Division algebras over function fields of surfaces … · Division algebras over function ... be the Brauer group of central simple algebras over K. ... (due to Rowen-Tignol-Sivatski

Division algebras over function fields ofsurfaces-d’apres Saltman

V. SureshUniversity Of Hyderabad

Emory University

16-20 May, 2011Ramifications in Algebra and Geometry

Emory University, Atlanta

V. Suresh (UoH/Emory) Division algebras over function fields of surfaces-d’apres Saltman

Introduction

Let K be a field.

By a central simple algebra over K we mean a finite dimensionalalgebra over K with center K and without any non-trivialtwo-sided ideals.

Theorem (Artin – Wedderburn)A central simple algebra A is isomorphic to the matrix algebraMn(D) for some n ≥ 1 and central division algebra D over K.

Let Br(K ) be the Brauer group of central simple algebras over K .

Every element of Br(K ) is represented by a central divisionalgebras over K .


Introduction

Let K be a field.






Introduction

Let K be a field.






Introduction

Let K be a field.






Introduction

Let K be a field.






Degree, index and period

Let A be a central simple algebra over K . Then we know that thedimension of A as a vector space over K is a square n2. Theinteger n is called the degree of A.

If A = Mm(D) with D a central division algebra, then the degreeof D is called the index of A.

For a division algebra degree = index.

The order of A in the Br(K ) is called the period of A.

We know that period divides index and the prime divisors of degreeand index are same.

In fact index = periodr for some r ≥ 1.










































Question. Does there exists a natural number r depending onlyon the given field K such that index(A) | period(A)r for all centralsimple algebras over K?

This question obviously has a negative answer if we takeK = C(t1, · · · , tn, · · · ).

Of course this field is not an interesting field.










Examples

Let us see some examples where the question 1 has an affirmativeanswer.

If K is a finite field or an algebraically closed field, then there areno non-trivial central division algebras.

Suppose K is a local field or a global field, the class field theorytells us that, index = period for every central simple algebra overK .


Examples





Examples





Let us modify the above question.

Question. Let K be a field. Suppose there exists a naturalnumber r such that index | period r for every central simple algebraover K . Does there exists a natural number t such thatindex | period t for every central simple algebra over K (t).


Let us modify the above question.

Question. Let K be a field. Suppose there exists a naturalnumber r such that index | period r for every central simple algebraover K . Does there exists a natural number t such thatindex | period t for every central simple algebra over K (t).


Saltman’s result

Theorem (Saltman (1997))Let k be a p-adic field and K be a function field of a curve over k.Let A be a central simple algebra of period n. Suppose that n iscoprime to p. Then index(A) | n2.

There are examples (due to Rowen-Tignol-Sivatski ) of centralsimple algebras over k(t), k-p-adic field, with index = period2.


Saltman’s result

Theorem (Saltman (1997))Let k be a p-adic field and K be a function field of a curve over k.Let A be a central simple algebra of period n. Suppose that n iscoprime to p. Then index(A) | n2.

There are examples (due to Rowen-Tignol-Sivatski ) of centralsimple algebras over k(t), k-p-adic field, with index = period2.


Cyclic algebras

Let L/K be a finite cyclic extension of degree n and σ ∈ Gal(L/K )be a generator.

Let b ∈ K ∗.

Let A = L⊕ Ly ⊕ · · · ⊕ Lyn−1.

Define yλ = σ(λ)y for all λ ∈ L and yn = b.

Then A is a central simple algebra and is called a cyclic algebra.

A is denoted by (L, σ, b).


Cyclic algebras


Let b ∈ K ∗.

Let A = L⊕ Ly ⊕ · · · ⊕ Lyn−1.





Cyclic algebras


Let b ∈ K ∗.

Let A = L⊕ Ly ⊕ · · · ⊕ Lyn−1.





Cyclic algebras


Let b ∈ K ∗.

Let A = L⊕ Ly ⊕ · · · ⊕ Lyn−1.





Cyclic algebras


Let b ∈ K ∗.

Let A = L⊕ Ly ⊕ · · · ⊕ Lyn−1.





Cyclic algebras


Let b ∈ K ∗.

Let A = L⊕ Ly ⊕ · · · ⊕ Lyn−1.





Assume that n is coprime with the characteristic of K and Kcontains a primitive nth root of unity.

