Cassels has described a pairing on the 2-Selmer group of an elliptic curve which shares some properties with the Cassels-Tate pairing. In this article, we prove that the two pairings are the same.

Manjul Bhargava has recently made a great advance in the arithmetic theory of elliptic curves. Together with his student, Arul Shankar, he determines the average order of the Selmer group Sel(E,m) for an elliptic curve E over Q, when m = 2, 3, 4, 5. We recall that the Selmer group is a finite subgroup of H(Q, E[m]), which is defined by local conditions. Their result (cf. [1, 2]) is that the ave...

Let E be a modular elliptic curve over [symbol, see text], without complex multiplication; let p be a prime number where E has good ordinary reduction; and let Finfinity be the field obtained by adjoining [symbol, see text] to all p-power division points on E. Write Ginfinity for the Galois group of Finfinity over [symbol, see text]. Assume that the complex L-series of E over [symbol, see text]...

Given an elliptic curve E1 over a number field K and an element s in its 2-Selmer group, we give two different ways to construct infinitely many Abelian surfaces A such that the homogeneous space representing s occurs as a fibre of A over another elliptic curve E2. We show that by comparing the 2Selmer groups of E1, E2 and A, we can obtain information about X(E1/K)[2] and we give examples where...

We study the Selmer variety associated to a canonical quotient of the Qp-pro-unipotent fundamental group of a smooth projective curve of genus at least two defined over Q whose Jacobian decomposes into a product of abelian varieties with complex multiplication. Elementary multivariable Iwasawa theory is used to prove dimension bounds, which, in turn, lead to a new proof of Diophantine finitenes...

Given an elliptic curve E defined over Q, we are motivated by the 2-part of the Birch and Swinnerton-Dyer formula to study the relation between the 2-Selmer rank of E and the 2-Selmer rank of an abelian variety A. This abelian variety A is associated to a modular form g of weight 2 and level Nq that is obtained by Ribet’s level raising theorem from the modular form f of level N associated to E....

In this paper, we make a study of the Iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary Zp-extension of a number field K in the case when p splits completely in K. Generalizing work of Kobayashi [9] and Perrin-Riou [17], we define restricted Selmer groups and λ±, μ±-invariants; we then derive asymptotic formulas describing the growth of the Selmer group in terms ...

Let k > 9 be an even integer and p a prime with p > 2k− 2. Let f be a newform of weight 2k− 2 and level SL2(Z) so that f is ordinary at p and ρ f,p is irreducible. Under some additional hypotheses, we prove that ordp(Lalg(k, f)) ≤ ordp(#S), where S is the Pontryagin dual of the Selmer group associated to ρ f,p ⊗ ε1−k with ε the p-adic cyclotomic character. We accomplish this by first constructi...

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose K/k is a quadratic extension of number fields, E is an elliptic curve defined over k, and p is an odd prime. Let K− denote the maximal abelian p-extension of K that is unramified at all primes where E has bad reduction and that is Galois over k with dihedral Galois group (i.e., the gene...

For certain algebraic Hecke characters χ of an imaginary quadratic field F we define an Eisenstein ideal in a p-adic Hecke algebra acting on cuspidal automorphic forms of GL2/F . By finding congruences between Eisenstein cohomology classes (in the sense of G. Harder) and cuspidal classes we prove a lower bound for the index of the Eisenstein ideal in the Hecke algebra in terms of the special L-...