On Tate’s conjecture for the elliptic modular surface of level N over a prime field of characteristic 1modN
Rémi Lodh1
1Springer 4 Crinan St. London N1 9XW, UK
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 p≡1modN 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 p≡1modN et on montre que le rang du groupe de Mordell–Weil est nul dans ce cas. Pour N≤4 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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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
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