Then L = K ( n√

a) for some a ∈ K ∗ and σ( n√

a) = ζ n√

a for someprimitive nth root of unity ζ ∈ K .

The algebra A is generated by two elements x , y with relations

xn = a, yn = b, and yx = ζxy .

In this case the algebra A is also denoted by (a, b)n.



Then L = K ( n√


a) = ζ n√







Then L = K ( n√


a) = ζ n√







Then L = K ( n√


a) = ζ n√






Let A be a central simple algebra of degree 2. Then it is easy tosee that A is a cyclic algebra.

Let A be a central simple algebra of degree 3. Then a classicalresult of Albert asserts that A is a cyclic algebra.

Question. Let A be a central simple algebra of prime degree. Is Aa cyclic algebra?

This question is open even for degree 5 algebras.

















Let k be a global field or a local field. Then the class field theoryasserts that every central simple algebra over k is cyclic.

We have the following:

Theorem (Saltman(2007))Let k be a p-adic field and K be a function field of a curve over k.Let ` be a prime not equal to p. Every central simple algebra overK of degree ` is cyclic.










Let K be a function field of a p-adic curve and q a prime not equalto p.

Let A be a central simple algebra of period q. To show that theindex of A divides q2, we need to produce an extension L/K ofdegree q2 such that A⊗ L is a split algebra.

Similarly to show that a degree q algebra is cyclic, we need to finda cyclic extension of degree q which splits the algebra.

In general it is difficult to verify when an extension splits thealgebra.

The condition whether the algebra A⊗ L is “unramified” can betranslated in to cohomological criterion.

Saltman’s idea is to split the ramification in good extensions andappeal to known theorems on unramified classes (for instance thevanishing of unramified classes) to obtain his results.

We explain these ideas now.
















































We explain these ideas now.V. Suresh (UoH/Emory) Division algebras over function fields of surfaces-d’apres Saltman

I R - a discrete valuation ring

I K - field of fractions of R

I k - residue field of R

I ν : K ∗ → Z - the discrete valuation given by R.

I n - a natural number which is a unit in R.

We have a homomorphism ∂ : nBr(K ) → H1(k, Z/nZ) called theresidue homomorphism.

We say that a central simple algebra A over K is unramified at Rif ∂(A) is trivial.


I R - a discrete valuation ring

I K - field of fractions of R

I k - residue field of R

I ν : K ∗ → Z - the discrete valuation given by R.

I n - a natural number which is a unit in R.

We have a homomorphism ∂ : nBr(K ) → H1(k, Z/nZ) called theresidue homomorphism.

We say that a central simple algebra A over K is unramified at Rif ∂(A) is trivial.


Suppose that R contains a primitive nth root of unity.

By fixing a primitive nth root of unity, we fix an isomorphismZ/nZ ' µn and identify H1(k, Z/nZ) with k∗/k∗n.

For A = (a, b)n, we have ∂(A) = u ∈ k∗/k∗n, whereu = (−1)ν(a)ν(b)aν(b)bν(a).

In particular if A = (π, u), where u is unit in R and π is parameterin R, then ∂(A) = u ∈ k∗/k∗n.

















I X - a regular integral scheme of dimension 2,

I X 1 - the set of codimension one points of X .

I For x ∈ X 1, the local ring at x is a discrete valuation ring.

I K - the function field of XI Assume that n is a unit on XI We have a residue homomorphism

∂x : nBr(K ) → H1(κ(x), Z/nZ).

We say that a central simple algebra A over K is unramified on Xif A is unramified at every point of X 1.


I X - a regular integral scheme of dimension 2,

I X 1 - the set of codimension one points of X .

I For x ∈ X 1, the local ring at x is a discrete valuation ring.

I K - the function field of XI Assume that n is a unit on XI We have a residue homomorphism

∂x : nBr(K ) → H1(κ(x), Z/nZ).

We say that a central simple algebra A over K is unramified on Xif A is unramified at every point of X 1.


From now onwards X denotes a non-singular surface (i.e. twodimensional separated excellent integral Noetherian schemequasi-projective over some affine scheme.

Let K be the field of fractions of X .

Let n be an integer which is a unit on X .

Theorem (Saltman)Let K be as above. Assume that K contains a primitive nth root ofunity. Let A be a central simple algebra of period n. Then thereexist f , g ∈ K ∗ such that A is unramified at every discretevaluation of K ( n

√f , n√

g).






