### A generalisation of Tate’s algorithm for hyperelliptic curves

Linfoot Number Theory Seminar

5th June 2024, 11:00 am – 12:00 pm

Fry Building, G.07

Tate's algorithm tells us that, for an elliptic curve $E$ over a discretely valued field $K$ with residue characteristic $\geq 5$, the dual graph of the special fibre of the minimal regular model of $E$ over $K^{\textup{unr}}$ can be read off from the valuation of $j(E)$ and $\Delta_E$. This is really important for calculating Tamagawa numbers of elliptic curves, which are involved in the refined Birch and Swinnerton-Dyer conjecture formula. For a hyperelliptic curve $C/K$, we can ask if we can give a similar algorithm that gives important data related to the curve and its Jacobian from polynomials in the coefficients of a Weierstrass equation for $C/K$. This talk will be split between being an introduction to cluster pictures of hyperelliptic curves, from which the important data can be gathered, and a presentation of how the cluster picture can be recovered from polynomials in the coefficients of a Weierstrass equation.

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