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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.

11 Nov 2020: The key idea is to use theLang–Weil estimate to relate the cardinality of JΣtop to the asymptotic behaviorof the number of points on Xµ(b) over a finite field, as ... Define the group scheme Jb over F by. Jb(R) ={g G(RF L) | g1bσ(g) = b.

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!

13 Nov 2012: 6. Let π : E B be a fibration over a path connected base B. ... Hint: quaternions.). 8. Prove that S is contractible. 9. Suppose that L1 and L2 are complex line bundles over B which are both locally trivialwith respect to an open cover {Uα}

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.

13 Dec 2009: Conclude thatπn1(SO(n)) is infinite for even n. 6. If V1 and V2 are two vector bundles over B, then V1 V2 is a bundle over B B. ... 1. PART B. 1. Show that real 1-dimensional vector bundles over B are in 1-1 correspondence withelements of H1(B; Z/2).

5 Nov 2009: b) If X = T 2, describe Ccell (X̃) as a complex over the group ring Z[Z2] ' Z[t1, s1].(c) Suppose that X̃ is contractible, and that π : X′ X ... homomorphism. Explain how φ can be used to make H into amodule over R.

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).

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.(.

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.

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

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.(.

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.

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.

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.

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.).

8 Oct 2021: We fix a totally indefinite quaternion algebra B over F which is split at all the places above p.Let S Σ Σp be a subset of even cardinality. ... We write BS for the quaternion algebra over F whose ramification set is precisely the union of Sand the

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.(.

3 Dec 2014: If H(M) is free over Z, we can choose a basis ofH2n(M) and write (a,b) = aTAb for some symmetric matrix A with entries in Z. ... 14. Construct a 3 dimensional real vector bundle over S4 which does not have a nonvanishingsection.

30 Mar 2022: Q and b Brs(k), write Jb for the Jacobian of the smooth projective curve Cb, an abelianvariety over k of dimension equal to the genus of Cb. ... J̄ B Compactified Jacobian 5.2H,G,V Extensions of above objects over Z 7.1N Sufficiently large integer 7.2S Z

29 Jan 2010: defined over F. Then there are commutative diagrams :. (1.2.1). Q[Q]0n1 Θ. ... Gm Z/NZ µµµN Z/NZ (Z/NZ)2. (ζaN,b) 7 (a,b). defined over Q(ζN ). 18. 7.3 The main theorem enables us to calculate the effect of

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 Feb 2023: and Φ(J) = ker(b)/ im(a).If X is semistable (meaning that Xs is smooth over k apart from double points with. ... v) (v′) Z[C]0,. the sum taken over all vertices v′ adjacent to v.In general, we have an isomorphism of Gal(k̄/k)-modules Φ(J) = ker(b)/

22 Oct 2020: let T̆ be its centralizer. By Steinberg’s theorem, G is quasi split over F̆. ... Let Z be a finite type affinescheme over OF and Z its generic fiber.

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.

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

9 Nov 2015: 14. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices. ... Deduce that if two n n real matrices P and Q are similar whenregarded as matrices over C, then they are similar as matrices over R.

14 Feb 2009: Show that thedual (over Q) of C is C+. (b) Show that H(C) is a direct sum of copies of Q[U ] and Q[U ]/(U ni ). (Use thestructure theorem ... d) Explain how this is formally similar to the homology of the Hopf bundle over CPn.

11 Oct 2022: 17. Let L/K be an extension, and x,y L transcendental over K. ... Show that x is algebraicover K(y) iff y is algebraic over K(x).

12 Aug 2008: 2. Let A and B be n n matrices over a field F. ... 3. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

7 May 2013: we can let B bethe local coefficient system which over {0} BΣ corresponds to the module Z[X],where X is a free transitive Σ-set, and over the other path ... There is an isomorphism of Σ-sets SX =. X,. and so (m )B is the local coefficient system which

26 Oct 2015: 3. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the basis{x1,x2,x3,x4} for ... 9. Let A and B be nn matrices over a field F.

