On the Rank of Elliptic Curves in Elementary Cubic Extensions

We give a method for explicitly constructing an elementary cubic extension L over which an elliptic curve ED:y2+Dy=x3 (D∈Q∗) has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of ED and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [L:Q] often becomes small.

Mean values connected with the Dedekind zeta-function of a non-normal cubic field

After Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K 3/ℚ, i.e. $$ S_{l,K_3 } (x) = \sum\nolimits_{m \leqslant x} {M^l (m)} $$, where M(m) denotes the number of integral ideals of the field K 3 of norm m and l ∈ ℕ. We improve the previous results for $$ S_{2,K_3 } (x) $$ and $$ S_{3,K_3 } (x) $$.

ELLIPTIC CURVES RELATED TO CYCLIC CUBIC EXTENSIONS

The aim of this paper is to study certain family of elliptic curves [Formula: see text] defined over a number field F arising from hyperplane sections of some cubic surface [Formula: see text] associated to a cyclic cubic extension K/F. We show that each [Formula: see text] admits a 3-isogeny ϕ over F and the dual Selmer group [Formula: see text] is bounded by a kind of unit/class groups attached to K/F. This is proven via certain rational function on the elliptic curve [Formula: see text] with nice property. We also prove that the Shafarevich–Tate group [Formula: see text] coincides with a class group of K as a special case.

Darcy’s Experiments and the Deviation to Nonlinear Flow Regime

Many important technological and natural processes involving flow through porous media are characterized by large filtration velocity. It is important to know when the transition from the linear flow regime to the quadratic flow regime actually occurs to obtain accurate models for these processes. By interpreting the quadratic extension of the original Darcy equation as a model of the macroscopic form drag, we suggest a physically consistent parameter to characterize the transition to quadratic flow regime in place of the Reynolds number, Re. We demonstrate that an additional data set obtained by Darcy, and so far ignored by the community, indeed supports the Darcy equation. Finally, we emphasize that the cubic extension proposed in the literature, proportional to Re3 and mathematically valid only for Re≪1, is irrelevant in practice. Hence, it should not be compared to the quadratic extension experimentally observed when Re⩾O1.[S0098-2202(00)01703-X]