Moduli space

In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. The use of the term modulus here for such a parameter goes back to the same source as in modular form: a modular form in general is some kind of differential form (or tensor density, since the forms come with a 'weight') on a moduli space, that is, a space whose co-ordinates are the moduli.

Related Topics:
Algebraic geometry - Parameter - Algebraic varieties - Modular form - Differential form - Tensor density

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In the case of elliptic curves, there is one modulus, so moduli spaces are algebraic curves. This is the quantity called k in Jacobi's elliptic function theory, which reduces elliptic integrals to a form involving

Related Topics:
Elliptic curve - Algebraic curve - Elliptic function

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:sqrt{(1-x^2)(1-k^2x^2),}.

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This modulus of the elliptic integral therefore was probably the first modulus to be recognised.

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The case of elliptic curves has been thoroughly studied, because of the great interest of the modular equations in this case. The j-invariant is a fundamental elliptic modular function. The moduli problem here is the prototype for moduli problems with level structure, meaning in this case some 'marking' of torsion groups of points on the curve. Each level structure gives rise to a subgroup of the modular group, and then its own modular curve. The j-invariant is called a Hauptmodul, traditionally, meaning that the modular curve has genus 0. There are other cases of genus 0, and other Hauptmoduls, which enter the remarkable monstrous moonshine theory.

Related Topics:
Elliptic curve - Modular equation - J-invariant - Elliptic modular function - Torsion group - Modular group - Modular curve - Genus - Monstrous moonshine

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In general a curve of genus g has

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:3g − 3

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moduli, for g > 1. This number was known classically as the number of parameters on which a compact Riemann surface depends. It agrees with the calculation of the dimension of the space of quadratic differentials on a fixed such Riemann surface, which is suggested by deformation theory combined with Serre duality. Except when g=2, this is larger than the number

Related Topics:
Compact - Riemann surface - Quadratic differential - Deformation theory - Serre duality

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:2g − 1

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of moduli of hyperelliptic curves.

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