√f , n√

g).






√f , n√

g).






√f , n√

g).


Corollary (Saltman (1997))Let k be a p-adic field and K be a function field of a curve over k.Let A be a central simple algebra of period n. Suppose that n iscoprime to p. Then index(A) | n2.

Proof. Proof if by induction on n the period of A.

If n = 1, there is nothing to prove. Assume that n ≥ 2.

Let q be a prime which divides n. Let B = qA ∈ Br(K ).

Then the period of B is nq

By the induction the index of B divides n2

q2 . Hence there exists an

extension L/K of degree n2

q2 which splits B. In particular the periodof A⊗ L is q. Thus it is enough to show that the index of A⊗ Ldivides q2.




















































Let M/L be the extension given by the qth roots of unity.

Since the degree of M/L is coprime to q, the index of A⊗ L issame as the index of A⊗M.

Since M is also a function field of a curve over a p-adic field, it isenough to prove the corollary when K contains all the qth roots ofunity and the period of A is q.

By the above theorem, there exists f , g ∈ K ∗ such that A isunramified over K ( q

√f , q√

g).

By a theorem of Grothendieck, the unramified Brauer group ofK ( q√

f , q√

g) is zero.

Hence A⊗ K ( q√

f , q√

g) is trivial. In particular the index of Adivides q2.






√f , q√

g).


f , q√

g) is zero.

Hence A⊗ K ( q√

f , q√







√f , q√

g).


f , q√

g) is zero.

Hence A⊗ K ( q√

f , q√







√f , q√

g).


f , q√

g) is zero.

Hence A⊗ K ( q√

f , q√







√f , q√

g).


f , q√

g) is zero.

Hence A⊗ K ( q√

f , q√







√f , q√

g).


f , q√

g) is zero.

Hence A⊗ K ( q√

f , q√



Theorem (Saltman (2008))Let K be as above and q a prime which is a unit on X . Assumethat K contains a primitive qth root of unity. Let A be a centralsimple algebra of degree q. Then there exists f ∈ K ∗ such that Ais unramified at every discrete valuation of K ( n

√f ).

Corollary (Saltman (2007))Let K be a function field of p-adic curve and q a prime not equal top. Then every central simple algebra over K of degree q is cyclic.

Proof. Let A/K be a central simple algebra of degree q. Then bythe above theorem, there exists f ∈ K ∗ such that A⊗ K ( q

√f ) is

unramified.

Once again by the theorem of Grothendieck, A⊗ K ( q√

f ) is trivial.Hence A is cyclic.



√f ).



√f ) is

unramified.





√f ).



√f ) is

unramified.





√f ).



√f ) is

unramified.




We now explain the general ideal behind the proof of these results.

X - a non-singular surface.

X 1 - set of codimension one points of X

K - function field of X .

n - an integer which is a unit on X

K -contains a primitive nth root of unity.

For x ∈ X 1, we have the residue homomorphism

∂x : nBr(K ) → κ(x)∗/κ(x)∗n









∂x : nBr(K ) → κ(x)∗/κ(x)∗n









∂x : nBr(K ) → κ(x)∗/κ(x)∗n









∂x : nBr(K ) → κ(x)∗/κ(x)∗n









∂x : nBr(K ) → κ(x)∗/κ(x)∗n









∂x : nBr(K ) → κ(x)∗/κ(x)∗n









∂x : nBr(K ) → κ(x)∗/κ(x)∗n


Two irreducible curves C1 and C2 on X said to have normalcrossings if they intersect at a point P, then the maximal idealmP at P is generated by π and δ, where π and δ define C1 and C2

at P respectively.

A divisor D on X is said to have normal crossings if any twodistinct irreducible curves in the support of D have normalcrossings.

Given a divisor D on X there exists a blow up X ′ of X such thatthe strict transform of D and the exceptional curves on X ′ havenormal crossings.



at P respectively.





at P respectively.




α ∈ Br(K ) of order n.

The union of the curves C where α is ramified is called theramification locus of α.

After blowing up X , we assume that the ramification locus of α isa union of regular curves Ci with normal crossings.

The intersection points among the Ci are called nodal points.

By the assumption every nodal point lies exactly on two curves Ci .

Any point on exactly one Ci is called a curve point.

