By Martin L. Brown
Heegner issues on either modular curves and elliptic curves over international fields of any attribute shape the subject of this learn monograph. The Heegner module of an elliptic curve is an unique thought brought during this textual content. The computation of the cohomology of the Heegner module is the most technical end result and is utilized to turn out the Tate conjecture for a category of elliptic surfaces over finite fields; this conjecture is resembling the Birch and Swinnerton-Dyer conjecture for the corresponding elliptic curves over worldwide fields.
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Extra info for Heegner Modules and Elliptic Curves
7, we have isomorphisms of κ-algebras M ∼ fΛ (EndM R (Λ)) = EndR (Λ) ⊗R R ∼ = κ[ ]/( 2 ). (π) Hence the elements of fΛ (EndM R (Λ)) have a unique 1-dimensional common eigenspace on Λ ⊗R κ namely the kernel of . 12(ii), there is a unique lattice Λ such that πΛ ⊂ Λ ⊂ Λ for which Exp(Λ ) ≤ Exp(Λ). 12(iii) this lattice has exponent equal to Exp(Λ) − 1. 12(ii) and (iii) Exp(Λ ) = Exp(Λ) + 1. As the lattices Λ verifying πΛ ⊂ Λ ⊂ Λ correspond to the elements of st([Λ]) this proves the statement above.
We may ﬁx the R-lattice Λ0 = R ⊕ sR. In both cases where M is or is not reduced, we obtain an equality of R-algebras n EndM R (Λ) = R ⊕ sπ R. where n = Exp(Λ). The exponent Exp(Λ) here is relative to Λ0 if M is not reduced and is relative to S if M is reduced. Suppose ﬁrst either that n ≥ 2 or that M is not reduced. We have (sπ n )2 = π 2 (π n s)(π n−2 s) ∈ π 2 EndM R (Λ), if n ≥ 2 0, if M is not reduced. 8) EndM R (Λ) ⊗R 2 = 0 (modulo π 2 ). We obtain R ∼ R [ ] = 2 2 . (π 2 ) (π ) ( ) This proves the lemma in these cases, as S/R is ´etale only if M is reduced.
We have therefore shown d([Λ], [Λn ]) ≤ d([Λ], [I]). Step 4. We have |Exp(Λ)| = minI d([Λ], [I]) where the minimum runs over all lattice ideals I of Λ0 of exponent 0. This follows from Steps 2 and 3. 21. Remark. (Maximal orders in the indeﬁnite quaternion algebra M2 (L)). For this remark suppose that L is a non-archimedean local ﬁeld. Let R be the ring of valuation integers of L and let π be a local parameter of R. Let M2 (L) be the non-commutative L-algebra of 2 × 2 matrices over L. Let V be a 2dimensional L-vector space on which M2 (L) acts with its usual action.
Heegner Modules and Elliptic Curves by Martin L. Brown