SYMMETRIC POWER CONGRUENCE IDEALS AND SELMER GROUPS

Symmetric Squares of Elliptic Curves: Rational Points and Selmer Groups

Congruences between Selmer groups ∗

The study of congruences between arithmetically interesting numbers has a long history and plays important roles in several areas of number theory. Examples of such congruences include the Kummer congruences between Bernoulli numbers and congruences between coefficients of modular forms. Many of these congruences could be interpreted as congruences between special values of L-functions of arith...

Selmer Groups and Quadratic Reciprocity

In this article we study the 2-Selmer groups of number fields F as well as some related groups, and present connections to the quadratic reciprocity law in F . Let F be a number field; elements in F× that are ideal squares were called singular numbers in the classical literature. They were studied in connection with explicit reciprocity laws, the construction of class fields, or the solution of...

Finding Large Selmer Groups

Raoul Bott has inspired many of us by the magnificence of his ideas, by the way he approaches and explains mathematics, and by his warmth, friendship, and humor. In celebration of Raoul’s eightieth birthday we offer this brief article in which we will explain how the recent cohomological ideas of Jan Nekovár̆ [N2] imply (under mild hypotheses plus the Shafarevich-Tate conjecture) systematic grow...

Selmer Groups as Flat Cohomology Groups

Given a prime number p, Bloch and Kato showed how the p8-Selmer group of an abelian variety A over a number field K is determined by the p-adic Tate module. In general, the p-Selmer group Selpm A need not be determined by the mod p Galois representation Arps; we show, however, that this is the case if p is large enough. More precisely, we exhibit a finite explicit set of rational primes Σ depen...