Mckenzie West - Research Statement
As research for my doctoral thesis, I have studied the Brauer-Manin obstruction to the Hasse
principle for a particular family of cubic surfaces in great depth. Though the details are intricate,
I find the motivating question here to have beauty in its simplicity:
Given an equation of polynomials, can one prove whether or not it has a rational solution?
To reach an answer, I make use of the geometry of the surfaces and class field theory. A rough
outline of the obstruction and other necessary definitions will be provided in section 1. Then in
section 2, I will describe my results on cubic surfaces.
Given the generality of the question above and the Brauer-Manin obstruction, I have been able
to work with others on similar research topics. Originating as a project at the 2015 Arizona Winter
School, I am working with several other graduate students to examine the geometry of certain K3
surfaces, in order to gain a more extensive insight on Brauer-Manin style questions. The Algebraic
Geometry BOOTCAMP in July 2015, has provided me with another chance for collaboration. Our
goal is to find interesting examples of rational degree 2 del Pezzo surfaces with a unique Brauer
quotient. The ongoing work for these projects will be outlined in sections 3 and 4.
Due to my arithmetic geometry background, I am an in a position to collaborate with mathematicians in a wide range of specialties. My research employs techniques of number theory, algebra,
and geometry, making me a well-rounded researcher. There are opportunities for undergraduates
interested in beginning mathematics research to study explicit examples which will provide new
insight on the area. Moreover, the rational point problem is an expanding area of research providing
a plethora of topics for graduate students to study. I am excited to mentor students of all levels
in their pursuit of mathematical knowledge. Additionally, I employ mathematical software such as
MAGMA and Sage, and look to develop new modules for these programs. For instance, Sage is
open-source and provides a setting where students with little to no background in programming
can build and implement applications.
1. Background
Let k be a number field. Denote by Ωk the set of absolute values, or places, on k, up to
equivalence. For each v ∈ Ωk , kv is the completion of k with respect to v. A projective k-variety X
is a set of non-zero solutions to finitely many homogeneous polynomial equations with coefficients
in k, and X(F ) is the subset of those solutions which lie in the field F . We call X a nice k-variety
if it is smooth, projective, and geometrically integral.
Q
Definition. A class of nice k-varieties, S, satisfies the Hasse principle if v∈Ωk X(kv ) 6= ∅ if and
only if X(k) 6= ∅ for every X ∈ S.
To see exactly what this means, take S to be a class of non-singular nice Q-varieties which
satisfy the Hasse principle. Take
Q any X ∈ S. By the Weil conjectures (cf. [Har77, Appendix C])
and Hensel’s Lemma, showing v∈Ωk X(Qv ) 6= ∅ can be reduced to the statements X(R) 6= ∅ and
X(Z/pZ) 6= ∅ for finitely many primes. If R and Z/pZ points exist, then X(Q) 6= ∅ by the Hasse
principle.
Generally, computing X(kv ) is much easier than computing X(k), so the Hasse principle is very
useful. Unfortunately, there are many classes of varieties which do not satisfy the Hasse principle.
The first published diagonal cubic surface that fails to satisfy the Hasse principle is that of Cassels
and Guy [CG66].
Theorem (Cassels and Guy). The surface defined by
5x3 + 12y 3 + 9z 3 + 10w3 = 0,
does not satisfy the Hasse principle.
As a cohomological and geometric generalization of quadratic reciprocity, Manin, [Man71,
Man74], noted a relationship between the Brauer group of a variety and its k-rational points,
1
X(k). The Brauer group of a nice k-variety is denoted by Br X. For a fixed A ∈ Br X, there is a
commutative diagram with the following shape:
X(k)
Y
X(kv )
v∈Ωk
evA
0
Br k
evA
M
Br kv
φA
inv
Q/Z
0
v∈Ωk
Each of the arrows in the picture represent a map, L
and the commutativity implies that no
matter which maps are taken from X(k), the image in
v∈Ωk Br kv is the same. Moreover, the
bottom row is exact, implying that the elements of Br k map to 0 in Q/Z. Therefore, if X(k) is
traced down and across its image is {0} in Q/Z so, abusing notation, we say φA (X(k)) = {0}.