By the assumption curve points are non-singular point on thecurve.


















































We begin with the following:

LemmaLet A be a two-dimensional regular local ring with maximal ideal mand field of fractions K. Let n be an integer which is a unit in A.Assume that K contains a primitive nth root of unity. Letα ∈ nBr(K ).

1) Suppose that m = (π, δ) and α is ramified only at π on A.Then α = α′ + (u, π) for some unit u ∈ A and α′ ∈ Br(A).

2) Suppose that m = (π, δ) and α is ramified only at π and δ onA. Then either α = α′ + (u, π) + (v , δ) or α = α′ + (uδs , π) forsome units u, v ∈ A, s coprime to n and α′ ∈ Br(A).

















Theorem (Saltman)Let K be as above. Assume that K contains a primitive nth root ofunity. Let α ∈ nBr(K ). Then there exist f , g ∈ K ∗ such that A isunramified at every discrete valuation of K ( n

√f , n√

g).

Proof. After blowing up, we assume that the ramification locus ofα is union of two regular curves C and E with normal crossings.Choose f ∈ K ∗ as follows:

divX (f ) = C + E + F

for some divisor F such that F does not contain any component ofeither C or E and F does not pass through the points of C ∩ E .


Theorem (Saltman)Let K be as above. Assume that K contains a primitive nth root ofunity. Let α ∈ nBr(K ). Then there exist f , g ∈ K ∗ such that A isunramified at every discrete valuation of K ( n

√f , n√

g).

Proof. After blowing up, we assume that the ramification locus ofα is union of two regular curves C and E with normal crossings.Choose f ∈ K ∗ as follows:

divX (f ) = C + E + F

for some divisor F such that F does not contain any component ofeither C or E and F does not pass through the points of C ∩ E .


Let P be closed point in C ∩ F .

By the above lemma, we have α = α′ + (π, u) for some α′

unramified at P, u a unit at P and π defines C at P. Let uP = u.

Similarly if P ∈ E ∩ F , we have α = α′ + (δ, v). Let uP = v .

Suppose that P ∈ C ∩ E .

We have f = πδw for some unit w at P and π and δ define C andE at P respectively.

We have α = α′ + (uδs , π) or α = α′ + (π, u) + (δ, v).

In the first case, let uP = uw and in the second case let uP = uv .
























































By the Chinese remainder theorem, we get a function g ∈ K ∗

which is a unit at all the above closed points and components ofC ,E ,F such that g(P)−1uP(P) is an nth power for allP ∈ C ∩ E ,C ∩ F ,E ∩ F .

Then α is unramified at every discrete valuation of K ( n√

f , n√

g).

CorollaryLet X ,K and n as above. Suppose that for every closed point P ofX the residue field κ(P) is algebraically closed. Let A be a centralsimple algebra over K of period n. Then there exists f ∈ K ∗ suchthat A is unramified over K ( n

√f ).





f , n√

g).


√f ).





f , n√

g).


√f ).


X - non-singular surface and K its field of fractions andα ∈ qBr(K ), q a prime.

Assume that the ramification locus of α on X is a union of regularcurves with normal crossings.

Let P be a nodal point and C1 and C2 are the two curves in theramification locus of α.

Let A = OX ,P . Then A is a two dimensional regular local ring withmaximal ideal mP = (π, δ) for some primes π, δ ∈ A defining C1

and C2 at P respectively. Let κ(P) be the residue field at P.

By the above lemma, we have either α = α′ + (u, π) + (v , δ) orα = α′ + (uδs , π) for some units u, v in A, s coprime to q andα′ ∈ Br(A).






























P is a cold point if α = α′ + (uδm, π) for some unit u at P and mcoprime to q.

Assume that α = α′ + (u, π) + (v , δ) for some units u, v ∈ A andα′ ∈ Br(A).

P is a cool point if u and v are qth powers modulo the maximalideal mP .

P is a hot point if images of u and v do not generate the samesubgroups of κ(P)∗/κ(P)∗q.

P is a chilly point if the images of u and v generate the samenon-trivial subgroups κ(P)∗/κ(P)∗q.

In the last case write v = usP up to an nth power. The integer sPis called the coefficient of P with respect to C1.





































Let P be a cool point. Let X ′ be the blow up of X at P. Then itis easy to see that α is unramified at the exceptional curve. Thusreplacing X by X ′, we can assume that there are no cool points.

