10
votes
3answers
534 views

What is the intuition behind the definition of cuspidal representations?

Let $\mathbb{G}$ be a reductive group defined over a number field $K$, let $Z$ be its center, and let $\mathbb{A}:=\mathbb{A}_K$ be the ring of adeles of $K$. Reasonably, we care about the ...
16
votes
1answer
663 views

Why does the Section Conjecture exclude curves of genus 1?

Let $X$ be an integral proper normal curve over a (perfect) field $F$, of genus $\geq 2$. One variant of Grothendieck's "section conjecture" states that the sections $G_F \rightarrow \pi_1(X)$ of the ...
54
votes
9answers
6k views

Why should I believe the Mordell Conjecture?

It was Faltings who first proved in 1983 the Mordell conjecture, that a curve of genus 2 or more over a number field has only finitely many rational points. I am interested to know why Mordell and ...
20
votes
2answers
3k views

Intuition behind the Eichler-Shimura relation?

The modular curve $X_0(N)$ has good reduction at all primes $p$ not dividing $N$. At such a prime, the Eichler-Shimura relation expresses the Hecke operator $T_p$ (as an element of the ring of ...
6
votes
1answer
719 views

Geometric Intuition for Big Monodromy

In various contexts, I have come across results referred to as "big monodromy." A standard arithmetic example is the open image theorem for the image of Galois action on non-CM elliptic curves. A ...
25
votes
6answers
3k views

Intuition for the last step in Serre's proof of the three-squares theorem

Serre's A Course in Arithmetic gives essentially the following proof of the three-squares theorem, which says that an integer a is the sum of three squares if and only if it is not of the form 4^m (8n ...
10
votes
5answers
3k views

Examples and intuition for arithmetic schemes

How should a beginner learn about arithmetic schemes (interpret this as you wish, or as a regular scheme, proper and flat over Spec(Z))? What are the most important examples of such schemes? Good ...