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 $ X(\mathbb{A}_k)$ be the adelic points of $ X$. Consider the subset of points $ P$ with the property that for every element $ z \in$   Br$ (X)$ the system of elements $ (z_v(P))_v$ has sum of invariants $ =0$.

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 $ X(\mathbb{A}_k)\neq \emptyset$ and $ X(\mathbb{A}_k)^{\rm Br}= \emptyset$ 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. $ X(\mathbb{A}_k)^{\rm Br}\neq \emptyset$ but still has no global point (Skorobogotav found first example).

After one glass of wine, McCallum advocates `` $ X(\mathbb{A}_k)^{\rm Br}$ should be called the set of Brauer points''.

Brauer-Severi variety :A twist of projective space $ \mathbf{P}^n$. Brauer-Severi varieties satisfy the Hasse principle.

BSD conjecture|BSD|Birch and Swinnerton-Dyer : Let $ A$ be an abelian variety over a global field $ K$ and let $ L(A,s)$ be the associated $ L$-function. The Birch and Swinnerton-Dyer conjecture asserts that $ L(A,s)$ extends to an entire function and ord$ _{s=1}
L(A,s)$ equals the rank of $ A(K)$. Moreover, the conjecture provides a formula for the leading coefficient of the Taylor expansions of $ L(A,s)$ about $ s=1$ in terms of invariants of $ A$.

Calabi-Yau variety :An algebraic variety $ X$ over $ \mathbf{C}$ 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 $ \mathbf{P}^1\times\mathbf{P}^1$ or a blowup of $ \mathbf{P}^2$ at up to $ 8$ 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 $ L/K$ and a variety $ X$ over $ L$, try to find a variety $ Y$ over $ K$ such that $ X=Y\times_K L$.

Diophantine set : Let $ R$ be a ring. A subset $ A\subset R^n$ is diophantine over $ R$ if there exists a polynomial $ f\in R[t_1,\ldots, t_n, x_1,\ldots, x_m]$ such that

$\displaystyle A=\{\vec{t}\in R^n :
\exists \vec{x} \in R^m$    such that $\displaystyle f(\vec{t},\vec{x})=0\}.

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

Equivalently, the normalization of the singular surface of degree $ 6$ in $ \mathbf{P}^3$ whose singularities are double lines that form a general tetrahedron.

Over $ \mathbf{C}$ an Enriques surface can be characterized cohomologically as follows: $ H^0(\Omega_X^2)=0$ and $ 2K_X=0$ but $ K_X\neq 0$.

Fano variety|Fano :Anticanonical divisor $ \omega^{\otimes -1}$ 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 $ x^d+y^d=z^d$. 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 $ \mathbf{P}^1$ ramified at 3 points, so they occur in the fund. group of...

More generally $ x_1^d+\cdots+x_n^d =0$ is sometimes called a Fermat variety.

General type :A variety $ X$ is of general type if there is a positive power of the canonical bundle whose global sections determine a rational map $ f:X\rightarrow \mathbf{P}^n$ with $ \dim f(X) = \dim X$. (If $ X$ 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 $ p$-adically for every $ p$, 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 $ R$ be a commutative ring. Hilbert's tenth problem for $ R$ is to determine if there is an algorithm that decides whether or not a given system of polynomial equations with coefficients in $ R$ has a solution over $ R$.

Jacobian :The Jacobian of a nonsingular projective curve $ X$ is an abelian variety whose points are in bijection with the group Pic$ ^0(X)$ 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 $ 2$).

Lang's conjectures :

  1. Suppose $ k$ is a number field and $ X$ is a variety over $ k$ of general type. Then $ X(k)$ is not Zariski dense in $ X$. (Also there are refinements where we specify which Zariski closed subset is supposed to contain $ X(k)$.)
  2. Suppose $ k$ is a number field and $ X$ is a variety over $ k$. All but finitely many $ k$-rational points on $ X$ lie in the special set.
  3. Let $ X$ be a variety over a number field $ k$. Choose an embedding of $ k$ into the complex number $ \mathbf{C}$, and suppose that $ X(\mathbf{C})$ is hyperbolic: this means that every holomorphic map $ \mathbf{C} \to X(\mathbf{C})$ is constant. Then $ X(k)$ 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 $ X$ and $ Y$ be curves and suppose $ f:X\rightarrow Y$ is a degree $ 2$ étale (unramified) cover. The associated Prym variety is the connected component of the kernel of the Albanese map Jac$ (X)\rightarrow$   Jac$ (Y)$. The Prym variety can also be defined as the connected component of the $ -1$ eigenspace of the involution on Jac$ (X)$ induced by $ f$.

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

  1. For any two points $ x,y\in X$ there exists a morphism $ \phi:\mathbf{P}^1\rightarrow X$ such that $ \phi(0)=x$ and $ \phi(\infty)=y$.
  2. For any $ n$ points $ x_1,\ldots, x_n\in X$ there exists a morphism $ \phi:\mathbf{P}^1\rightarrow X$ such that $ \{x_1,\ldots, x_n\}$ is a subset of $ \phi(\mathbf{P}^1)$.
  3. For any two points $ x,y\in X$ there exist morphisms $ \phi_i:\mathbf{P}^1\rightarrow X$ for $ i=1,\ldots, r$ such that $ \phi_1(0)=x$, $ \phi_r(0)=y$, and for each $ i=1,\ldots,r-1$ the images of $ \phi_i$ and $ \phi_{i+1}$ have nontrivial intersection.

Schinzel's Hypothesis :Suppose $ f_1,\ldots,f_r\in
\mathbf{Z}[x]$ are irreducible and no prime divides

$\displaystyle f_1(n) f_2(n) \cdots f_r(n)$

for all $ n\in \mathbf{Z}$. Then there are infinitely many integers $ n$ such that $ \vert f_1(n)\vert,\ldots, \vert f_r(n)\vert$ are simultaneously prime.

Selmer group : Given Galois cohomology definition for any $ A\subset B$. Example $ A=\ker(\phi)$ where $ \phi$ 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 $ X$ by a congruence subgroup of an algebraic group $ G$ that acts transitively on $ X$. Examples include moduli spaces $ X_0(N)$ 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 $ X$ is the Zariski closure of the union of all positive-dimensional images of morphisms from abelian varieties to $ X$. Note that this contains all rational curves (since elliptic curves cover $ \mathbf{P}^1$).

Torsor : Let $ B$ be a variety over a field $ k$ and let $ G$ be an algebraic group over $ k$. A left $ B$-torsor under $ G$ is a $ B$-scheme $ X$ with a $ B$-morphism $ G\times_k X \rightarrow X$ such that for some étale covering $ \{U_i \rightarrow B\}$ there is a $ G$-equivariant isomorphism of $ U_i$-schemes from $ X\times U_i$ to $ G\times U_i$, for all $ i$. If $ B=$Spec$ (k)$ these are also called principal homogenous spaces.

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

Weak approximation : For a projective variety $ X$ over a global field, say weak approximation holds if $ X(K)$ is dense in the adelic points $ X(\mathbb{A}_K)$. Simplest example where it holds: $ \mathbf{P}^0$, also $ \mathbf{P}^1$. It does not hold for an elliptic curve over $ K$. (For example, if $ E$ has rank 0 it clearly doesn't hold... but more generally could divide all generators by $ 2$ 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 $ X$ 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 $ \mathbf{Q}$ with certain behavior at $ 3$, $ 5$, $ 13$, 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 $ \mathbf{P}^1$ satisfies weak approximation).'

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