Multiplicities in Mordell–Weil groups
David E. Rohrlich1
1 Department of Mathematics and Statistics Boston University Boston, MA 02215 U. S. A.
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 125-142

Let $K$ be a finite Galois extension of $\mathbb{Q}$ and let $\rho $ be an irreducible self-dual complex representation of $\mathrm{Gal}(K/\mathbb{Q})$. For an elliptic curve $E$ over $\mathbb{Q}$ let $W(E,\rho )$ be the root number in the functional equation of $L(s,E,\rho )$. We give an example where $\rho $ is of dimension 4 and Schur index 1 but $W(E,\rho )=1$ for all $E$ over $\mathbb{Q}$. The image of $\rho $ has order 32.

Soient $K$ une extension galoisienne finie de $\mathbb{Q}$ et $\rho $ une représentation irréductible autoduale de $\mathrm{Gal}(K/\mathbb{Q})$ sur $\mathbb{C}$. Si $E$ est une courbe elliptique sur $\mathbb{Q},$ notons $W(E,\rho )$ le signe de l’équation fonctionnelle de la fonction $L$ tordue $L(s,E,\rho )$. Nous donnons un exemple où $W(E,\rho )$ est égal à 1 pour toute courbe elliptique $E$ sur $\mathbb{Q}$ bien que l’indice de Schur de $\rho $ soit $1$. La dimension de $\rho $ est 4 et son image est un groupe d’ordre 32.

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DOI : 10.5802/jtnb.1315
Classification : 11G05, 11R32
Keywords: elliptic curve, Mordell–Weil group, Artin representation

David E. Rohrlich 1

1 Department of Mathematics and Statistics Boston University Boston, MA 02215 U. S. A.
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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David E. Rohrlich. Multiplicities in Mordell–Weil groups. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 125-142. doi: 10.5802/jtnb.1315

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