Subsequently, one can make the following definitions:
Q
Definition. Suppose X is a nice variety and X(Ak ) = v∈Ωk X(kv ). The Brauer Set of X over k
is the set
\
X(Ak )Br :=
(1)
φ−1
A (0).
A∈Br X
We say X has a Brauer-Manin obstruction to the Hasse principle if X(Ak ) 6= ∅ and X(Ak )Br = ∅.
The Brauer-Manin obstruction is the only obstruction to the Hasse principle for the family S of
nice k-varieties if for every X ∈ S either X satisfies the Hasse principle or X has a Brauer-Manin
obstruction to the Hasse principle.
2. Cubic Surfaces
Colliot-Télène and Sansuc, [CTS80], conjectured that the Brauer-Manin obstruction is the only
obstruction to the Hasse principle for cubic surfaces, and more generally for rationally connected
nice varieties. In [BSD75], Birch and Swinnerton-Dyer used Manin’s method to construct several
counterexamples to the Hasse Principle. A portion of these examples have the following form
linear ∗ quadratic norm = cubic norm,
(2)
which will be refered to as BSD cubic surfaces. Through a few examples, they provided some intuition as to why the Brauer-Manin obstruction might exist for such surfaces. Birch and SwinnertonDyer’s argument however, did not provide general proof of the existence of Brauer-Manin obstructions; they simply impart an intuition through a few specific examples. My results provide proof
of their sketches. Consider the following setup:
Let L be degree 6 extension of the number field k with Galois group Gal(L/k) = S3 .
Suppose K = k(θ) is the unique quadratic extension of k contained in L and that
θ is the quadratic conjugate of θ. Take any φ, ψ ∈ L so that (1, φ, ψ) is a basis for
a cubic extension over k. Let φ0 , φ1 , φ2 and ψ0 , ψ1 , ψ2 be the Galois conjugates
of φ and ψ, respectively, over k. Then every one of the BSD cubic surfaces in (2) is
isomorphic to one of the form
(?)
2
Y
(x + φi z + ψi w),
dy(x + θy)(x + θy) =
(3)
i=0
for some d ∈ k where x, y, z, and w are variables.
Note that we are working in projective space, ie. for any non-zero constant λ, there is an equality
(x : y : z : w) = (λx : λy : λz : λw) which makes a seemingly four-dimensional space into a threedimensional one. The shape of (3) is incredibly general; for instance, the product (x + θy)(x + θy)
can represent any quadratic polynomial.
2
The Brauer Group: Given the definition of the Brauer set, (1), it is logical that the first step
in computing the Brauer-Manin obstruction is to determine the algebras, A ∈ Br X. For a cubic
surface, as in the BSD case, one must find the exactly 27 lines lying on X and consider their relation
to one another. Generally the coefficients in these linear equations will not be defined over k but
instead over some larger field. The Galois orbit of the coefficients will determine the Brauer group
of X. Though lines are rather simple objects, finding all 27 in generality is difficult. I was able to
do so with the help of MAGMA, ultimately proving a general statement about the BSD surfaces.
Theorem (West). Forthe surfaces X described in (?), Br X/ Br k ' Z/3Z or {1}. Moreover, in
∈ Br X/ Br k is non-trivial.
the case of Z/3Z, A = L/K, x+θy
y
Notice that x + θy and y appear on the left-hand side of (3). This is no coincidence, they
define two planes which intersect the surface at exactly 3 lines each; one of the few methods of
determining Brauer elements, A ∈ Br X, takes advantage of such nice shapes.
The Invariant Map: After having found some A ∈ Br X and thus ev
PA , we must determine the
map inv. This is done one absolute value, v, at a time. That is inv = v∈Ωk invv , a finite sum.
