# Characterization of tori/elliptic curve isogenies

I am reading Chapter 11 of Dale Husemöller's Elliptic Curves Springer book and I got stuck on Theorem (1.4) (c.f., image below).

Notation and definitions: Let $$L$$ and $$L'$$ be two complex lattices spanned, respectively, by periods $$\{\psi_1,\psi_2\}$$ and $$\{\psi_1',\psi_2'\}$$. Set $$\tau:=\psi_2/\psi_1\in\mathfrak{H}$$ and $$\tau':=\psi_2'/\psi_1'\in\mathfrak{H}$$, where $$\mathfrak{H}$$ is the upper-half complex plane. Finally, we introduce two tori $$T_\tau:=\mathbb{C}/L$$ and $$T_{\tau'}:=\mathbb{C}/L'$$.

By definition, $$T_{\tau}$$ and $$T_{\tau'}$$ are said to be isogenic iff there's a complex number $$\lambda$$ such that $$\lambda L'\subset L$$.

Below, $$GL^+_2(\mathbb{K})$$ denotes the group of two-by-two invertible matrices with positive determinant and with entries in the field $$\mathbb{K}$$.

My question regarding the proof of (2) below is the following: Why do we need to restrict to the subgroup $$GL^+_2(\mathbb{Q})$$ of $$GL^+_2(\mathbb{R})$$ and not just take $$GL^+_2(\mathbb{R})$$ itself? I see no obstruction in replacing $$\mathbb{Q}$$ by $$\mathbb{R}$$ in the proof.

Am I missing something here?

• $a,b,c,d$ are integers. Commented Apr 9, 2022 at 21:49

The lattice $$L$$ is defined by taking integer combinations of the basis vectors $$1$$ and $$\tau$$. If you have arbitrary $$a,b,c,d\in\mathbb{R}$$ and set $$\lambda=c\tau+d$$, then $$\lambda\cdot 1\in\lambda L'$$ will be a real combination of $$1$$ and $$\tau$$, but it will not be an integer combination unless $$c,d$$ themselves are integers.
• Sorry I am not following you. Why would $\lambda$ need to be a integer combination of 1 and $\tau$? We just want to show that the prime lattice is contained in the unprimed one under $\lambda$-scaling for some choice of complex $\lambda$, no ? Where would the need from your condition come from then? Commented Apr 9, 2022 at 22:50
• $\lambda$ is - a priori - allowed to be any complex number. But the claim that $\lambda L'\subseteq L$ is saying something very specific: it's saying that $\lambda\cdot 1$ and $\lambda\cdot \tau'$ are both in $L$. This means that $\lambda\cdot 1$ and $\lambda\cdot \tau'$ must both be elements of $\{m+n\tau:m,n\in\mathbb{Z}\}$. Commented Apr 9, 2022 at 22:52
• I think an example could potentially help. Let's take two lattices, $L'$ spanned by $\{1,i\sqrt{2}\}$ (rectangular lattice) and $L$ by $\{1,i\}$ (square lattice). We want to find any $\lambda\in\mathbb{C}$ that satisfies $\lambda L'\subseteq L$. This means $\lambda$ has to take both $1$ and $i\sqrt{2}$ to elements of $\mathbb{Z}[i]$. But the fact that $\lambda\cdot 1\in\mathbb{Z}[i]$ means that we actually must have $\lambda\in\mathbb{Z}[i]$, and this implies $\lambda\cdot i\sqrt{2}$ can't be in $\mathbb{Z}[i]$. (continued) Commented Apr 9, 2022 at 23:05
• Hence the two corresponding tori are not isogenous, despite the fact that $\begin{pmatrix} \sqrt{2}&0\\0&1\end{pmatrix}\in GL_2^+(\mathbb{R})$ takes $i$ to $i\sqrt{2}$. Commented Apr 9, 2022 at 23:05