On Tate’s conjecture for the elliptic modular surface of level N over a prime field of characteristic 1modN
Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 193-204.

Assuming partial semisimplicity of Frobenius, we show Tate’s conjecture for the reduction of the elliptic modular surface E(N) of level N at a prime p satisfying p1modN and show that the Mordell–Weil rank is zero in this case. This extends a result of Shioda to N>4. Furthermore, we show that for every number field L partial semisimplicity holds for the reductions of E(N) L at a set of places of density 1.

Modulo une hypothèse de semi-simplicité partielle, on démontre le conjecture de Tate pour la surface elliptique modulaire E(N) de niveau N sur un corps premier de cardinalité p1modN et on montre que le rang du groupe de Mordell–Weil est nul dans ce cas. Pour N4 c’est un résultat de Shioda. De plus, on démontre que l’hypothèse de semi-simplicité vaut en dehors d’un ensemble de nombres premiers p de densité nulle.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1117
Classification: 11G05,  11F11,  14F30
Keywords: elliptic curves, modular forms, p-adic cohomology, zeta function
Rémi Lodh 1

1 Springer 4 Crinan St. London N1 9XW, UK
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Rémi Lodh. On Tate’s conjecture for the elliptic modular surface of level $N$ over a prime field of characteristic $1\ \protect \mathrm{mod}\ N$. Journal de Théorie des Nombres de Bordeaux, Volume 32 (2020) no. 1, pp. 193-204. doi : 10.5802/jtnb.1117. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1117/

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