Capitulation in theZ2 extension of CM number fields, Math.
Proc. Camb. Phil. Soc., 166, 371-380.
[Ne 1] Neukirch, J. (1999)Algebraic Number Theory, Grundlehren der mathematis-chen Wissenschaften (A series of comprehensive studies in mathematics) volume 322, Springer, Berlin-Heidelberg.
[Ne 2] Neukirch, J. (2013) Class field Theory, Springer Berlin-Heidelber
[Ne-Sc-Wi] Neukirch, J. , Schmidt, A. and Wingberg, A. (2008)Cohomology of Num-ber Fields, Grundlehren der mathematischen Wissenschaften (A series of compre-hensive studies in mathematics) volume 323 2nd edition, Springer, Beerlin Hei-delberg.
[Ou] Oukhaba, H. (2012)On Iwasawa Theory of elliptic units and 2-class groups J.
Ramanujan Math. Soc. 27, no 3, pp 255-373.
[O-V] Oukhaba, H., Vigui´e, S. (2016) On theµ-invariant of Katzp-adic L-functions attached to imaginary quadratic fields: Forum Math. 28, no. 3, pp. 507-525.
[Oz-Ta] Ozaki, M. and Taya, H. (1997) On the Iwasawa λ2-invariants of certain families of real quadratic fieldsmanuscripta math. 94, 437-444.
[Qi 1] Yue,Q.(2011)8-ranks of class groups of quadratic number fields and their den-sities, Acta Math. Sin. 17 , 1419–1434.
[Qi 2] Q. Yue, (2009)The generalized R´edei-matrix, Mathematische Zeitschrift 261, 23–37.
[Ra-Wi] (2018) Ramdorai, S. and Witte, M. Fine Selmer Groups and Isogeny in-variance In: Akbary A., Gun S. (eds) Geometry, Algebra, Number Theory, and Their Information Technology Applications. GANITA 2016. Springer Proceedings in Mathematics & Statistics, vol 251. Springer.
[Ra-Ra] (2020) Ray, A. and Ramdorai, S. Euler Characteristics and their Congru-ences for Multisigned Selmer groups preprint arXiv:2011.05387.
[Ro] Robert, G. (1973).Unit´es elliptiques et formules pour le nombre de classes des extensions ab´eliennes d’un corps quadratique imaginaire, Bulletin Sociente Math-ematique de France 36, pp. 5-77.
[Ru 1] Rubin, K. (1988)On the main conjecture of Iwasawa theory for imaginary quadratic fields: Invent. Math. 93, pp 701-7013
[Ru 2] Rubin, K. (1991).The ”main conjectures” of Iwasawa Theory for imaginary quadratic fields: Invent. Math., 103, pp. 25-68.
[Ru 3] Rubin, K. (1990). The Main Conjecture Appendix to the second edition of S.
Lang: Cyclotomic Fields I and II Graduate Texts in Mathematics 121, Springer.
[Sch] Schneps, L. (1987).On theµ-invariant of p-adic L-functions attached to elliptic curves with complex multiplication: J. Number Theory 25, no. 1, pp. 20-33.
[Shi] Shimura, G. (1971). Introduction to the Arithmetic Theory of Automorphic Functions: Publications of the Mathematical Society of Japan.
[Sie] Siegel, C.L. (1961).Lectures on advanced analytic number theory: Tata Institute of Fundamental Research.
BIBLIOGRAPHY 157 [Sil 1] Silverman, J.H. (1986).The Arithmetic of Elliptic Curves, Graduate Texts in
Mathematics 106: Springer.
[Sil 2] Silverman, J.H. (1994).Advanced Topics in the Arithmetic of Elliptic Curves:
Springer.
[Si] Sinnott, W. (1984).On the µ-invariant of theΓ-transform of a rational function:
Invent. Math., 75, pp. 273-282.
[Ts] Tsuji, T. (1999)Semi-Local Units modulo Cyclotomic UnitsJ. Nubmer Theory, Volume 78, Issue 1, pp 1-26.
[Ve] Venjakob, O. (2002)On the Structure Theory of the Iwasawa Algebra of a p-adic Lie GroupJ. Eur. Math. Soc. (4) pp. 271-311.
[Vi-1] Vigui´e, S (2012)Global Units modulo elliptic units and 2class groups Interna-tionl Journal of number Theory, volume 8, nr. 3, pp 569-588.
[Vi-2] Vigui´e, S. (2016) On the classical main conjecture for imaginary quadratic fields Pioneer Journal of Algebra, Number Theory and its applications, Volume 12, Issue 1-2, pp. 1-27. See also arXiv:1103.1125 (2011).
[Wa] Wada,H. (1966) On the class number and the unit group of certain algebraic number fields, J. Fac. Sci. Univ. Tokyo, 13, 201–209.
[Wash] Washington, L.C. (1997)Introduction to cyclotomic fields, 2nd Edition, Grad-uate Texts in Mathematics 83: Springer, New York.
[We] Weil, A. (1955).On a certain type of characters of the id`ele class group of an algebraic number field: Proc. Int. Symp. on Alg. Number Th. Tokyo, pp. 1-7.