Descent and Selmer groups

Let E be an elliptic curve over a number field K and let n be at least two. The multiplication map is separable in characteristic zero. Its geometric n-torsion gives a finite Galois module, and the Kummer connecting map injects E(K)/nE(K) into the first Galois cohomology group with coefficients in that module. Informally, the class records how a chosen geometric n-th division point fails to descend to a K-rational division point.

For every completion K_v, there is a corresponding local Kummer image. The n-Selmer group consists of global cohomology classes whose restrictions belong to those local images at every place, including archimedean places. It is finite, and contains the actual global Kummer image. Although the definition uses all places, ramification restrictions and local structure reduce a concrete descent to finitely many relevant calculations. This reduction is a theorem to establish, rather than permission to test a few arbitrarily selected primes.

The Tate–Shafarevich group is the kernel of restriction from the first cohomology of E over K to the product of its local counterparts. Geometrically, its elements represent torsors under E that have points over every completion but may have no point over K. A torsor with a K-point becomes trivial after choosing that point. Thus local solubility is a necessary compatibility test, not a general local–global principle for genus-one curves.

An n-covering used in descent carries more data than a naked equation: it represents a class relative to multiplication by n. Distinct classes can survive every local calculation without producing distinct rational generators. The exact sequence separates the classes arising from rational points from those in Sha[n]. Proving that a specific Selmer class comes from a rational point may require an explicit covering point, another descent, or an independent theorem eliminating its obstruction.

Subtracting the rational torsion dimension from the Selmer dimension gives an upper bound on rank. Independently verified points give a lower bound, for example through a positive height-pairing determinant. When these bounds meet, the rank is determined and the displayed Selmer calculation shows that the corresponding Sha[ell] vanishes. This does not show that every prime-primary part of Sha vanishes, nor that the chosen points form a saturated integral basis of the free group.

If the bounds do not meet, the gap should remain visible. More searching can improve the lower bound, and higher descents or isogeny descents can sharpen the upper bound. Neither step is guaranteed to settle every curve unconditionally by a fixed small computation. Software may offer conditional conclusions based on BSD or other assumptions; an arithmetic report must label those assumptions separately from the finite local calculations that have actually been certified.