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Selmer group
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(Definition)
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Given an elliptic curve  we can define two very interesting and important groups, the Selmer group and the Tate-Shafarevich group, which together provide a measure of the failure of the Hasse principle for elliptic curves, by measuring whether the curve is everywhere locally soluble. Here we present the construction of these groups.
Let  be elliptic curves defined over
 and let
 be an algebraic closure of
 . Let
 be an non-constant isogeny (for example, we can let  and think of  as being the “multiplication by  ” map,
![$ [n]\colon E\to E$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img10.png "$ [n]\colon E\to E$") ). The following standard result asserts that  is surjective over
 :
Proof. See
 , Chapter II.6.8. 
Since
 is non-constant, it must be surjective and we obtain the following exact sequence:
where
![$ E(\bar{\mathbb{Q}})[\phi]=\operatorname{Ker}\phi$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img21.png "$ E(\bar{\mathbb{Q}})[\phi]=\operatorname{Ker}\phi$") . Let
 , the absolute Galois group of
 , and consider the  -cohomology group
 (we abbreviate by  ). Using equation  we obtain the following long exact sequence (see Proposition 1 in group cohomology):
Note that
and similarly
From  we can obtain an exact sequence:
We could repeat the same procedure but this time for  defined over
 ,for some prime number  , and obtain a similar exact sequence but with coefficients in
 which relates to the original in the following commutative diagram (here
 ):
The goal here is to find a finite group containing
 . Unfortunately
![$ H^1(G,E(\bar{\mathbb{Q}})[\phi])$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img48.png "$ H^1(G,E(\bar{\mathbb{Q}})[\phi])$") is not necessarily finite. With this purpose in mind, we define the  -Selmer group:
Equivalently, the  -Selmer group is the set of elements  of
![$ H^1(G,E(\bar{\mathbb{Q}})[\phi])$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img53.png "$ H^1(G,E(\bar{\mathbb{Q}})[\phi])$") whose image  in
![$ H^1(G_p,E(\bar{\mathbb{Q_p}})[\phi])$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img55.png "$ H^1(G_p,E(\bar{\mathbb{Q_p}})[\phi])$") comes from some element in
 .
Finally, by imitation of the definition of the Selmer group, we define the Tate-Shafarevich group:
The Tate-Shafarevich group is precisely the group that measures the Hasse principle in the elliptic curve  . It is unknown if this group is finite.
- 1
- J.P. Serre, Galois Cohomology, Springer-Verlag, New York.
- 2
- James Milne, Elliptic Curves, online course notes.
- 3
- Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
- 4
- R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.
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"Selmer group" is owned by alozano.
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(view preamble)
Cross-references: image, finite, commutative diagram, coefficients, similar, prime number, group cohomology, proposition, equation, absolute Galois group, exact sequence, algebraic map, morphism, field, algebraically closed, surjective, map, isogeny, algebraic closure, locally soluble, curve, Hasse principle, measure, groups, elliptic curve
There are 2 references to this entry.
This is version 3 of Selmer group, born on 2003-08-14, modified 2003-08-14.
Object id is 4586, canonical name is SelmerGroup.
Accessed 4435 times total.
Classification:
| AMS MSC: | 14H52 (Algebraic geometry :: Curves :: Elliptic curves) |
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Pending Errata and Addenda
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![$\displaystyle 0\to E(\bar{\mathbb{Q}})[\phi]\to E(\bar{\mathbb{Q}})\to E'(\bar{\mathbb{Q}})\to 0 \quad\quad (1) $](http://images.planetmath.org:8080/cache/objects/4586/l2h/img20.png "$\displaystyle 0\to E(\bar{\mathbb{Q}})[\phi]\to E(\bar{\mathbb{Q}})\to E'(\bar{\mathbb{Q}})\to 0 \quad\quad (1) $"){kind=small}
![$\displaystyle 0 \to H^0(G,E(\bar{\mathbb{Q}})[\phi]) \to H^0(G,E)\to H^0(G,E') \to H^1(G,E(\bar{\mathbb{Q}})[\phi])\to H^1(G,E)\to H^1(G,E') \quad\quad (2)$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img28.png "$\displaystyle 0 \to H^0(G,E(\bar{\mathbb{Q}})[\phi]) \to H^0(G,E)\to H^0(G,E') \to H^1(G,E(\bar{\mathbb{Q}})[\phi])\to H^1(G,E)\to H^1(G,E') \quad\quad (2)$"){kind=small}
![$\displaystyle H^0(G,E(\bar{\mathbb{Q}})[\phi])={(E(\bar{\mathbb{Q}})[\phi])}^G=E(\mathbb{Q})[\phi]$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img29.png "$\displaystyle H^0(G,E(\bar{\mathbb{Q}})[\phi])={(E(\bar{\mathbb{Q}})[\phi])}^G=E(\mathbb{Q})[\phi]$"){kind=small}
![$\displaystyle 0\to E'(\mathbb{Q})/\phi(E(\mathbb{Q})) \to H^1(G,E(\bar{\mathbb{Q}})[\phi]) \to H^1(G,E)[\phi]\to 0$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img32.png "$\displaystyle 0\to E'(\mathbb{Q})/\phi(E(\mathbb{Q})) \to H^1(G,E(\bar{\mathbb{Q}})[\phi]) \to H^1(G,E)[\phi]\to 0$"){kind=small}
![$\displaystyle H^1(G,E(\bar{\mathbb{Q}})[\phi])$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img39.png "$\displaystyle H^1(G,E(\bar{\mathbb{Q}})[\phi])$"){kind=small}
![$\displaystyle \to H^1(G,E)[\phi]\to 0$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img40.png "$\displaystyle \to H^1(G,E)[\phi]\to 0$"){kind=small}
![$\displaystyle H^1(G_p,E(\bar{\mathbb{Q}_p})[\phi])$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img45.png "$\displaystyle H^1(G_p,E(\bar{\mathbb{Q}_p})[\phi])$"){kind=small}
![$\displaystyle \to H^1(G_p,E)[\phi]\to 0$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img46.png "$\displaystyle \to H^1(G_p,E)[\phi]\to 0$"){kind=small}
![$\displaystyle S^{\phi}(E/\mathbb{Q})=\operatorname{Ker}\left(H^1(G,E(\bar{\mathbb{Q}})[\phi])\to \prod_p H^1(G_p,E)\right)$](http://images.planetmath.org:8080/cache/objects/4586/l2h/img50.png "$\displaystyle S^{\phi}(E/\mathbb{Q})=\operatorname{Ker}\left(H^1(G,E(\bar{\mathbb{Q}})[\phi])\to \prod_p H^1(G_p,E)\right)$")
