**Abelian variety :**
A smooth projective geometrically integral group variety over a field.
Over the complex numbers abelian varieties are tori.

**Brauer-Manin obstruction :**
The terminology is utterly awful!
Many families don't satisfy Hasse Principle.
One explanation of Manin (see his paper):
a cohomological obstruction using the Brauer group of the
variety.

If a variety has a local point everywhere then it has an adelic point. Manin defined, using a cohomological condition involving Brauer group, a subset of the adelic points that must contain the global points. Let be the adelic points of . Consider the subset of points with the property that for every element Br the system of elements has sum of invariants .

The B-M is an interesting construction in English. It is a nounal-phrase defined purely in terms of the sentences in which in which it may occur. There is no such actual object ``the Brauer-Manin obstruction''.

Example: A variety that satisfies and is a counterexample to the Hasse principle explained by the Brauer-Manin obstruction.

For a long time people were interested in whether there are counterexamples ot Hasse principle not explained by the Brauer-Manin obstruction. but still has no global point (Skorobogotav found first example).

After one glass of wine, McCallum advocates `` should be called the set of Brauer points''.

**Brauer-Severi variety :**A twist of projective space
. Brauer-Severi varieties satisfy the Hasse principle.

**BSD conjecture|BSD|Birch and Swinnerton-Dyer :** Let be an
abelian variety over a global field and let be the
associated -function. The Birch and Swinnerton-Dyer conjecture
asserts that extends to an entire function and
ord equals the rank of . Moreover, the conjecture provides
a formula for the leading coefficient of the Taylor expansions of
about in terms of invariants of .

**Calabi-Yau variety :**An algebraic variety over
is a Calabi-Yau variety if it has trivial canonical sheaf (i.e., the
canonical sheaf is isomorphic to the structure sheaf).
[Noriko just deleted the simply connected assumption.]

**Del Pezzo surface :**
A Del Pezzo surface is a Fano variety of dimension two.

It can be shown that the Del Pezzo surfaces are exactly
the surfaces that are geometrically either
or a
blowup of
at up to points in general position.
By *general position* we mean that
no three points lie on a line, no six points lie on a conic,
and no eight lie points lie on a singular cubic with one of
the eight points on the singularity.

**Descent :**

- The process of expressing the rational points on a variety as the union of images of rational points from other varieties.
- The descent problem is as follows: Given a field extension and a variety over , try to find a variety over such that .

**Diophantine set :**
Let be a ring. A subset
is diophantine over
if there exists a polynomial
such that

such that

**Enriques Surface :**
A quotient of a K3 surface by a fixed-point free involution.

Equivalently, the normalization of the singular surface of degree in whose singularities are double lines that form a general tetrahedron.

Over an Enriques surface can be characterized cohomologically as follows: and but .

**Fano variety|Fano :**Anticanonical divisor
is ample. This class of varieties is ``simple'' or
``close to rational''. For example, one conjectures that Brauer-Manin
is only obstruction. Manin-Batyrev conjecture: asymptotic for number
of points of bounded height. A Fano variety of dimension two is
also called a Del Pezzo surface.

**Fermat curve :**A curve of the form
.
Good examples of many phenomenon. Good source
of challenge problems. (E.g., FLT.) Lot of symmetry so you can
compute a lot with them. Computations are surprising and nontrivial.
They're abelian covers of
ramified at 3 points, so they occur
in the fund. group of...

More generally is sometimes called a Fermat variety.

**General type :**A variety is
of general type if there is a positive
power of the canonical bundle whose global
sections determine a rational map
with
.
(If is of general type then there exists
some positive power of the canonical bundle such
that the corresponding map is birational to its image.)

``It is a moral judgement of geometers that you would be wise to stay away from the bloody things.'' - Swinnerton-Dyer

**Hardy-Littlewood circle method :**
An analytic method for obtaining asymptotic formulas for the
number of solutions to certain equations satisfying certain
bounds.

**Hasse principle :**A family of varieties satisfies the Hasse principle if
whenever a variety in the family has points everywhere locally
it has a point globally. Here ``everywhere locally'' means
over the reals and -adically for every , and ``globally''
means over the rationals.