Consider the following graph:vertices are the curves in the ramification locus of α. The edgesare the chilly points. Two vertices have an edge if both the curvesintersect at a chilly point.

A loop in this graph is called a chilly loop

After blowing up several times, we ensure that there are no chillyloops and no cool points.

Suppose Ci are all the curves in the ramification locus. Then wecan choose, for each Ci , a non-zero si ∈ Z/nZ such that: SupposeP is a chilly point on Ci and Cj with coefficient s with respect toCi . Then s = sj/si ∈ Z/nZ.


























We would like to find f ∈ K ∗ such that all the ramification of α iskilled by the extension K ( q

√f ).

Let Ci be the curves in the ramification locus and si be integerscoprime to q. Let νCi

be the discrete valuation given by Ci .

Let f ∈ K ∗ be such that νCi(f ) = si .

Let Ri be the discrete valuation ring at Ci and Si be the integralclosure of R in K ( q

√f ).



√f ).





√f ).



√f ).





√f ).



√f ).





√f ).


Let κ(Ci ) be the residue field of Ri . Since νCi(f ) = si coprime to

q, Si is a discrete valuation ring unramified over Ri , with residuefield κ(Ci ).

α is unramified at Si . Hence the specialization of α gives anelement βCi

∈ Br(κ(Ci )).

βCiis called the residual Brauer class of α with respect to f . Let

Li/κ(Ci ) be the residue of α at Ci . If α has index q, then theresidual Brauer class of α is split by the extension Li .

Suppose that we find f ∈ K ∗ such that α is unramified overK ( q√

f ). Then it is easy to see that all the residual Brauer class βCi

is unramified on the curve Ci .





∈ Br(κ(Ci )).










∈ Br(κ(Ci )).










∈ Br(κ(Ci )).







Using a local patching argument, one chooses f ∈ K ∗ such thatdivX (f ) =

∑siCi + F for some divisor F on X which does not

contain any of the curves Ci and does not pass through the nodalpoints. Further all the residual Brauer classes with respect to f aretrivial (in particular unramified).

This choice of f kills most of the ramification of α, except thosevaluations centered at the closed points which are in theintersection of F and Ci .

Finally one needs a further modification of f such that α isunramified over K ( q

√f ).

We skip the details of this and go to the consequences of theresults and the methods.







√f ).








√f ).








√f ).



Galois cohomology

Let k be a field and ` a prime not equal to the characteristic of k.

For n ≥ 0, let Hn(k, µ`) denote the nth Galois cohomology groupwith coefficients in the group µ` of `th roots of unity.

We have

H0(k, µ`) = µ`(k)

H1(k, µ`) ' k∗/k∗`

H2(k, µ`) ' `Br(k)


Galois cohomology



We have

H0(k, µ`) = µ`(k)

H1(k, µ`) ' k∗/k∗`

H2(k, µ`) ' `Br(k)


Galois cohomology



We have

H0(k, µ`) = µ`(k)

H1(k, µ`) ' k∗/k∗`

H2(k, µ`) ' `Br(k)


Galois cohomology



We have

H0(k, µ`) = µ`(k)

H1(k, µ`) ' k∗/k∗`

H2(k, µ`) ' `Br(k)


Galois cohomology



We have

H0(k, µ`) = µ`(k)

H1(k, µ`) ' k∗/k∗`

H2(k, µ`) ' `Br(k)


Symbols

For a ∈ k∗, let (a) denote its image in H1(k, µ`)

Assume that k contains all the `th roots of unity.

Fix an isomorphism Z/`Z with µl .

a1, · · · , an ∈ k∗. The cup product gives an element

(a1) · (a2) · · · · (an) ∈ Hn(k, µ`)

An element of the form (a1) · (a2) · · · · (an) is called a symbol.

The image of (a) · (b) ∈ H2(k, µ`) in `Br(k) is the cyclic algebra(a, b)`.


Symbols





(a1) · (a2) · · · · (an) ∈ Hn(k, µ`)




Symbols





(a1) · (a2) · · · · (an) ∈ Hn(k, µ`)




Symbols





(a1) · (a2) · · · · (an) ∈ Hn(k, µ`)




Symbols





(a1) · (a2) · · · · (an) ∈ Hn(k, µ`)




Symbols





(a1) · (a2) · · · · (an) ∈ Hn(k, µ`)




Related results

Let K be a function field of a p-adic curve and D a central divisionalgebra over K .

