We say that two elliptic curves E1, E2 over a number field K are n-Selmer companions for a positive integer n if for every quadratic character χ of K, there is an isomorphism Seln(E χ 1 /K) ∼= Seln(E 2 /K) between the nSelmer groups of the quadratic twists E 1 , E χ 2 . We give sufficient conditions for two elliptic curves to be n-Selmer companions, and give a number of examples of non-isogenou...

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...

We characterize all infinite-dimensional graded virtual modules for Thompson's sporadic simple group whose traces are weight 32 weakly holomorphic modular forms satisfying certain special properties. then use these to detect the non-triviality of Mordell–Weil, Selmer, and Tate-Shafarevich groups quadratic twists elliptic curves.

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...

In this paper, we consider the average order of the 2-Selmer groups of elliptic curves over Q given by the equation E : y2 = x(x + a)(x + b), where a and b are integers. We show that, with a being fixed, the average order of the 2-Selmer groups of such curves closely depends on a. More exactly, we show that the average order is bounded if |a| is not a square and unbounded if |a| is the square o...

Let $\\rho$ be the $p$-adic Galois representation attached to a cuspidal, regular algebraic automorphic of $\\operatorname{GL}\_n$ unitary type. Under very mild hypotheses on $\\rho$, we prove vanishing (Bloch–Kato) adjoint Selmer group $\\rho$. We obtain definitive results for groups associated non-CM Hilbert modular forms and elliptic curves over totally real fields.

1. Quick Review of Elliptic Curves 2 2. Elliptic Curves over C 4 3. Elliptic Curves over Local Fields 6 4. Elliptic Curves over Number Fields 12 5. Elliptic Curves with Complex Multiplication 15 6. Descent 22 7. Elliptic Units 27 8. Euler Systems 37 9. Bounding Ideal Class Groups 43 10. The Theorem of Coates and Wiles 47 11. Iwasawa Theory and the “Main Conjecture” 50 12. Computing the Selmer G...

Let E be a modular elliptic curve over a totally real cubic field. We have a cubic-triple product motive over Q constructed from E through multiplicative induction; it is of rank 8. We show that, under certain assumptions on E, the non-vanishing of the central critical value of the L-function attached to the motive implies that the dimension of the associated Bloch–Kato Selmer group is 0.

Abstract Let p be a fixed odd prime. E an elliptic curve defined over number field F with either good ordinary reduction or multiplicative at each prime of above . We shall study the characteristic element Selmer group -adic Lie extension. In particular, we relate order vanishing these elements evaluated Artin representations to coranks and their twists in intermediate subextensions