5 Nov 2006: 6. Modify the construction of the Hopf bundle given in Lectures, replacing C everywhere by R toobtain a rank one real vector bundle over S1. ... Show that the tangent bundle TS2 induces on Y a structureof a principal S1-bundle over S2.(ii) Show further

6 Oct 2019: map(B,M) = {families of (.) over B}. Then. H(M) = {characteristic classes of such families}. ... B. n0. BGLn(R).  =: K(R), algebraic K-theory13. The Madsen–Weiss theorem.

9 Nov 2023: Integrate. d[δQ[ξ] ξ θ(φ,δφ)] = 0. over a hypersurface extending from the bifurcation surface B tospatial infinity. ... d[δQ[ξ] ξ θ(φ,δφ)] = ξaδCaIntegration over the region of the horizon bounded by.

27 Feb 2008: Moreover, the restriction of a finitely generated A-module is finitelygenerated over B. ... action explained in the proof of Proposition 2.3. The latter module is just a directsum of dim V copies of ResAB(M) and is hence finitely generated over B.

15 Oct 2021: Exercise A.6. Let us take X = Rn, thought of as a vector space over R and define:. ... 1p. 3. a) Show that X is a vector space over R, where scalar multiplication and vector additionare defined pointwise.

27 Oct 2019: Hence i and j arehomotopy inverses. Corollary 3.2. All fibres of a fibration π : E B over a path-connected base are homotopy equivalent. ... 3.5. B as delooping. Recall that a principal G-bundle π : E B over a CW complex is a fibrationso there is a

5 Jun 2020: n1 obtained byminimizing the ideal Shannon code-lengths over the family {Qθ}θΘ,. ... Integrating over distortion-balls effectively changes a continuous alphabet intoa discrete one and log Q(B(xn1,D)) still makes sense as an idealized code-length.

7 Nov 2016: What happens over. R? 4. (i) Let V be a vector space, let π1,π2,. , ... 14. Let C be an nn matrix over C, and write C = A iB, where A and B are real nn matrices.

27 Oct 2014: Concretely we have. Objects Over b B: x E(b′) with u : b′ b.Maps From y with v : c′ c to x with u : b′ b :. c′v. - c. y ... have. Objects Over b B: x E(b′) with u : b′ b.Maps From y with v : c′ c to x with u : b′ b :. f b′F. - c′v. -

26 Oct 2016: 3. Let V be a 4-dimensional vector space over R, and let {ξ1,ξ2,ξ3,ξ4} be the basis of V dual to the basis{x1,x2,x3,x4} for ... en) is dual. S.J.Wadsley@dpmms.cam.ac.uk - 1 - October 2016. 9. Let A and B be n n matrices over a field F.

26 Jan 2023: Let A B C be rings. (i) Show that if B is finite over A, and C is finite over B, then C is finite over A. ... ii) Show that if B is integral over A, and C is integral over B, then C is integralover A.

22 Oct 2020: We can alsoconsider the corresponding objects over L. Then for x B(G,L), we have Gx(OL) = Gx(OL)ker κ̃Gwhere. ... If G isan unramified group (i.e. G is quasi-split and split over an unramified extension of F), then everyparahoric subgroup is connected;

21 May 2005: 2. Let A and B be n n matrices over a field F. ... 3. Let C be an n n matrix over C, and write C = A iB, where A and B are real n n matrices.

25 Dec 2010: w = z α(x z) β(y z),where α 6= 1 and β is transcendental over Q(α). ... Moreover, R(β) = 0. Henceβ is algebraic over A(α) and so is algebraic over Q(α).

25 Nov 2005: Leta K. Show that Xp a is irreducible over K if and only if it is irreducible over K′. ... Suppose further that L = K(x, y), where y is algebraic over K.

19 May 2003: n then. B = P1AP. Corollary 3.3 Let V be an n dimensional vector space over F. ... 15. Theorem 3.10 If A and B are n n matrices over F then detAdetB =det(AB).