Suppose the period of D is 2 and p 6= 2. By the Saltman’s result,the degree of A is at most 4. In particular, either a quaternionalgebra or A is isomorphic to a tensor product of two quaternionalgebras (a theorem of Albert).

Theorem (Suresh(2007))Let K be a function field of a p-adic curve. Let D be a centraldivision algebra over K of degree q. Suppose that q is a prime notequal to p and K contains a primitive qth root of unity. Then D iseither a cyclic algebra or a tensor product of two cyclic algebras.


Related results





Related results





Proof. Suppose that the degree of D is q. Then by the theorem ofSaltman, D is cyclic.

Assume that the degree of D is bigger than q.

We find two elements f , g ∈ K such that D ⊗ (f , g)−1 has no hotpoints.

Hence D ⊗ (f , g)−1 = (a, b) and D ' (a, b)⊗ (f , g).

















A local-global principle

Using the methods of Saltman on the study of ramification ofalgebras over surfaces, we have proved the following local-globalprinciple.

I X - a regular, integral surface

I K - the function field of XI ` - a prime not equal to the characteristic of K

I Assume that ` is a unit in OXI Also assume that K contains a primitive `th root of unity.

I X 1 - the set of all codimension one points of XI For x ∈ X 1, Kx - completion of K with respect to the discrete

valuation νx given by x .

I κ(x) -the residue field at x


A local-global principle

Using the methods of Saltman on the study of ramification ofalgebras over surfaces, we have proved the following local-globalprinciple.

I X - a regular, integral surface

I K - the function field of XI ` - a prime not equal to the characteristic of K

I Assume that ` is a unit in OXI Also assume that K contains a primitive `th root of unity.

I X 1 - the set of all codimension one points of XI For x ∈ X 1, Kx - completion of K with respect to the discrete

valuation νx given by x .

I κ(x) -the residue field at x


Unramified cohomology

Hnnr (K/X , Z/`Z) = {ζ ∈ Hn(K , Z/`Z | ζ ∈ Image(Hn(OX ,x , Z/`Z)

→ Hn(K , Z/`Z)) for all x ∈ X 1}

Hnnr (K/X , Z/`Z) is called the unramified cohomology.

For a field F , let

Hnnr (F , Z/`Z) = {ζ ∈ Hn(F , Z/`Z | ζ ∈ Image(Hn(Ov , Z/`Z) →

Hn(F , Z/`Z)) for all discrete valuations v of F}






For a field F , let








For a field F , let




local-global principle continued


Theorem (Parimala-Suresh (2010))Let X and K be as above. Suppose that for every irreducibleclosed curve C on X , κ(C ) is a global field or a local field. Letζ ∈ H3(K , Z/`Z) and α ∈ H2(K , Z/`Z) is a symbol. If for everyx ∈ X 1, there exists fx ∈ K ∗

x such thatζ − α · (fx) ∈ H3

nr (Kx , Z/`Z), then there exists f ∈ K ∗ such thatζ − α · (f ) ∈ H3

nr (K/X , Z/`Z).


local-global principle continued


Theorem (Parimala-Suresh (2010))Let X and K be as above. Suppose that for every irreducibleclosed curve C on X , κ(C ) is a global field or a local field. Letζ ∈ H3(K , Z/`Z) and α ∈ H2(K , Z/`Z) is a symbol. If for everyx ∈ X 1, there exists fx ∈ K ∗

x such thatζ − α · (fx) ∈ H3

nr (Kx , Z/`Z), then there exists f ∈ K ∗ such thatζ − α · (f ) ∈ H3

nr (K/X , Z/`Z).


Applications

Using the above local-global principle and the results of Saltmanmentioned above, we have proved the following:

Theorem (Parimala-Suresh(2007))Let K be the function field of a p-adic curve and q a prime notequal to p. Suppose that K contains a primitive nth root of unity.The every element in H3(K , Z/qZ) is a symbol.

Theorem (Parimala-Suresh(2007))Let K be the function field of a p-adic curve. If p 6= 2, then everyquadratic form over K in at least 9 variables is isotropic.


Applications





Applications





Documents

Division algebras over function fields of surfaces … · Division algebras over function ... be the Brauer group of central simple algebras over K. ... (due to Rowen-Tignol-Sivatski