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Erich Selder 《manuscripta mathematica》1983,41(1-3):45-74
We give explicit equations for the calculation of Chern classes of holomorphic line bundles on a complex torus X. As easy applications we deduce properties of the Picard numbers ρ(X) of n-dimensional tori, when the complex structure changes. The tori with ρ(X)≥k form a countable union of analytic subsets in a moduli space M; furthermore the set of tori with ρ(X)=k is empty or dense in M. For n-dimensional tori one has O≤ρ(X)≤n2, but for n≥3 not all numbers 0≤k≤n2 occur as Picard numbers. We conclude our considerations with a list of examples and with some remarks about this gap phenomenon in the distribution of Picard numbers of complex tori. 相似文献
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Plane quartic curves given by equations of the form y
2=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied
by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction
establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an
algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity.
We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law
on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds
to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals
of the form ò(1/?{P(x)}) dx\int (1/\sqrt{P(x)})\,\mathrm{d}x with a quartic polynomial P can be derived directly from this addition law. 相似文献
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Plane quartic curves given by equations of the form y 2=P(x) with polynomials P of degree 4 represent singular models of elliptic curves which are directly related to elliptic integrals in the form studied by Euler and for which he developed his famous addition formulas. For cubic curves, the well-known secant and tangent construction establishes an immediate connection of addition formulas for the corresponding elliptic integrals with the structure of an algebraic group. The situation for quartic curves is considerably more complicated due to the presence of the singularity. We present a geometric construction, similar in spirit to the secant method for cubic curves, which defines an addition law on a quartic elliptic curve given by rational functions. Furthermore, we show how this addition on the curve itself corresponds to the addition in the (generalized) Jacobian variety of the curve, and we show how any addition formula for elliptic integrals of the form \(\int (1/\sqrt{P(x)})\,\mathrm{d}x\) with a quartic polynomial P can be derived directly from this addition law. 相似文献
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Mathematische Semesterberichte - We study the question at which relative distances four squares of rational numbers can occur as terms in an arithmetic progression. This number-theoretical problem... 相似文献
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