Theorem (West). Assume the notation of (?) and that L is unramified over K, ie. primes
behave Q
nicely. All places v ∈ Ωk for which |d|v = 1 have invv (evA (Pv )) = 0 for all points
(Pv ) ∈ v∈Ωk X(kv ).
The arguments used in the proof cannot easily be extended to those v for which |d|v < 1.
However, specifying d to have certain absolute values has allowed me to find an infinite class of
surfaces X that have a Brauer-Manin obstruction to the Hasse principle.
The Rational Case and Examples:
In the case k = Q, the set {p : p is prime} ∪ {∞}
enumerates Ωv . Specifically the prime p has |pn · qr |p = 1/pn if p - rq.
Theorem (West). Assume the notation of (?). Let k = Q and L be unramified over K. Suppose
there is a prime p dividing θθ such that p factors into the product of two primes in the integers of
L. Then
2
Y
py(x + θy)(x + θy) =
(x + φi z + ψi w),
i=0
has a Brauer-Manin obstruction to the Hasse principle.
To verify that my research corresponds to the ideas of BSD, we must consider the examples
they provide in this new context.
Example. Suppose X is given by
32y(x2 − 13xy + 48y 2 ) = x3 + 2x2 w − xz 2 − 3xzw + xw2 + z 3 − zw2 + w3 .
Then X has a Brauer-Manin obstruction to the Hasse
principle because inv2 (evA (P2 )) 6= 0 and
Q
invv (evA (Pv )) = 0 otherwise, for all points (Pv ) ∈ v∈Ωk X(Qv ).
There are few published examples where the invariant map has two or more non-zero summands.
Given my results, examples of this can be found quickly.
Example. Suppose the φi satisfy φ3i = −φi − 1 and θ, θ are the roots of T 2 − 4T + 35.
Then the surface X defined by
52 · 7y(x + θy)(x + θy) =
2
Y
(x + φi z + φ2i w),
i=0
has a Brauer-Manin obstruction to the Hasse principle with the invariant map, inv, being
inv5 (evA (P )) + inv7 (evA (P )) 6= 0,
3
for all points P ∈
Q
v∈Ωk
X(Qv ). In particular, X(Q) ⊆ X(AQ )Br =
T
A∈Br X
φ−1
A ({0}) = ∅.
Future Work: The work described above comprises [Wes15], yet there are further questions to
consider for these Birch and Swinnerton-Dyer surfaces.
(Q2.1) If d = 1, do the surfaces satisfy the Hasse principle?
(Q2.2) What can be said if the extension L is ramified over K?
(Q2.3) Is there a more general statement to be made about cubic surfaces?
3. K3 Surfaces
A K3 surface is a surface which is diffeomorphic to a quartic surface over C. Therefore, examining the Fermat quartic,
x4 + y 4 + z 4 + w4 = 0,
will provide some insight on the geometry of a general K3 surface. However, the question of
rational points cannot be inferred by looking over C. There is little known about the BrauerManin computations in terms of K3 surfaces, so any new classes of examples are worth examining.
Fix a non-zero constant D and consider the surface, XD , given by
w2 = x6 + y 6 + z 6 + Dx2 y 2 z 2 ,
(4)
in weighted projective space, meaning (x : y : z : w) = (λx : λy : λz : λ3 w) for any non-zero
constant λ. As in section 2, this computation must begin by determining the exceptional curves on
the surface, which are analogous to lines in the cubic case. For K3 surfaces the exceptional curves
are not necessarily lines, so we must be more creative in finding them.
The Picard Group of a surface X, Pic(X) is the equivalence classes of exceptional curves which
lie on X in the complex plane. Our first task is to find the Picard number of XD , which is the value
r such that Zr is isomorphic to Pic(XD ). For K3 surfaces the Picard number is always bounded
above by 20.
Theorem (BCFLNVW). For XD as in (4), the Picard number is 19 for all but finitely many cases.
In those cases the Picard number is 20.
After we had this information, I was able to lead the task of finding curves on XD . With the
help of MAGMA, I developed code that would take a general surface, XD and compute a large
number of curves which lie on it. Through my expertise in this area, my collaborators and I were
able to find 19 curves which generate Pic XD . This list brings us one step closer to Brauer-Manin
style computations includeing the following theorem.
