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 $X(\mathbb{A}_k)$ 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

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

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

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

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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