1
1
Spinor class fields for lattices in function fields
Luis Arenas-Carmona
Departamento de Matematicas, Universidad de Chile learenas@uchile.cl
June 3, 2009
Abstract
The theory of spinor class fields allows the study of the set of maxi- mal orders in a central simple algebra over a number field, or the set of quadratic lattices that are isometric at every completion of the number field. We extend this theory to the function field case through the use os schemes to study how the spinor class field depends on the choice of an affine subset of a projective curve.
1 Summary
Let n be a positive integer and let k be a number field. There exists a correspondence between the group hk/hnk, where hk is the class group of k, and the set of conjugacy classes of maximal orders in the matrix algebra Mn(k). This correspondence sends an ideal class of an idealIto the maximal order Dn(I), where
D2(I) =
Ok I I−1 Ok
, D3(I) =
Ok I I I−1 Ok Ok I−1 Ok Ok
, . . . .
Since the group hk/hnk, is in correspondence with the Galois group of the maximal extension Σ/k of exponent n contained in the Hilbert class field, the same is true of the set of maximal orders. More generally, IfAis a central simple algebra of degree greater than 2 over a number field k, or if A is a quaternion algebra splitting at some infinite place, there is a correspondence between conjugacy classes of maximal orders and the elements of the Galois group of some abelian extension ΣA/k of exponent n that is un-ramified at every finite place in k. In this context we have studied the existence of subfieldsFof ΣAdescribing the set of maximal orders containing a conjugate of a certain non-maximal order H, in the sense that a maximal orderD of
∗Supported by Fondecyt grant 1085017
2
Acontains a conjugate of Hif and only if the corresponding element of the Galois group is trivial on F. We have proved that this field always exists for some important families of maximal abelian orders or orders generated by projective representations of the abelian group Cn×Cn ([?] and [?]), although counterexamples do exist whenH of generalized Eichler orders in algebras of degree 3 or greater ([?]).
Similarly, if Λ is a lattices over the ring of integers of a number field k provided with a non-singular bilinear form b, the set of isometry classes of lattices that are isometric to (Λ, b) for every completion of k (i.e. the genus of Λ) is in correspondence with the Galois group of an abelian ex- tension Σ(Λ)/k. For instance, if Λ is a free lattice of rank 2n with a basis x1, y1, x2, y2, . . . , xn, ynsuch thatb(xi, yi) = 1 fori= 1, . . . , nand 0 in every other case, every lattice in the genus of Λ is isometric to a lattice of the form
hx1, y1, . . . , xn−1, yn−1i ⊥Ixn⊥I−1yn, whereI is a fractional ideal ofk andI−1 its inverse.
These are particular cases of a general theory of spinor genera and spinor class fields, which also allows, for instance, the study of lattices in skew- hermitian spaces. It applies to structures whose groupGof automorphisms is a group with spinor norm in the sense defined in ([?], p.2025). The spinor class field is an abelian extension Σ of the base field k whose galois group has a free and transitive action on theset of equivalence classes of lattices (e.g., isometry classes of quadratic lattices or conjugacy classes of maximal orders) that are locally equivalent to a given lattice. This action is defined in terms of the Artin map on ideles.
In all the preceeding statements, the ring of integers of the field k can be replace with the ring ofS-integers for any finite setS of places ofkcontain- ing the infinite places. In this context we can define spinor class fields for S-orders and quadratic or skew-hermitianS-lattices. If S ⊆S0, the Spinor Class Field corresponding to S0 in contained in the Spinor Class Field cor- responding to S. It follows that the spinor class field for an S-lattice Λ is contained in the spinor class field of any lattice over the ring of full integers that generates Λ as an S-lattice. To be precise, if ΣS0,ΣS are the spinor class fields corresponding to the two setsS and S0, then ΣS0 is the largest subfield of ΣS where the places inS0\S split completely. One way to extend these results to a global function fieldKis fixing a non-empty setSof places ofK and defining spinor class fields for S-lattices. Then one might ask how this field depends on the choice of S.
Note that a global function field is isomorphic to the field of rational func- tions over a smooth projective curve X. The choice of a non-empty set S of places is equivalent to the choice of an affine open set in X. Lattices over rings of regular functions of affine subsets ofX paste together to give a scheme of lattices provided they satisfy some natural compatibility condi-
3
tions. We can therefore define a spinor class field for these schemes in the same way it was done for in the number field case.
For example, There exists a correspondence between P(X)/P(X)n, where P(X) is the Picard group ofX, and the set of conjugacy classes of maximal orders in the matrix algebra Mn(K). This correspondence sends a divisor B to the maximal order Dn(B), where
D2(B) =
OX LB L−B OX
, D3(B) =
OX LB LB L−B OX OX L−B OX OX
, . . . .
Here OX denotes the structure sheaf of the projective curve X and LB denotes the sheaf of functions defined by
LB(U) ={f ∈K|vp(f)≥ −vp(B)∀p∈U}.
Then the correspondence between orders and elements of a Galois group is defined via Artin map as over number fields. Similar constructions can be defined for quadratic forms.
References
[1] L.E. Arenas-Carmona, Applications of spinor class fields: embed- dings of orders and quaternionic lattices, Ann. Inst. Fourier 53(2003), 2021-2038.
[2] L.E. Arenas-Carmona, Integral Springer Theorem for quaternionic forms. Nagoya Math. J.187 (2007), 157-174.
[3] L.E. Arenas-Carmona, An embedding theorem for orders in central simple algebras, Submitted.
[4] O.T. O’Meara, Introduction to Quadratic Forms, Spriger-Verlag, Berlin, 1973.
4
