Franklin's Notes

Weierstrass P Function

The Weierstrass P Function $\wp:\mathbb C\to\hat{\mathbb C}$ is a classical attempt at constructing a nonconstant doubly periodic function with periods $\omega_1,\omega_2$. Since any meromorphic doubly periodic function must have poles, and in fact at least two poles in any fundamental parallelogram, we may as well assume that $\wp$ has a pole of order $2$ at $z=0$, so that $\wp(z)\sim 1/z^2$ as $z\to 0$. This means that $f$ must also have poles of order $2$ at each point $m\omega_1+n\omega_2$ for $m,n\in\mathbb N$. We might be tempted to write something like except that this sum does not converge absolutely, so its meaning isn't even unambiguous. (The value of the sum could depend on the order in which its terms are summed.) However, if we instead define This gives an absolutely convergent sum, for we have that as $|\omega|\to\infty$. This means that we can rearrange terms of the summation without worrying about convergence issues, allowing us to show using a simple rearrangement that $\wp(z+\omega_1)=\wp(z+\omega_2)=\wp(z)$ as desired.

Laurent Series for $\wp$

We can expand each term of the series definition of the $\wp$ function as follows: which allows us to write where $\Lambda$ is the lattice $\mathbb Z\omega_1 + \mathbb Z\omega_2$. We can switch the order of summation because of absolute convergence: Notice that when $n$ is odd, the infinite sum over $\omega$ vanishes, because $1/\omega^{n+2}$ is an odd function of $\omega$ and the lattice $\Lambda$ is symmetric with respect to the origin. Hence, we only have even-numbered terms: which can also be written as where $G_{2n+2}$ is an Eisenstein Series . The first couple terms are

Differential equation for $\wp$

From the first few terms of the Laurent series for $\wp$ listed above, we may compute the following: It is possible to find a linear combination of $\wp'^2, \wp^3,$ and $\wp$ that yields a function with no terms in its Laurent series with negative exponent. The coefficients can be determined by solving a system of linear equations, yielding the identity Notice that the LHS of this equation is a doubly periodic function , but the RHS shows that it has no pole at $z=0$ and therefore no poles anywhere (since $\wp$ only has a pole at $0$). But this is impossible, unless both sides are constant functions. Hence, we must have that or It is conventional to denote the quantities $60G_4$ and $140G_6$ by $g_2$ and $g_3$, so that we have This is the differential equation for the Weierstrass $\wp$ function. Since the coefficients of $\wp$ are multiples of the Eisenstein Series , equating coefficients in this differential equation yields nontrivial relationships between the values of these Eisenstein Series for fixed $\omega_1,\omega_2$, and in particular, it allows us to express each $G_{2k}$ as a polynomial in $G_4$ and $G_6$.



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