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# Séminaire de Théorie Algorithmique des Nombres

Responsables : Fredrik Johansson et Aurel Page

Page du séminaire

• Le 25 janvier 2022 à 10:00
• Online
Céline Maistret (University of Bristol)
Parity of ranks of abelian surfaces

Let $K$ be a number field and $A/K$ an abelian surface. By the Mordell-Weil theorem, the group of $K$-rational points on $A$ is finitely generated and as for elliptic curves, its rank is predicted by the Birch and Swinnerton-Dyer conjecture. A basic consequence of this conjecture is the parity conjecture: the sign of the functional equation of the $L$-series determines the parity of the rank of $A/K$.
Assuming finiteness of the Shafarevich-Tate group, we prove the parity conjecture for principally polarized abelian surfaces under suitable local constraints. Using a similar approach we show that for two elliptic curves $E_1$ and $E_2$ over $K$ with isomorphic $2$-torsion, the parity conjecture is true for $E_1$ if and only if it is true for $E_2$. In both cases, we prove analogous unconditional results for Selmer groups.

• Le 8 février 2022 à 10:00
• Online
Elisa Lorenzo Garcia (Université de Neuchâtel)
Reduction type of hyperelliptic curves in terms of the valuations of their invariants.

In this talk we will first review the classical criteria to determine the (stable) reduction type of elliptic curves (Tate) and of genus 2 curves (Liu) in terms of the valuations of some particular combinations of their invariants. We will also revisit the theory of cluster pictures to determine the reduction type of hyperelliptic curves (Dokchitser’s et al.). Via Mumford theta constants and Takase and Tomae’s formulas we will be able to read the cluster picture information by looking at the valuations of some (à la Tsuyumine) invariants in the genus 3 case. We will also discuss the possible generalization of this strategy for any genus and some related open questions.

• Le 8 mars 2022 à 10:00
• Online
Elena Berardini (Télécom Paris)
TBA

TBA

• Le 15 mars 2022 à 10:00
• Salle 2
Pierrick Dartois (Corps des mines, Rennes 1)
TBA

TBA

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