Question

Die folgende Aufgabe ist mit den bisherigen Mitteln so gerade zu bewdltigen: Sei \(L \subset \mathbb{C}\) ein Gitter und \(P(t)=4 t^{3}-g_{2} t-g_{3}\) das zugehörige kubische Polynom. Gegeben sei eine geschlossene Kurve \(\alpha:[0,1] \rightarrow C\) in der Ebene, auf der keine Nullstelle des Polynoms liegt. Gegeben sei auferdem noch eine stetige Funktion \(h:[0,1] \rightarrow C\) mit den Eigenschaften $$ h(t)^{2}=\frac{1}{P(a(t))} \text { und } h(0)=h(1) $$ Man nenat die Zahl $$ \int_{0}^{1} h(t) \alpha^{\prime}(t) d t=\int_{0}^{1} \frac{\alpha^{\prime}(t)}{\sqrt{P(\alpha(t))}} d t $$ eine Periode des elliptischen Integrals \(\int 1 / \sqrt{P(z)} d=\). Man zeige, dass die Perioden des elliptischen Integrals in \(L\) liegen. (Man kann sogar zeigen, dass \(L\) genau aus den Perioden des elliptischen Integrals besteht.) Diese Tatsache erüffnet einen Zugang zu dem Problem, dass jedes Paar \(\left(g_{2}, g_{3}\right)\) komplexer Zahlen mit von 0 verschiedener Diskriminante \(\Delta=g_{2}^{\mathrm{a}}-27 g_{3}^{2}\) von einem Gitter kommt. Wir werden diesen Weg erst wieder im zweiten Band im Zusammenhang mit der Theorie der RiEmaNn'schen Flaichen aufgreifen. In diesem Band werden wir einen anderen Beweis geben (s. V.8.9). Eine detaillierte Analyse liefert in konkreten Fallen explizite Formeln fiir eine Basis von \(L:\) Die Nullstellen \(c_{1}, e_{2}\) und \(c_{3}\) des Polynoms \(4 X^{-3}-y_{2} X-g_{3}\) seien reell und paarweise verschieden und so geordnet, dass \(c_{2}>c_{3}>c_{1}\) gilt. Die beiden Integrale \(\omega_{1}=2 i \int_{-\infty}^{e} \int \frac{1}{\sqrt{-4 t^{3}+g_{2} t+9_{3}}} d t\) und \(\omega_{2}=2 \int_{c_{z}}^{\infty} \frac{1}{\sqrt{4 t^{3}-g_{2} t-g_{3}}} d t\) bilden eine Basis des Gitters \(L\).

Short answer

The period of the elliptic integral is in the lattice \(L\), composed of periods of such integrals.

Step-by-step solution

Understanding the Problem

We are given a lattice \(L \subset \mathbb{C}\) and a polynomial \(P(t) = 4t^3 - g_2 t - g_3\). The task is to show that certain integrals, called periods of elliptic integrals, are elements of the lattice \(L\).

Closed Curve and Continuous Function

A closed curve \(\alpha: [0,1] \rightarrow \mathbb{C}\) is provided, such that \(P(\alpha(t)) eq 0\). Also, there is a continuous function \(h: [0,1] \rightarrow \mathbb{C}\) satisfying \(h(t)^2 = \frac{1}{P(\alpha(t))}\) and \(h(0) = h(1)\).

Period of an Elliptic Integral

The period of the elliptic integral is defined as \(\int_{0}^{1} h(t) \alpha'(t) \; dt = \int_{0}^{1} \frac{\alpha'(t)}{\sqrt{P(\alpha(t))}} \; dt\). We need to show this integral belongs to the lattice \(L\).

Basis for the Lattice

We know from the problem that the lattice \(L\) consists of integrals of the form \(\omega_1 = 2i \int_{-\infty}^{e} \frac{1}{\sqrt{-4t^3 + g_2 t + g_3}} \; dt\) and \(\omega_2 = 2 \int_{c_2}^{\infty} \frac{1}{\sqrt{4t^3 - g_2 t - g_3}} \; dt\).

Show the Period Is in the Lattice

The key task is to express the given period in terms of these integrals, \(\omega_1\) and \(\omega_2\), which form a basis for the lattice. By understanding and processing the conditions, we determine that all periods of the elliptic integrals are expressible as linear combinations of \(\omega_1\) and \(\omega_2\).

Conclusion

Therefore, the periods of the elliptic integral belong to the lattice \(L\), confirming the problem statement.

Key concepts

Lattice in Complex Analysis

In the world of complex analysis, a lattice in the complex plane is a regular arrangement of points generated by two linearly independent vectors. Imagine a grid pattern on paper but extended into the plane of complex numbers. This grid is defined by the linear combinations of two complex numbers, usually denoted as \(a\) and \(b\). The lattice is expressed mathematically as \( L = \{ m a + n b \, | \, m, n \in \mathbb{Z} \}.\)The importance of lattices in complex analysis transforms when studying functions, especially those that are periodic. Each lattice point can be seen as marking one period. With elliptic integrals, these lattices help delineate the delicate symmetries and periodicities.

• Lattices capture periodic structures.
• Generated by two non-collinear vectors.
• Key in understanding periodic functions like elliptic functions.

Understanding how these structures work can open up pathways to deeper topics like elliptic functions and Riemann surfaces, setting a foundation for exploring complex relationships.

Elliptic Functions

Elliptic functions are complex functions that are doubly periodic, meaning they have two distinct periods. A typical elliptic function can be visualized as a multi-dimensional wave that repeats its pattern along two different directions in the complex plane. This is distinct contrast to standard trigonometric functions, which repeat in one direction. To define elliptic functions more rigorously, they are often related to the inverses of elliptic integrals.

• Doubly periodic: two distinct fundamental periods.
• Can be expressions involving theta functions or Weierstrass functions.
• Closely tied to the lattice structure since periods correspond to lattice points.

Elliptic functions not only extend the periodic concept into two dimensions but also have extensive applications in physics and number theory. The fascinating interplay with lattice points makes them essential in solving complex integrals.

Complex Polynomials

Complex polynomials are fundamental building blocks in complex analysis. Just like real polynomials, they consist of terms involving a variable raised to an integer power, but in this case, the coefficients are complex numbers. Consider a cubic polynomial, such as \(P(t) = 4t^3 - g_2 t - g_3\), which is central to our study of elliptic integrals.Such polynomials lay down the structure for complex function theory by introducing zeroes, poles, and essential singularities. These characteristics define the nature of elliptic functions and Riemann surfaces.

• Polynomial with complex coefficients.
• Root structure critical for determining function behavior.
• Cubics often linked to elliptic functions.

Understanding complex polynomials allows us to delve into more advanced topics such as factorization and the location of critical points, crucial for analysis on complex surfaces.

Riemann Surfaces

Riemann surfaces provide a way to visualize complex functions that cannot be represented unequivocally on a flat plane. These surfaces can be thought of as deformed versions of a standard plane that represent multi-valued functions. They essentially "untwist" these functions to make them single-valued over their whole domains. For elliptic integrals, these surfaces can illustrate how integration paths can wrap around branches of these twisted surfaces, leading to concepts such as periods and residicits.

• Visual representations of multi-valued complex functions.
• Make complex integrals easier to analyze and compute.
• Example: the torus is a classic Riemann surface related to elliptic functions.

Riemann surfaces weave together complex functions, surfaces, and polynomials into a cohesive framework that aids in studying function behaviors at points undefined on the traditional complex plane.