Theorem (BCFLNVW). For XD as in (4) and for certain D, we have
ker(Br XD → Br X D )/ Br Q ' (Z/2Z)3 .
This is ongoing work with a number of outstanding questions.
(Q3.1) What exact algebras can be written down in the Brauer quotient?
(Q3.2) Can the invariant map be calculated generically?
(Q3.3) Can the methods used here be extended to other examples or general K3 surfaces?
4. Degree 2 del Pezzo Surfaces
At the 2015 Algebraic Geometry BOOTCAMP, I investigated degree 2 del Pezzo surfaces with
several other young mathematicians. The quotient Br X/ Br k was mentioned briefly in section 2;
computing this quotient is necessary when determining the existence of a Brauer-Manin obstruction.
For del Pezzo surfaces of degree 2, Br X/ Br k is isomorphic to one of the following [Cor05, 1.4.1]:
{1}, (Z/2Z)s for 1 ≤ s ≤ 6, (Z/3Z)t for 1 ≤ t ≤ 2, (Z/2Z)u × Z/4Z for 0 ≤ u ≤ 2, or (Z/4Z)2 .
Published examples of degree 2 del Pezzo surfaces defined over Q with Brauer quotients equal to
4
(Z/2Z)s for 3 ≤ s ≤ 6, (Z/2Z)t × Z/4Z for 1 ≤ t ≤ 2, or (Z/Z)2
are either rare or non-existent. Our goal is to find and study the geometry of such examples. We
would like to determine whether or not a degree 2 del Pezzo surface over Q with the given Brauer
quotient exists and if not, is there a number field k over which they do exist?
During the week of the workshop, the members of my group worked at these questions from
many different perspectives. I chose to work toward the task of writing down as many surfaces, X,
as possible and computing Br X/ Br Q.
Suppose f4 (x, y, z) is a homogeneous polynomial of degree 4 with rational coefficients. For our
purposes, a del Pezzo surface of degree 2 is a surface in weighted projective space defined by the
equation
w2 = f4 (x, y, z),
where w has weight 2. Then the exceptional curves on X are again lines with the following
construction:
Consider the projection into 2 dimensions, forgetting the w coordinate. The curve
0 = f4 (x, y, z) has 28 bi-tangents, lines which intersect it twice tangentially. Take the
preimage of those lines in 3-dimensional space. These 56 preimages generate Pic(X).
I wrote MAGMA code that runs through this construction for any degree 2 del Pezzo surface. It is
able to find the 56 lines, compute the number field L over which they are defined, and determine
the Brauer quotient.
Thus far, none of the interesting examples have been found. I am currently working with a
fellow graduate student at Emory University to determine whether or not a surface of the form
w2 = ax4 + by 4 + cz 4
can possibly have an interesting Brauer quotient. In addition to this work, we have several remaining
questions:
(Q4.1) Do interesting examples exist over Q?
(Q4.2) What geometric structure will force or prevent the existence of an interesting example?
5. Future Work
The three sets of questions listed above provide a firm basis for the development for my research
program. They are both tactile and expandable. For (Q2.1)-(Q2.3), a number of directions may
be taken. To study more about the Brauer-Manin obstruction for cubic surfaces, I must use my
observations in the BSD case to infer about all cubic surfaces. To answer (Q3.1)-(Q3.3), I plan to
study what are called transcendental Brauer elements, as it is likely they are necessary in BrauerManin computations. In the case of (Q4.1) and (Q4.2), my focus will be on the computational
elements of the stated problem. As I see more examples, a pattern in the structure will become
apparent.
Beyond the projects listed above are a number of related mathematical quandries. I plan to
expand my study on K3 surfaces and to develop an understanding of even more classes of surfaces,
such as Enriques surfaces. Brauer groups are among the most relevant research topics in algebra and
number theory because rational solutions are so important to many areas of research. Moreover,
the geometry utilized when computing Brauer-Manin obstructions can be applied in many other
contexts. My research program is well-founded and my arithmetic geometry background is highly
relevant; this makes me a valuable asset for any mathematics department.
