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Maximal Orders of

Alyson Deines

Quaternion Algebras Maximal Orders of Quaternions

Maximal Orders

Norms, Traces, and Alyson Deines Integrality

Local Fields and Hilbert Symbols December 8th Math 581b Final Project Uses

A non-commutative analogue of Rings of Integers of Number Abstract

Maximal Orders of Quaternions Alyson Deines Abstract I will define maximal orders of quaternion algebras, give Algebras

Maximal examples, discuss how they are similar to and differ from rings Orders of integers, and then I will discuss one of their applications . Norms, Traces, and Integrality

Local Fields and Hilbert Symbols

Uses Quaternion Algebras over Number Fields

Maximal Orders of Definition Quaternions

Alyson Deines Let F be a field and a, b ∈ F .   A quaternion algebra over F is an F -algebra B = a,b Quaternion F Algebras with basis Maximal 2 2 Orders i = a, j = b, and ij = −ji.

Norms, Traces, and Integrality Examples: Local Fields  −1,−1  and Hilbert = Symbols H R √ Uses  −1,−1  F = Q( 5), B = F  1,b  ∼ Let F be any number field, F = M2(F ) via  1 0   0 b  i 7→ and j 7→ 0 −1 1 0 Maximal Orders

Maximal Orders of Quaternions Lattices Alyson Deines Let F be a number field with of integers R and let V

Quaternion be a finite dimensional F -vector space. Algebras An R-lattice of V is a finitely generated R-submodule Maximal Orders O ⊂ V such that OF = V . Norms, Traces, and Integrality Orders Local Fields and Hilbert Let F be a number field with R and let B Symbols be a finite dimensional F -algebra. Uses An R-order O ⊂ B is an R-lattice of B that is also a . A maximal R-order O ⊂ B is an order which is not properly contained in any other orders. Examples

Maximal Orders of Quaternions Alyson Deines Example

Quaternion Let F be a number field with ring of integers R and take Algebras   B = 1,1 =∼ M (F ). Maximal F 2 Orders Then M2(R) is a maximal order. Norms, Traces, and Integrality

Local Fields Non-maximal Example and Hilbert √ Symbols  −1,−1  Let F = Q( 5) and B = F . Uses √ 1+ 5 Then R = Z[γ] where γ = 2 . O = R ⊕ Ri ⊕ Rj ⊕ Rij is a non-maximal order. Examples

Maximal Orders of Quaternions Alyson Deines Maximal Example

Quaternion Let Algebras 1 e1 = (1 − γi + γj) Maximal 2 Orders

Norms, 1 Traces, and e2 = (−γi + j + γk) Integrality 2

Local Fields 1 and Hilbert e3 = (γi − γj + k) Symbols 2 Uses 1 e = (i + γj − γk) 4 2

OB = Re1 ⊕ Re2 ⊕ Re3 ⊕ Re4 is a maximal order. Norm and Trace

Maximal Orders of Quaternions Definition Alyson Deines Let x = u + vi + wj + zij ∈ B. Quaternion x = u − vi − wj − zij Algebras 2 2 2 2 Maximal nrd(x) = xx = u − av − bw + abz Orders

Norms, trd(x) = x + x = 2u Traces, and Integrality

Local Fields Definition and Hilbert Symbols We say an element x ∈ B is integral over R if it satisfies a Uses monic polynomial with coefficients in R. If x ∈ B and x 6∈ F , then x has minimal polynomial t2 − nrd(x)t + trd(x). So x ∈ B is R-integral if and only if nrd(x), trd(x) ∈ R. Integrality Issues

Maximal Orders of Quaternions Now that we have a concept of integrality, we can see ways in which maximal

Alyson Deines orders fail as an analogue to rings of integers and Dedekind domains

Quaternion Definition Algebras   Maximal 0 0 Orders Take B = M ( ) with elements x = and 2 Q 1/2 0 Norms,   Traces, and 0 1/2 Integrality x = . 0 0 Local Fields and Hilbert 2 2 Symbols Then x = y = 0, so x and y are integral over Z but not Uses in M2(Z). Additionally nrd(x + y) = 1/4 so x + y is not integral. Thus the set of all integral elements is not closed under addition and can’t be a ring. Local Fields

Maximal Orders of Quaternions

Alyson Deines Local Quaternion Algebra Quaternion Algebras Let v be a valuation of F . Maximal Orders Then Bv = B ⊗ Fv is a quaternion algebra over Fv . Norms, Traces, and If O ⊂ B is an R-order, Ov = O ⊗ Rv is an Rv -order. Integrality O is maximal if and only if O is maximal for every prime Local Fields p and Hilbert p of R. Symbols Uses For v a noncomplex valuation there is up to Fv -algebra isomorphism, a unique division ring Bv over Fv . Ramification/Splitting

Maximal Orders of Definition Quaternions We say that B is ramified at v if B is a division ring. Alyson Deines v ∼ Otherwise, we say that B is split at v (i.e. Bv = M2(Fv )). Quaternion Algebras The number of places of F where B is ramified is finite. Maximal Orders

Norms, Discriminant Traces, and Integrality The discriminant of B (relative to R) is the R-ideal Q Local Fields discR (B) = p ⊂ R. and Hilbert p ramified Symbols Let O ⊂ B be an R-order. Then the discriminant of O is Uses the ideal disc(O) of R generated by {d(x1, x2, x3, x4): x1, ..., xn ∈ O} and d(x1, x2, x3, x4) = det(trd(xi xj ))i,j=1,2,3,4. Let O be an R-order in B. Then O is maximal if and only if discred(O) = discR (B). The Hilbert Symbol

Maximal Orders of Quaternions

Alyson Deines

Quaternion Hilbert Reciprocity Algebras The Hilbert symbol Maximal   Orders a,b 1 when B = F is split. Norms, (a, b)F = Traces, and −1 otherwise. Integrality × Local Fields If F is a number field and a, b ∈ F then and Hilbert Symbols Y Uses (a, b)Fv = 1. v Hilbert Modular Forms

Maximal Orders of There are two ways to compute Hilbert Modular Forms, both Quaternions of which involve quaternion algebras and their maximal orders. Alyson Deines

Quaternion Computing Maximal Orders Algebras Using the Hilbert symbol you can compute maximal orders Maximal Orders Can use maximal orders to work with Shimura curves and Norms, Traces, and compute Hilbert Modular Forms Integrality

Local Fields and Hilbert Hecke Operators Symbols Let O be the intersection of two maximal orders Uses c We can use this order to create a space which we can compute Hecke operators one. These Hecke operators correspond to the Hecke operators of certain Hilbert modular forms.