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The following is from Diamond and Shurman's A First Course in Modular Forms book:

enter image description here

I have two questions (as underlined above):

1- Which and how the theory of finite Abelian groups are related to the basis of $m \Lambda$? A simple explanation for whole of the sentence starting from "By the theory of finite Abelian groups..." would be much appreciated.

2- What does $\text{deg(ϕ)}$ mean?

Edit. I think $\text{deg(ϕ)} = |\text{ker(ϕ)}|$.

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  • Do you see how a $2 \times 2$ integer matrix acts on a lattice $\subset \mathbb{C}$ ? What is the inverse of a $2 \times 2$ integer matrix ? $m$ is the determinant. – reuns Nov 02 '18 at 22:45
  • Is "degree of an isogeny" not defined earlier in the book? – Gerry Myerson Nov 02 '18 at 22:57
  • @GerryMyerson, I Ctrl+F.ed "degree" in the book and degree of an isogeny appears for the first time in the image at OP uploaded! But most probably it would be the order of ker(?) –  Nov 02 '18 at 23:01
  • @Edi Does my answer here answer your first question? (Take $R = \mathbb{Z}$ in that answer since a $\mathbb{Z}$-module is the same thing as an abelian group.) – Viktor Vaughn Nov 02 '18 at 23:12
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    See https://math.stackexchange.com/questions/271602/what-does-degree-of-an-isogeny-mean for definition of degree of an isogeny. – Gerry Myerson Nov 02 '18 at 23:14
  • @André3000, unfortunately your answer is too advanced to me; I am on Ch4 of Allufi's Algebra (parallel studying with MF).. –  Nov 02 '18 at 23:27
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    @Edi Really it's just the result in the second paragraph that's essential. The paper of Keith Conrad I linked is also a good reference. – Viktor Vaughn Nov 02 '18 at 23:31

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The result about abelian groups is the following:

(i): let $M$ be a finitely generated, free abelian group. Then there exists a uniquely determined integer $r \geq 0$, called the rank of $M$, and a basis $v_1, ... , v_r$ of $M$: which is to say that the $v_i$ are elements of $M$ such that every element $m \in M$ can be expressed as $c_1v_1 + \cdots + c_rv_r$ for unique integers $c_1, ... , c_r$.

(ii): let $N$ be a subgroup of a finitely generated, free abelian group $M$. Let $d$ and $r$ be the ranks of $N$ and $M$. Then $d \leq r$, and there exists a basis $v_1, ... , v_r$ of $M$, as well as nonzero integers $a_1, ... , a_d$ such that $a_1v_1, ... , a_dv_d$ is a basis of $N$.

You can find a proof in most comprehensible linear algebra or abstract algebra textbooks. In the context you are seeing the result used, $\Lambda'$ and $m\Lambda$ are both finitely generated free abelian groups, each with rank $2$.

I don't immediately know the answer to your second question.

D_S
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