Everywhere local solubility is necessary for global solubility. Hasse proved that it is also sufficient in the case of quatratic forms.

**Hilbert's tenth problem :**Let be a commutative ring.
Hilbert's tenth problem for is to determine if there is an
algorithm that decides whether or not a given system of polynomial
equations with coefficients in has a solution over .

**Jacobian :**The Jacobian of a nonsingular projective
curve is an abelian variety whose points are in bijection with
the group
Pic of isomorphism classes of invertible
sheaves (or divisor classes) of degree 0.

**K3 surface :**
A surface with trivial canonical bundle and trivial fundamental
group (i.e., a Calabi-Yau variety of dimension ).

**Lang's conjectures :**

- Suppose is a number field and is a variety over of general type. Then is not Zariski dense in . (Also there are refinements where we specify which Zariski closed subset is supposed to contain .)
- Suppose is a number field and is a variety over . All but finitely many -rational points on lie in the special set.
- Let be a variety over a number field . Choose an embedding of into the complex number , and suppose that is hyperbolic: this means that every holomorphic map is constant. Then is finite.

**Local to global principle :**Another name for the Hasse principle.

**Picard group :**The Picard group
of a variety is the group of isomorphism classes of invertible
sheaves.

**Prym variety :**A Prym variety is an abelian
variety constructed in the following way. Let and
be curves and suppose
is a degree
étale (unramified) cover. The associated Prym variety is
the connected component of the kernel of the Albanese
map
Jac Jac.
The Prym variety can also be defined as the connected component
of the eigenspace of the involution on
Jac induced by .

**Rationally connected variety :**
There are three definitions of rationally connected. These are equivalent
in characteristic zero but not in characteristic .

- For any two points there exists a morphism such that and .
- For any points there exists a morphism such that is a subset of .
- For any two points there exist morphisms for such that , , and for each the images of and have nontrivial intersection.

**Schinzel's Hypothesis :**Suppose
are irreducible and no prime divides

**Selmer group :**
Given Galois cohomology definition for any
.
Example
where is an isogeny of
abelian variety.
Accessible.
It's what we can compute, at least in theory.

**Shimura variety :** A variety having a Zariski open subset
whose set of complex points is analytically isomorphic to a quotient
of a bounded symmetric domain by a congruence subgroup of an
algebraic group that acts transitively on . Examples include
moduli spaces of elliptic curves with extra structure and
Shimura curves which parametrize quaternionic multiplication abelian
surfaces with extra structure.

**Special Set :**
The (algebraic) special set of a variety is the Zariski closure
of the union of all positive-dimensional images of morphisms
from abelian varieties to . Note that this contains
all rational curves (since elliptic curves cover
).

**Torsor :**
Let be a variety over a field and let be an algebraic
group over . A left -torsor under is a -scheme
with a -morphism
such that for some étale covering
there is a -equivariant isomorphism of -schemes
from
to
, for all .
If
Spec these are also called principal homogenous
spaces.

**Waring's problem :**Given , find the smallest number
such that every positive integer is a sum of positive th powers.
The ``easier'' Waring's problem refers to the analogous problem where
the th powers are permitted to be either positive or negative.
Modification: Given , find the smallest number such
that every sufficiently large positive integer is a sum of
positive th powers.

**Weak approximation :**
For a projective variety over a global field, say weak
approximation holds if is dense in the adelic points
. Simplest example where it holds:
,
also
. It does not hold for an elliptic curve over .
(For example, if has rank 0 it clearly doesn't hold... but more
generally could divide all generators by and choose a prime that
splits completely.)

Example: ``Weak approximation does not hold for cubic surfaces.''

Example: ``The theory of abelian descent in some cases reduces the question of whether the Brauer-Manin obstruction is the only obstruction to Hasse on a base variety to the question of whether weak approximation holds for a universal torsor.''

Example: ``Weak approximation on a moduli space of varieties yields the existence of varieties over a global field satisfying certain local conditions. For example, we want to know there is an elliptic curve over with certain behavior at , , , as long as can do it over local fields with that behavior, weak approximation on the moduli space gives you a global curve that has those properties (because satisfies weak approximation).'

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