Selmer group
The Selmer group, often denoted as Sel(φ)(E/K) for an isogeny φ: E → E' of elliptic curves E and E' over a number field K, is an algebraic group that plays a crucial role in studying the arithmetic of elliptic curves and other abelian varieties. It provides a computable approximation to the Tate-Shafarevich group, which measures the failure of the Hasse principle for elliptic curves.
Definition and Construction:
The Selmer group arises from considering the long exact sequence in Galois cohomology associated to the isogeny φ: E(K) → E'(K):
0 → E(K)[φ] → E(K) --φ--> E'(K) → H1(K, E[φ]) → H1(K, E) --φ--> H1(K, E') → ...
where E(K)[φ] denotes the φ-torsion points of E defined over K, and H1(K, E) is the first Galois cohomology group.
The Selmer group Sel(φ)(E/K) is defined as the kernel of the map
H1(K, E[φ]) → H1(K, E)
More practically, it can be defined via the following local conditions:
Sel(φ)(E/K) = { s ∈ H1(K, E[φ]) : resv(s) ∈ Im(E(Kv)/φ) for all places v of K }
Here, resv is the restriction map from H1(K, E[φ]) to H1(Kv, E[φ]), and Kv denotes the completion of K at the place v. The image of E(Kv) under the isogeny φ in E'(Kv) is denoted Im(E(Kv)/φ). In essence, an element of the Selmer group is a Galois cohomology class which becomes locally trivial, in the sense that its restriction to each local field Kv lies in the image of the local points under the isogeny.
Relationship to other Groups:
The Selmer group sits in the following short exact sequence:
0 → E'(K)/φ(E(K)) → Sel(φ)(E/K) → Sha(E/K)[φ] → 0
where Sha(E/K) (often written as Ш(E/K)) denotes the Tate-Shafarevich group of E over K, and Sha(E/K)[φ] denotes its φ-torsion subgroup. This sequence shows that the Selmer group is an extension of the group of rational points on E' modulo the image of rational points on E, by the φ-torsion in the Tate-Shafarevich group.
Properties and Importance:
- The Selmer group is finite.
- It provides an upper bound for the rank of the elliptic curve E.
- The Selmer group is effectively computable in many cases. This computability makes it a valuable tool for studying the arithmetic of elliptic curves.
- It is closely related to the Birch and Swinnerton-Dyer conjecture, which connects the arithmetic of an elliptic curve to the analytic properties of its L-function. Specifically, understanding the size of the Selmer group is crucial for verifying this conjecture.
- Studying the behavior of Selmer groups as the elliptic curve or the number field varies has led to deep insights into the arithmetic of elliptic curves.
Generalizations:
The notion of the Selmer group has been generalized to other abelian varieties, motives, and Galois representations. In each case, the Selmer group provides a way to study global solutions to arithmetic problems by considering local solutions and measuring the obstruction to global solvability.