# Glossary

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 :

1. The process of expressing the rational points on a variety as the union of images of rational points from other varieties.
2. 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 :

1. 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 .)
2. Suppose  is a number field and  is a variety over . All but finitely many -rational points on  lie in the special set.
3. 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 .

1. For any two points there exists a morphism such that and .
2. For any points there exists a morphism such that is a subset of .
3. 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

for all . Then there are infinitely many integers  such that are simultaneously prime.

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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