Let \(\Lambda_1,\Lambda_2\subset\mathbb C\) be lattices. Suppose \[ \sum_{\lambda\in\Lambda_1\setminus\{0\}}\lambda^{-4} = \sum_{\lambda\in\Lambda_2\setminus\{0\}}\lambda^{-4} \] and \[ \sum_{\lambda\in\Lambda_1\setminus\{0\}}\lambda^{-6} = \sum_{\lambda\in\Lambda_2\setminus\{0\}}\lambda^{-6}. \] Must \(\Lambda_1=\Lambda_2\)?
Solution
Yes.
1. The two lattices have the same Weierstrass invariants \(g_2\) and \(g_3\).
For a lattice \(\Lambda\), put \[ G_2(\Lambda) =\sum_{\lambda\in\Lambda\setminus\{0\}}\lambda^{-4}, \qquad G_3(\Lambda) =\sum_{\lambda\in\Lambda\setminus\{0\}}\lambda^{-6}, \] and the Weierstrass invariants are \[ g_2(\Lambda)=60G_2(\Lambda), \qquad g_3(\Lambda)=140G_3(\Lambda). \] The two equalities in the hypothesis therefore imply \[ g_2(\Lambda_1)=g_2(\Lambda_2), \qquad g_3(\Lambda_1)=g_3(\Lambda_2). \] Denote the common values by \(g_2,g_3\).
2. The pair \((g_2,g_3)\) determines the lattice itself.
The lattice with invariants \(g_2,g_3\) is the period lattice of the differential \(dx/y\) on the cubic \(y^2=4x^3-g_2x-g_3\): it is generated by the values \[ \pm\int_\gamma \frac{dx}{\sqrt{4x^3-g_2x-g_3}}, \] where \(\gamma\) ranges over oriented loops in the \(x\)-plane enclosing exactly two roots of \[ 4x^3-g_2x-g_3. \] The cubic is nonsingular because it comes from a lattice, so these period integrals are well-defined with the branch chosen continuously near each such loop.
These period integrals depend only on the pair \((g_2,g_3)\). Since \(\Lambda_1\) and \(\Lambda_2\) have the same pair by 1, both equal the same period lattice. Hence \[ \boxed{\Lambda_1=\Lambda_2}. \]
3. Equivalently, equality can be seen directly from the Weierstrass functions.
Both \(\wp_{\Lambda_1}\) and \(\wp_{\Lambda_2}\) satisfy \[ (\wp')^2=4\wp^3-g_2\wp-g_3 \] and have the same normalized principal part \[ \wp(z)=\frac1{z^2}+O(z^2) \] at \(0\). The differential equation recursively determines the Laurent series, so the two functions agree near \(0\), and hence everywhere by analytic continuation.
Finally, the full period group of \(\wp_\Lambda\) is exactly \(\Lambda\): every element of \(\Lambda\) is a period, while any period \(a\) sends the pole at \(0\) to a pole at \(a\), and the poles of \(\wp_\Lambda\) are precisely the lattice points. Thus equality of the two Weierstrass functions also forces \[ \Lambda_1=\Lambda_2. \]