Proposition 3. If T is essentially positive, then its closure T is positive.
Proposition 3. For any domain V , aV ba can be UC-realized with statistical security in the a-smt- hybrid model, in expected-constant rounds and against an adaptive and malicious t-adversary, pro- vided t < n .
Proposition 3. Let K be a cyclic sextic CM field with a primitive CM type Φ. It holds I0(Φr) = IKr if and only if h∗K = 2tK−1 and hk = 1, where tK is the number of primes in F that are ramified in K. Let K be a cyclic sextic CM field with G := Gal(K/Q) = y . In this notation, complex conjugation is y3. Then K has a totally real cubic subfield F and an imaginary quadratic subfield k. So K = kF .
Proposition 3. Let K be a cyclic sextic CM field and let Φ be a primitive CM type of K. Suppose I0(Φr) = IKr . Then we have h∗K = 2tK−1, where tK is the number of primes in F that are ramified in K. Proof. Recall that without loss of generality we have Φ = {id, y, y−1} and Kr = K. B y Lemma 3.2.3, we have h = 2tK−1[IK : IHPK ], where IH = {b ∈ IK | b = b}. So it is enough to show [IK : IHPK ] = 1 under the assumption I0(Φr) = IKr . For any b ∈ IK , we have the following equality NΦr (y−1(b)/y−2(b)) = bb−1. By the assumption I0(Φr) = IKr , we get bb− = (β), where β ∈ K× and
Proposition 3. Let K be a non-normal sextic CM field containing an imaginary quadratic field k. Let Φ be a primitive CM type of K. Let F be the totally real cubic subfield of K. Then I0(Φr) = IKr holds if and only if h∗K = 2tK−1 and hk = 1, where tK is the number of primes in F that are ramified in K. Let K be a non-normal sextic CM field containing Q(√−d), where d ∈ Z>0. The normal closure N of K is a dihedral CM field of degree 12
Proposition 3. Let K be a non-normal sextic CM field containing an imaginary quadratic field and let Φ be a primitive CM type of K. Suppose I0(Φr) = IKr . Then h∗K = 2tK−1, where tK is the number of primes in F that are ramified in K. Proof. The idea is similar to the proof of Proposition 3.3.5. Put IH = {b ∈ IK | b = b} and PH = PK ∩IH, where H := Gal(K/F ). Then by Lemma 3.2.3 we have h∗K = 2tK−1[IK : IHPK ]. On the other hand, Lemma 2.2.3 tells us the following. If for every b ∈ IK , we have NΦr NΦ(b) = (β)bb−1 and ββ ∈ Q (3.4.1) with β ∈ K×, then [IK : IHPK ] ≤ [IKr : I0(Φr)]. Without loss of generality, we can take Φ = {id, y|K, y−1|K}. Then for any a ∈ IK , we have the following equality NΦr NΦ(a) = NK/Q(a)NΦr (a)aa−1. (3.4.2) By the assumption I0(Φr) = IKr , there exists α ∈ K× such that NΦr (a) = (α) and αα ∈ Q. Moreover, the assumption I0(Φr) = IKr also implies that there is a β ∈ K× such that NK/Q(a)aa−1(α) = (β) and ββ ∈ Q. So the CM type (K, Φ) satisfies (3.4.1), by the assump- tion I0(Φr) = IKr . Therefore, Lemma 2.2.3 implies [IK : IHPK ] ≤ [IKr : I0(Φr)] = 1. Hence the result follows.
Proposition 3. Let K be a CM class number one non-normal sextic CM field containing an imaginary quadratic field k. Let F be the totally real cubic subfield of K. Then we have k = Q(√−d) with d ∈ {3, 4, 7, 8, 11, 19, 43, 67, 163} and an upper bound on dF is given in Table 3.2. |dk| dF ≤ 9 4 3 · 1010
Proposition 3. Let µ be a p-restricted dominant weight and µ' a dominant weight for SL(n). If V (µ)|G is a composition factor of V (µ')|G then ƒ (µ) ≤ ƒ (µ'). Σ Σ
