Number Theory Seminar
Let E be an elliptic curve defined over ℚ. Fix <code style="text-decoration:overline">ℚ̅</code> as an algebraic closure of <code>ℚ</code>. We get a Galois representation
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ρ_E: Gal(ℚ) → GL₂(<code style="text-decoration:overline">ℤ̂</code>)
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by choosing a compatible bases for the N-torsion subgroups of E(<code style="text-decoration:overline">ℚ̅</code>). Let G be an open subgroup of GL₂(<code style="text-decoration:overline">ℤ̂</code>) such that det(G)=<code style="text-decoration:overline">ℤ̂</code> and -I ∈ G. Associated to G we have the modular curve X_G which loosely parametrises elliptic curves E such that im(ρ_E) ⊆ G. In this talk I will start with the description of modular curves and then discuss some techniques to classify genus 0 modular curves with a rational point.