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.

**Read Online or Download Heegner Modules and Elliptic Curves PDF**

**Best algebraic geometry books**

**Introduction to modern number theory : fundamental problems, ideas and theories**

This version has been referred to as ‘startlingly up-to-date’, and during this corrected moment printing you may be yes that it’s much more contemporaneous. It surveys from a unified standpoint either the trendy country and the traits of continuous improvement in quite a few branches of quantity conception. Illuminated by means of straightforward difficulties, the relevant principles of contemporary theories are laid naked.

From the studies of the 1st printing of this ebook, released as quantity 6 of the Encyclopaedia of Mathematical Sciences: ". .. My common impact is of a very great publication, with a well-balanced bibliography, steered! "Medelingen van Het Wiskundig Genootschap, 1995". .. The authors provide the following an up-to-the-minute advisor to the subject and its major functions, together with a couple of new effects.

**An introduction to ergodic theory**

This article offers an advent to ergodic idea appropriate for readers understanding simple degree thought. The mathematical must haves are summarized in bankruptcy zero. it's was hoping the reader can be able to take on study papers after interpreting the publication. the 1st a part of the textual content is anxious with measure-preserving variations of likelihood areas; recurrence houses, blending homes, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy concept are mentioned.

- Algebraic Geometry: An Introduction, 1st Edition
- An Introduction to Ergodic Theory (Graduate Texts in Mathematics)
- Undergraduate Algebraic Geometry (London Mathematical Society Student Texts)
- Introduction to the Mori Program (Universitext)
- Basic Algebraic Geometry 1: Varieties in Projective Space
- Symmetric Functions, Schubert Polynomials and Degeneracy Loci (Smf/Ams Texts and Monographs, Vol 6 and Cours Specialises Numero 3, 1998)

**Additional info for Heegner Modules and Elliptic Curves**

**Sample text**

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 .