Geodesic
Disparity in Selmer ranks of quadratic twists of elliptic curves
Zev Klagsbrun1 ; Barry Mazur2 ; Karl Rubin3
1 Department of Mathematics, University of Wisconsin - Madison, 480 Lincoln Dr., Madison, WI 53706
2 Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138-2901
3 Department of Mathematics, University of California Irvine, Irvine, CA 92697-3875
Annals of mathematics, Tome 178 (2013) no. 1, pp. 287-320.

Voir la notice de l'article provenant de la source Annals of Mathematics website

We study the parity of $2$-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve $E$ over an arbitrary number field $K$. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even $2$-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve $E$ such that as $K$ varies, these fractions are dense in $[0, 1]$. More generally, our results also apply to $p$-Selmer ranks of twists of $2$-dimensional self-dual $\mathbf{F}_p$-representations of the absolute Galois group of $K$ by characters of order $p$.
DOI : 10.4007/annals.2013.178.1.5

Zev Klagsbrun 1 ; Barry Mazur 2 ; Karl Rubin 3

1 Department of Mathematics, University of Wisconsin - Madison, 480 Lincoln Dr., Madison, WI 53706
2 Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138-2901
3 Department of Mathematics, University of California Irvine, Irvine, CA 92697-3875 
@article{10_4007_annals_2013_178_1_5,
     author = {Zev Klagsbrun and Barry Mazur and Karl Rubin},
     title = {Disparity in {Selmer}  ranks of quadratic twists of elliptic curves},
     journal = {Annals of mathematics},
     pages = {287--320},
     publisher = {mathdoc},
     volume = {178},
     number = {1},
     year = {2013},
     doi = {10.4007/annals.2013.178.1.5},
     mrnumber = {3043582},
     zbl = {06190561},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.1.5/}
}
TY  - JOUR
AU  - Zev Klagsbrun
AU  - Barry Mazur
AU  - Karl Rubin
TI  - Disparity in Selmer  ranks of quadratic twists of elliptic curves
JO  - Annals of mathematics
PY  - 2013
SP  - 287
EP  - 320
VL  - 178
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.1.5/
DO  - 10.4007/annals.2013.178.1.5
LA  - en
ID  - 10_4007_annals_2013_178_1_5
ER  - 
%0 Journal Article
%A Zev Klagsbrun
%A Barry Mazur
%A Karl Rubin
%T Disparity in Selmer  ranks of quadratic twists of elliptic curves
%J Annals of mathematics
%D 2013
%P 287-320
%V 178
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.1.5/
%R 10.4007/annals.2013.178.1.5
%G en
%F 10_4007_annals_2013_178_1_5
Zev Klagsbrun; Barry Mazur; Karl Rubin. Disparity in Selmer  ranks of quadratic twists of elliptic curves. Annals of mathematics, Tome 178 (2013) no. 1, pp. 287-320. doi : 10.4007/annals.2013.178.1.5. http://geodesic.mathdoc.fr/articles/10.4007/annals.2013.178.1.5/

Cité par Sources :