Lemma 3. (Termination) For each run, every process pi ∈ Correct of the sys- tem HASf [L, ∅, n] eventually decides some value.
Lemma 3. Let C/Q be a hyperelliptic curve of genus g ≥ 4, with a rational Weierstrass point, geometrically simple Jacobian with r ≤ 1, good reduction at 3, which satisfies condition (†). Let P1, P2 ∈ C(Q) be conjugate quadratic points, with P1, P2 ∈ CF3 (F9) \ CF3 (F3), and P3 ∈ C(Q) a rational point. If n(ΛC, P1) = 1, there are at most 26 ordered triples (Q1, Q2, Q3) of conjugate cubic points in DP1 × DP2 × DP3 . If n(ΛC, P1) = 0, there
Lemma 3. Let K be a sextic CM field containing an imaginary quadratic field k and Φ be a CM type of K. Let F be totally cubic subfield of K. Put IH = {b ∈ IK | b = b}, where H := Gal(K/F ). Then we have h∗K = 2tK−1[IK : IHPK ]. Proof. Lemma 2.2.2 tells us that if × = WK ×, then h∗ IHPK ]. Combine this with Lemma 3.2.2. = 2tK−1[IK : Proof of Proposition 3.2.1. Identify K with Kr via an isomorphism. By Lemma 3.2.3, we have h∗K = 2tK−1 if and only if IK = IHPK. For any b ∈ IF , we have NΦr (b) = (NF/Q(b)), where NF/Q(b) ∈ Z. Hence IF PK ⊂ I0(Φr). We can see from the exact sequence (2.2.1) p prime of F that the elements of IH/IF are represented by the products of the primes in K that are ramified in K/F . For any such prime P, let pZ = P∩ F and p = p∩ Q. Then the following holds 2 NΦr (P) = NΦr (p0K) = NF/Q(p)0K, (3.2.1) where NF/Q(p) p, p2, p3 depending on the splitting behavior of p in F . The prime P lies over a rational prime p that is ramified in k, see Xxxx [21, Proposition 4.8-(ii) in II]. Moreover, the prime p is the unique ramified prime in k/Q. Indeed, by genus theory, if the class number of an imaginary quadratic field k is odd then there is one and only one ramified prime in k/Q. By (3.2.1), we have NΦr (P) = qNF/Q(p)0K. (3.2.2) If NF/Q(p) = p, then the right hand side of (3.2.2) is generated by √ p if k /= Q(i) and generated by i + 1 if k ∼= Q(i). Therefore, in both cases we have a generator π in k of NΦr (P) such that ππ ∈ Q. Similarly, in cases NF/Q(p) = p2 that ππ ∈ Q. or p3 , there exists a generator π in k of NΦr (P) such Hence every element of IHPK , which is IK , is in I0(Φr). In particular, we get IK = I0(Φr).
Lemma 3. The space Vp(Rn) of p-frames in Rn is a compact smooth manifold of dimension nk + k(k−1) .
Lemma 3. The vector AX with A = (GX⊤ − XG⊤) represents the action of DFX on the tangent space TXVp(Rn) Proof. Consider the differential G = ∂F ∂Xi,j ∈ Rn×p as well as some vector Z ∈ TXVp(Rn). Each can be represented in the coordinate form at X as G = XGA +X⊥GB and Z = XZA + Z⊥ZB with the restriction that ZA is skew symmetric. The directional derivative DFX(Z) is thus given by the substitution as DFX(Z) = Tr(G⊤Z) = Tr(GA⊤ZA) + Tr(G⊤BZB)
Lemma 3. 2.1. (Convergence of private empirical marginal distribution). limn→∞ F˜n(t) = limn→∞ Fˆn(t) = F (t) almost surely, where F˜n(t) is the empirical CDF based on the
3.2. DPCOPULA 33 private histogram, Fˆn(t) is the empirical CDF based on the original histogram, and F (t) is the population CDF when n tends to be infinity.
Lemma 3. 2.1. Suppose that A is a C∗-algebra of real rank zero, and let U, V ∈ A be unitary elements such that —1 ∈/ σ(U ), —1 ∈/ σ(V ). Then for any ε > 0 there exists a unitary element Wε such that UV — Wε ≤ ε and —1 ∈/ σ(Wε).
Lemma 3. 5 For a tripartite 3-uniform hypergraph H the following state- ments are equivalent:
Lemma 3. For 1 m ≤ o(n1/5), almost surely we have m ≥ F3([n]m) ≥
Lemma 3. For any 1 m ≤ n, almost surely we have F3([n]m) = Ω(m1/3). For this proof, it will be convenient to use the model [n]p with p = m/n rather than [n]m (recall Remark 3.1.6). Without loss of generality we assume that 1/p, pn, pn/3 ∈ N. Our strategy here follows that of [31]. In order to show the existence of a Sidon set of order (pn)1/3 in a typical instance of a random set [n]p we will use the following theorem of Xxxx and Xxxxxx [10] (with the statement adapted for our purposes).
