We show that the average size of $2$-Selmer group family Jacobians non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by natural height, is bounded above $3$. achieve this interpreting elements as integral orbits representation associated stable $\mathbb{Z}/2\mathbb{Z}$-grading on Lie algebra type $E_6$ and using Bhargava's orbit-counting techniques. use res...

Abstract The Chabauty–Kim method is a tool for finding the integral or rational points on varieties over number fields via certain transcendental p -adic analytic functions arising from Selmer schemes associated to unipotent fundamental group of variety. In this paper we establish several foundational results curves fields. two main ingredients in proof these are an unlikely intersection result...

Generalizing results of Stroeker and Top we show that the 2-ranks of the TateShafarevich groups of the elliptic curves y = (x + k)(x + k) can become arbitrarily large. We also present a conjecture on the rank of the Selmer groups attached to rational 2-isogenies of elliptic curves. 1991 Mathematics Subject Classification: 11 G 05

The proper treatment of computationalism, as the thesis that cognition is computable, is presented and defended. Some arguments of James H. Fetzer against computationalism are examined and found wanting, and his positive theory of minds as semiotic systems is shown to be consistent with computationalism. An objection is raised to an argument of Selmer Bringsjord against one strand of computatio...

Let K be a number field, let A an abelian variety defined over and $$K_\infty /K$$ uniform p-adic Lie extension. We compare several arithmetic invariants of Iwasawa modules ideal class groups on the one side fine Selmer varieties other side. If $$ contains sufficiently many p-power torsion points A, then we can ranks $$\mu -invariants these algebra. In special cases (e.g. multiple $$\mathbb {Z}...

Let E : y = F (x) be an elliptic curve over Q defined by a monic irreducible integral cubic polynomial F (x) with negative and square-free discriminant −D. We determine its 2-Selmer rank in terms of the 2-rank of the class group of the cubic field L = Q[x]/F (x). We then interpret this result as a mod 2 congruence between the Hasse-Weil L-function of E and a degree two Artin L-function associat...

This paper consists of two parts. In the first we present a general theory of Euler systems. The main results (see §§3 and 4) show that an Euler system for a p-adic representation T gives a bound on the Selmer group associated to the dual module Hom(T, μp∞). These theorems, which generalize work of Kolyvagin [Ko], have been obtained independently by Kato [Ka1], Perrin-Riou [PR2], and the author...