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18 Feb 2015: H = B is the multiplicative group of a quaternion division algebra B over F (sothat Y is compact). ... over R or C).• (Log-)syntomic cohomology Hisyn(,sQp(j)) (for varieties over a p-adic.

27 Oct 2014: 0 3 1. . 3. Let A and B be nn matrices over a field F. ... Show that A can bediagonalised. 11. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.

29 Jan 2010: E! X, and D is the divisor over which fails to be smooth. ... X is dened over K. This determines direct and inverse image maps: Wk̀!

23 May 2024: Gm(F ) over B(F ), so that it suffices to count elements with invariants in Σ to prove Theorem1.2. ... Then, the set Λ (B(1)) is a fundamental domain for the action onO over B(F), which we take as our Σi,.

11 Nov 2019: The ψβα’s can be taken from some vector bundle E over B, then P will be‘constructed from E’. ... The construction can be reversed, so as to start from a principalG-bundle P over a base manifold B and obtain the vector bundle E over B.

20 Nov 2015: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

14 Oct 2016: If not, then get on with the questions below and don’t turn over. ... 0a) Write down a definition of multiplication for natural numbers. Prove that multipli-cation is distributive over addition: that is, for all natural numbers a, b and c we havea(b

28 Jan 2024: We fix T B Ga maximal torus and Borel pair in G defined over F which are in good positionrelative to s, and let I = Gal(F/F). ... We let B Tbe the Borel subgroup defined over F determined by BH and let X(T).

4 Dec 2014: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

21 Apr 2013: A) g : X X′ is étale over Y ′(B) Γ(X,OX) = K is a field. ... ar,. where sum ranges over complete set of representatives of congruence classes modulop in Q(ζm).

20 Oct 2004: in symbolic form. (For this question the letters m, n, a, b should be understood as ranging. ... over all positive integers - for instance, m really means m N.).

23 Jan 2019: F ; Z)/tors over B.Then. ... ξπ := (Hd/2(F; R) Hd/2(F; R)) C K0(B).In both cases these virtual vector bundles are defined over BO(H,λ), giving a.

12 Sep 2002: The size of A A is 20, which is over. three times larger than the size of A. ... B B = {2, 3, 4, 5, 6, 7, 8, 10, 12} ,. which has size 9 and is therefore only just over twice the size of B.

29 Sep 2023: elliptic curves over Q in 2001 by Breuil, Conrad,Diamond, and Taylor [8]. ... 2] P. B. Allen, C. Khare, J. A. Thorne, Modularity of GL2(Fp)-representations over CMfields, arXiv:1910.12986.

21 Nov 2016: 1. The square matrices A and B over the field F are congruent if B = PTAP for some invertible matrixP over F. ... Which, if any, over Q?) Try to get away with the minimum calculation.(.

19 Dec 2014: ZFC. (a) G is generic in P over M ;(b) GD 6= for every dense open set D P in M ;(c) GC 6= for every dense subset C P in ... The intended interpretationof the second-order variables ranges over arbitrary subsets of the domain.

10 Oct 2023: Suppose that α has minimal polynomial f over Q, and let β = α4.Find a, b, c Q such that β = a bα cα2. ... The elements α andβ are then said to be algebraically dependent over K.].

10 Oct 2013: In whichof these cases is U a vector space over R?(a) x1 > 0.(b) either x1 = 0 or x2 = 0.(c) x1 x2 = 0.(d) x1 x2 = 1.(e) x1 ... 2. Determine which of the following sets of sequences of real numbers (xn) form vector spaces over R.(a) xn is bounded.(b) xn

6 Oct 2011: In whichof these cases is U a vector space over R?(a) x1 > 0.(b) either x1 = 0 or x2 = 0.(c) x1 x2 = 0.(d) x1 x2 = 1.(e) x1 ... 2. Determine which of the following sets of sequences of real numbers (xn) form vector spaces over R.(a) xn is bounded.(b) xn

29 Nov 2022: 10. (i) Find the Galois group of f = T4 4T 2 over Q,(ii) Find the Galois group of f over Q(i). ... X2(X 1)2. 13. Let K be any field and L = K(X) the field of rational functions over K.