Level structure (algebraic geometry)

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In algebraic geometry, a level structure on a space X is an extra structure attached to X that shrinks or eliminates the automorphism group of X, by demanding automorphisms to preserve the level structure; attaching a level structure is often phrased as rigidifying the geometry of X.[1][2]

In applications, a level structure is used in the construction of moduli spaces; a moduli space is often constructed as a quotient. The presence of automorphisms poses a difficulty to forming a quotient; 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 (see #Example: an abelian scheme). There is a level structure attached to a formal group called a Drinfeld level structure, introduced in (Drinfeld 1974).[3]

Level structures on elliptic curves

Classically, level structures on elliptic curves E = C / Λ {\displaystyle E=\mathbb {C} /\Lambda } {\displaystyle E=\mathbb {C} /\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 Z ⊕ Z ⋅ τ {\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau } {\displaystyle \mathbb {Z} \oplus \mathbb {Z} \cdot \tau } for τ ∈ h {\displaystyle \tau \in {\mathfrak {h}}} {\displaystyle \tau \in {\mathfrak {h}}} in the upper-half plane. Then, the lattice generated by 1 / n , τ / n {\displaystyle 1/n,\tau /n} {\displaystyle 1/n,\tau /n} gives a lattice which contains all n {\displaystyle n} {\displaystyle n}-torsion points on the elliptic curve denoted E [ n ] {\displaystyle E[n]} {\displaystyle E[n]}. In fact, given such a lattice is invariant under the Γ ( n ) ⊂ SL 2 ( Z ) {\displaystyle \Gamma (n)\subset {\text{SL}}_{2}(\mathbb {Z} )} {\displaystyle \Gamma (n)\subset {\text{SL}}_{2}(\mathbb {Z} )} action on h {\displaystyle {\mathfrak {h}}} {\displaystyle {\mathfrak {h}}}, where

Γ ( n ) = ker ( SL 2 ( Z ) → SL 2 ( Z / n ) ) = { M ∈ SL 2 ( Z ) : M ≡ ( 1 0 0 1 )  (mod n) } {\displaystyle {\begin{aligned}\Gamma (n)&={\text{ker}}({\text{SL}}_{2}(\mathbb {Z} )\to {\text{SL}}_{2}(\mathbb {Z} /n))\\&=\left\{M\in {\text{SL}}_{2}(\mathbb {Z} ):M\equiv {\begin{pmatrix}1&0\\0&1\end{pmatrix}}{\text{ (mod n)}}\right\}\end{aligned}}} {\displaystyle {\begin{aligned}\Gamma (n)&={\text{ker}}({\text{SL}}_{2}(\mathbb {Z} )\to {\text{SL}}_{2}(\mathbb {Z} /n))\\&=\left\{M\in {\text{SL}}_{2}(\mathbb {Z} ):M\equiv {\begin{pmatrix}1&0\\0&1\end{pmatrix}}{\text{ (mod n)}}\right\}\end{aligned}}}

hence it gives a point in Γ ( n ) ∖ h {\displaystyle \Gamma (n)\backslash {\mathfrak {h}}} {\displaystyle \Gamma (n)\backslash {\mathfrak {h}}}[4] called the moduli space of level N structures of elliptic curves Y ( n ) {\displaystyle Y(n)} {\displaystyle Y(n)}, which is a modular curve. In fact, this moduli space contains slightly more information: the Weil pairing

e n ( 1 n , τ n ) = e 2 π i / n {\displaystyle e_{n}\left({\frac {1}{n}},{\frac {\tau }{n}}\right)=e^{2\pi i/n}} {\displaystyle e_{n}\left({\frac {1}{n}},{\frac {\tau }{n}}\right)=e^{2\pi i/n}}

gives a point in the n {\displaystyle n} {\displaystyle n}-th roots of unity, hence in Z / n {\displaystyle \mathbb {Z} /n} {\displaystyle \mathbb {Z} /n}.

Example: an abelian scheme

Let X → S {\displaystyle X\to S} {\displaystyle X\to S} be an abelian scheme 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 σ 1 , … , σ 2 g {\displaystyle \sigma _{1},\dots ,\sigma _{2g}} {\displaystyle \sigma _{1},\dots ,\sigma _{2g}} such that[5]

  1. for each geometric point s : S → X {\displaystyle s:S\to X} {\displaystyle s:S\to X}, σ i ( s ) {\displaystyle \sigma _{i}(s)} {\displaystyle \sigma _{i}(s)} form a basis for the group of points of order n in X ¯ s {\displaystyle {\overline {X}}_{s}} {\displaystyle {\overline {X}}_{s}},
  2. m n ∘ σ i {\displaystyle m_{n}\circ \sigma _{i}} {\displaystyle m_{n}\circ \sigma _{i}} is the identity section, where m n {\displaystyle m_{n}} {\displaystyle m_{n}} is the multiplication by n.

See also: modular curve#Examples, moduli stack of elliptic curves.

See also

Notes

  1. Mumford, Fogarty & Kirwan 1994, Ch. 7.
  2. Katz & Mazur 1985, Introduction
  3. Deligne, P.; Husemöller, D. (1987). "Survey of Drinfeld's modules" (PDF). Contemp. Math. 67 (1): 25–91. doi:10.1090/conm/067/902591.
  4. Silverman, Joseph H. (2009). The arithmetic of elliptic curves (2nd ed.). New York: Springer-Verlag. pp. 439–445. ISBN 978-0-387-09494-6. OCLC 405546184.
  5. Mumford, Fogarty & Kirwan 1994, Definition 7.1.

References

Further reading