Heegner Modules and Elliptic Curves by Martin L. Brown

By Martin L. Brown

Heegner issues on either modular curves and elliptic curves over international fields of any attribute shape the subject of this examine monograph. The Heegner module of an elliptic curve is an unique idea brought during this textual content. The computation of the cohomology of the Heegner module is the most technical consequence and is utilized to end up the Tate conjecture for a category of elliptic surfaces over finite fields; this conjecture is akin to the Birch and Swinnerton-Dyer conjecture for the corresponding elliptic curves over worldwide fields.

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8). Case 3. Assume that κ has characteristic 2 and S is ´etale over R; we have that a is not divisible by π and hence that a ∈ R∗ . 9) to the form EndM R (Λ) ⊗R R ∼ R = 2 (π 2 ) (π ) ( 2 [ ] . −π ) This algebra contains a nilpotent element of order 4 and hence it is not R-isomorphic to R[ ]/(π 2 , 2 ). 10. Definition. The star st(x) of a vertex x of the Bruhat-Tits tree ∆(SL2 (L)) is the set of adjacent vertices to x, distinct from x, in ∆(SL2 (L)). That is to say, the star st(x) is the set of vertices of ∆(SL2 (L)) whose distance from x is exactly 1, with respect to the normalised standard metric.

2) Let M be a 2-dimensional commutative L-algebra. 2). A Bruhat-Tits building with complex multiplication by M is a triple (∆(SL2 (L)), ExpM,Λ0 , M ) where ExpM,Λ0 is the map ExpM,Λ0 : L → Z. defined as follows. Let x ∈ L. Select an R-lattice Λ of M whose equivalance class [Λ] is equal to x. 3) EndM R (Λ) = {m ∈ M | mΛ ⊂ Λ}. 1(iii)). The abelian group M ∗ acts, as a subgroup of GL2 (L), on the vertices of ∆(SL2 (L)) and preserves the function ExpM,Λ0 ; hence M ∗ is a group of automorphisms of the triple (∆(SL2 (L)), ExpM,Λ0 , M ).

1). 5). 2. 1 or [Bro2, Ch. VI,§3]). 2)). 2. Definition. Let N be an R-subalgebra of M . Then an ideal of the ring N which is also an R-lattice of M is called a lattice ideal of N . 2). 3. Theorem. If M is a reduced algebra then for any R-lattice Λ of M we have Exp(Λ) = d([Λ], [I]) where I is the unique lattice ideal of S, up to multiplication by an element of L∗ , for which the distance d([Λ], [I]) is minimum. 4) Suppose that M is not reduced. 1). We fix the lattice Λ0 to be the R-subalgebra R ⊕ R of the integral closure S = R ⊕ L of R; Λ0 then depends only on the choice of .

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