Level structure (algebraic geometry)
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In
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a level structure on a
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...
''X'' is an extra structure attached to ''X'' that shrinks or eliminates the
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of ''X'', by demanding automorphisms to preserve the level structure; attaching a level structure is often phrased as rigidifying the geometry of ''X''. In applications, a level structure is used in the construction of
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
s; a moduli space is often constructed as a quotient. The presence of automorphisms poses a difficulty to forming a
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
; thus introducing level structures helps overcome this difficulty. There is no single definition of a level structure; rather, depending on the space ''X'', one introduces the notion of a level structure. The classic one is that on an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
(see #Example: an abelian scheme). There is a level structure attached to a
formal group In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one o ...
called a Drinfeld level structure, introduced in .


Level structures on elliptic curves

Classically, level structures on elliptic curves E = \mathbb/\Lambda are given by a lattice containing the defining lattice of the variety. From the moduli theory of elliptic curves, all such lattices can be described as the lattice \mathbb\oplus \mathbb\cdot \tau for \tau \in \mathfrak in the upper-half plane. Then, the lattice generated by 1/n, \tau/n gives a lattice which contains all n-torsion points on the elliptic curve denoted E /math>. In fact, given such a lattice is invariant under the \Gamma(n) \subset \text_2(\mathbb) action on \mathfrak, where
\begin \Gamma(n) &= \text(\text_2(\mathbb) \to \text_2(\mathbb/n)) \\ &= \left\ \end
hence it gives a point in \Gamma(n)\backslash\mathfrak called the moduli space of level N structures of elliptic curves Y(n), which is a
modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular ...
. In fact, this moduli space contains slightly more information: the
Weil pairing Weil may refer to: Places in Germany *Weil, Bavaria *Weil am Rhein, Baden-Württemberg *Weil der Stadt, Baden-Württemberg *Weil im Schönbuch, Baden-Württemberg Other uses * Weil (river), Hesse, Germany * Weil (surname), including people with ...
e_n\left(\frac, \frac\right) = e^
gives a point in the n-th roots of unity, hence in \mathbb/n.


Example: an abelian scheme

Let X \to S be an
abelian scheme In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular function ...
whose geometric fibers have dimension ''g''. Let ''n'' be a positive integer that is prime to the residue field of each ''s'' in ''S''. For ''n'' ≥ 2, a level ''n''-structure is a set of sections \sigma_1, \dots, \sigma_ such that # for each geometric point s : S \to X, \sigma_(s) form a basis for the group of points of order ''n'' in \overline_s, # m_n \circ \sigma_i is the identity section, where m_n is the multiplication by ''n''. See also: modular curve#Examples,
moduli stack of elliptic curves In mathematics, the moduli stack of elliptic curves, denoted as \mathcal_ or \mathcal_, is an algebraic stack over \text(\mathbb) classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves \mathcal_. In parti ...
.


See also

*
Siegel modular form In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular form ...
*
Rigidity (mathematics) In mathematics, a rigid collection ''C'' of mathematical objects (for instance sets or functions) is one in which every ''c'' ∈ ''C'' is uniquely determined by less information about ''c'' than one would expect. The above statement do ...
*
Local rigidity Local rigidity theorems in the theory of discrete subgroups of Lie groups are results which show that small deformations of certain such subgroups are always trivial. It is different from Mostow rigidity and weaker (but holds more frequently) than ...


Notes


References

* * * *


Further reading


Notes on principal bundles
* J. Lurie
Level Structures on Elliptic Curves.
Algebraic geometry {{algebraic-geometry-stub