Proposition 2. 3.10. Let ε > 0 be fixed sufficiently small, let A > 3 be fixed. Let Q0 be defined as in (2.3.5) and let (log X)19+ε ≤ H ≤ X log−A X. Then for all but at most O(HQ−1/3) values of 0 < |h| ≤ H we have that ∫ 2 , Σ 1 X |a(α)| e(−hα)dα = S(h)X P<p≤P 1+δ logη X , for some η = η(ε) > 0, where we define the singular series S(h) as in (1.0.3). !
Proposition 2. 3.5. Let K be a non-normal quartic CM field and let F be its real quadratic subfield. Let Φ be a primitive CM type of K. Suppose I0(Φr) = IKr . Then F = Q(√p) and Fr = Q(√q), where p and q are prime numbers with q 3 (mod 4) and (p/q) = (q/p) = 1. Moreover, all the rational primes (distinct from p and q) that are ramified in Kr/F r are inert in F and Fr. Proof. We first prove that if a prime l ramifies in both F and Fr, then it is equal to p, where F = Q(√p). Indeed, by Lemma 2.3.7-(i), the prime l is totally ramified in Kr/Q and hence by Corollary 2.3.9, we get F = Q( l), so l = p. Now we see that there are four types of prime numbers that ramify in N/Q:
Proposition 2. 3.7 (Major Arc Integral). Let A > 3 be fixed and let ε > 0 be fixed sufficiently small. Let (log X)19+ε ≤ H ≤ X log−A X. With M defined as in (2.3.4) and δ > 0 sufficiently small, there exists some η = η(ε) > 0 such that for all but at most O(HQ−1/3) values of 0 < |h| ≤ H we have that ∫ 2 , Σ 1 X |S(α)| e(−hα)dα = S(h)X p + O P<p≤P 1+δ logη X , where S(h) is the singular series given in (1.0.3). Assuming Proposition 2.3.6 and Proposition 2.3.7, we can now prove The- orem 2.3.3. Proof of Theorem 2.3.3. We follow the arguments in [27, Pages 32-34]. By (2.3.3), we have that Σ Σ ϖ2(n)ϖ2(n + h) − ∫ |S(α)|2e(−hα)dα 0<|h|≤H X<n≤2X−h M ≪ 0<Σ|h|≤H ∫m |S(α)|2e(−hα)dα . We now apply a smoothing; we multiply the above by an even non-negative
Proposition 2. A, h, ρ |=SSL ϕ if and only if A, h, ρ |=FSL finh ∧ T (ϕ).
Proposition 2. For all c ≥ 0 and n ≥ 1, we have cn log2(n) ≤ fc(n) ≤ (c + 1)n[log2(n)|.
Proposition 2. If in a T1 triopoly, the branded firm’s profit are nearly constant in first mover advantage (i.e., ∂ΠT1/∂κ 0), and there exists a κ∗ [0, 1] such that the net surplus from lunching AG at κ∗ is zero, then under take-it or leave-it offer for the licensing fee, the threat to launch an AG is credible for all κ ≥ κ∗. (Proof in Appendix (A.2)). 0 The condition ∂ΠT1/∂κ ≈ 0, that the equilibrium profit for the branded firm in T1 is nearly constant, is stronger than needed. What we need for net surplus to be increasing in κ is the condition |∂ΠT1/∂κ| < |∂ΠT1/∂κ|, i.e., the branded firm’s equilibrium profit is decreasing in first mover advantage at a slower rate than the increase in the equilibrium profit of the first generic entrant so that the overall net surplus still keeps on increasing in κ (recall that ΠT0 does not change with κ, but ΠT1 can decrease in κ due to price coordination between the brand and the AG, see lower-left panel in Figure A-3 for the shape of ΠT0). Next, we can provide conditions – or values of cost θ – under which a branded firm would prefer to launch an in-house AG. To do so, we fist define two threshold values. Let θ∗(κ) = (ΠT1 +ΠT1 − ΠT0) 0 and θ∗∗(κ) = (ΠT1 + ΠT0 − ΠD0) + δ · θ∗(κ).
Proposition 2. Let N1, N2 ∈ Mm(E) be two nilpotent matrices. Then N1 ⪯ N2 if and only if, for all i ∈ Z≤1, we have rank(Ni) ≤ rank(Ni). Given a Weil–Deligne inertial type τ = (ρτ , Nτ ) of L over E, and a fi- nite dimensional irreducible Qp-representation θ : IL → GL(Vθ) with open kernel, we can consider the θ-isotypic component ρτ [θ] : IL → GL(Vτ [θ]) of ρτ ⊗E Qp. As Nτ commutes with the action of IL, it restricts to a nilpotent endomorphism Nτ [θ] ∈ End(Vτ [θ]).
Proposition 2. Let v be a finite place of F, unramified above the place v¯ of F +. Then the unnormalised Satake transform S : H(G˜(Fv¯ ), G˜(OF+ )) → H(G(Fv¯ ), G(OF+ )) + + sends P˜v(X) to Pv(X)qn(2n−1)P ∨c (q1−2nX).
Proposition 2. 1.60 allows a slightly different reformulation if we group the isometric factors together. Namely, let M = M0 × Ml1 × · · · × Mlk be a de Rham-like li decomposition where Mi ̸≃ Mj for i j and M simply means the product of li copies of The group Sl≃ (here l = li) then consists of those permutations that shuffle the first Mi. Then each φij used in the construction of Sk≃ ‹→ I(M ) would have to be an isometry between two copies of MΣs for some s ∈ {1, . . . , k}, so we can take it to be the identity. l1 elements with each other, the next l2 elements with each other, and so on. Hence, Sl≃ ≃ Sl1 × · · · × Slk . The embedding Sl≃ ‹→ I(M ) then looks like σ(p0, p1, . . . , pl) = (p0, pσ(1), . . . , pσ(l)), and decomposition (2.1.7) becomes I(M ) = I(M0) × I(M1)l1 × · · · × I(Mk)lk ⋊ Sl≃.