I am excited to work with students on projects related to my own. Undergraduates can quickly
become involved with examples; students who have completed a linear algebra or a multivariable
calculus course have the necessary background to start working with explicit surfaces. With the
aid of a computer, they will be able to find local points, exceptional curves, and Brauer elements.
For graduate students interested in these topics, there are limitless published works concerning the
Brauer-Manin obstruction. A vast array of topics remain available for study because it is clear that
we do not yet have all of the answers.
5
References
[BSD75]
Bryan Birch and Peter Swinnerton-Dyer. The Hasse problem for rational surfaces. J.
reine angew. Math, 274(275):164–174, 1975.
[CG66]
John Cassels and Michael Guy. On the Hasse principle for cubic surfaces. Mathematika,
13(02):111–120, 1966.
[Cor05]
Patrick Corn. Del Pezzo surfaces and the Brauer-Manin obstruction. PhD thesis,
Harvard University, 2005.
[CTKS87]
Jean-Louis Colliot-Thélène, Dimitri Kanevsky, and Jean-Jacques
Arithmétique des surfaces cubiques diagonales. Springer, 1987.
[CTS80]
Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc. La descente sur les variétés
rationnelles. Journées de géométrie algébrique d’Angers, pages 223–237, 1980.
[Gro64]
Alexander Grothendieck. Le groupe de Brauer: I. Algèbres d’Azumaya et interprétations diverses. Séminaire Bourbaki, 9:199–219, 1964.
[Gro65]
Alexander Grothendieck. Le groupe de Brauer: Ii. Théories cohomologiques. Séminaire
Bourbaki, 9:287–307, 1965.
[Har77]
Robin Hartshorne. Algebraic geometry, volume 52. Springer Science & Business Media,
1977.
[Jah14]
Jorg Jahnel. Brauer groups, Tamagawa measures, and rational points on algebraic
varieties, volume 198. American Mathematical Soc., 2014.
[Man71]
Yuri Manin. Le groupe de Brauer-Grothendieck en géométrie diophantienne. In Actes
du Congres International des Mathématiciens (Nice, 1970), Tome, volume 1, pages
401–411. World Scientific, 1971.
[Man74]
Yuri Manin. Cubic forms: algebra, geometry, arithmetic, volume 4. North-Holland
Publishing Co., Amsterdam, 1974.
[Mil80]
James Milne. Etale cohomology (PMS-33). Number 33 in Princeton Mathematical
Series. Princeton University Press, 1980.
Sansuc.
[MSTVA14] Kelly McKinnie, Justin Sawon, Sho Tanimoto, and Anthony Várilly-Alvarado. Brauer
groups on K3 surfaces and arithmetic applications. arXiv preprint arXiv:1404.5460,
2014.
[Poo08]
Bjorn Poonen. Rational points on varieties. Notes available at http://www-math. mit.
edu/ poonen/papers/Qpoints. pdf, 2008.
[SD93]
Peter Swinnerton-Dyer. The Brauer group of cubic surfaces. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 113-03, pages 449–460. Cambridge
Univ Press, 1993.
[SD99]
Peter Swinnerton-Dyer. Brauer-Manin obstructions on some del Pezzo surfaces. In
Mathematical Proceedings of the Cambridge Philosophical Society, volume 125-02,
pages 193–198. Cambridge Univ Press, 1999.
[Sko01]
Alexei Skorobogatov. Torsors and rational points, volume 144. Cambridge University
Press, 2001.
[VAV14]
Anthony Várilly-Alvarado and Bianca Viray. Arithmetic of del Pezzo surfaces of degree
4 and vertical Brauer groups. Advances in Mathematics, 255:153–181, 2014.
[Wes15]
Mckenzie West. On a family of norm form cubic surfaces. Preprint, arXiv: 1510.03769,
2015.
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