We have put together some cool examples and exercises.
I've arranged some playful examples and standard textbook problems.

list:

Example 1

● What are some examples of expansion of branching index e and remainder index f?

● Benefits of projection

● Supersingular elliptic curves

● The Tamagawa number is the number of connected components of the special fiber of the Neron model.

● If more than 62.5% of the pairs of elements are commutative, we can conclude that it is an abelian group!

● Infinite dimensional Galois theory

Example 2

● Is the completion of the union of perfectoid bodies a perfectoid?

● Witt rings that are not perfect cannot be DVR.

● Alternating matrices on F_2 and 2-Selmer groups

● Selmer groups of representations and classical Selmer groups

Quiz: What are some examples of expansion of ramified index e and residue field degree f?

Benefits of Projection

This is a problem to find the number of rational points on the unit circle Fp. The problem itself is very elementary number theory, and there are many different solutions, but it is easier to understand if you think of it as an affine algebraic variety, project it, and subtract the extra points at infinity. The answer is divided into cases by the remainder when p is divided by 4, because there is a number that becomes -1 when squared, which means there will be two points at infinity or none at all.

By the way, when calculating the order of a rational point after projection, the following holds for an algebraic curve C of genus g: |#C(Fq)-(q＋1)|≦2g√q, so it can also be calculated if g＝0.

Supersingular elliptic curves

Continuing from above, here is the case of g = 1. This was a question asked in the oral examination for graduate school. An example of an elliptic curve for which the left side of the Hasse inequality for elliptic curves is 0 is a supersingular elliptic curve, as shown above. Conversely, if the left side is 0, then it is a supersingular elliptic curve, as can be seen from the fact that φ^ is non-separable.

When p is 2 or 3, even with supersingular, there are cases such as E/F3:y^2＝x^3-x−1, where E(F3)＝{0} and it is supersingular but #E(F3)＝4 does not hold.

The Tamagawa number is the order of Q_p rational points of the component group of the special fiber of the Neron model

There is a proposition called the finiteness of Tamagawa numbers. Let K be a discrete valuation field. E_0(K) is a subgroup of E(K) consisting of all points that transition to non-singular points when reduced at a point on an elliptic curve, and this proposition states that this is a subgroup with a finite index. There is a criterion by Neron ogg shafarevich that shows how an elliptic curve is reduced by using n-part points on an elliptic curve and the action of an inertia group on a Tate module, and this is used to prove it.

Furthermore, while E(Qp) is an infinite group, E(Qp)/mE(Qp) (the group that appears in the proof of the weak Mordell Weil theorem) is a finite set, which can be easily seen from the finiteness of the Tamagawa numbers.

Although this is a proposition that can be understood at an elementary level, the Neron model is used to prove it.

Let R be a DVR and K be its quotient field. A Néron model of an elliptic curve E/K is a group scheme on Spec R whose generic fiber is isomorphic to E/K and satisfies the universality property called the Neron mapping property (NMP). Néron models exist. For example, let W be a closed subscheme of P^2_R defined by the Weierstrass equation. If W is smooth on R, then W/R is a Néron model. In fact, W×_{SpecR} Spec K is isomorphic to E/K and satisfies the NMP. In fact, let x/R be a smooth group scheme on Spec R with generic fibern X/K. For any rational map X/K\to E/K, the rational map x\to W between the original schemes extends to a morphism because W is proper on R.

E(K) can be identified with {arrow Spec K\to E=Spec K[x,y]/(y^2=f(x)) \cong {arrow K[x,y]/(y^2-f(x))\to K}\cong {(x,y)\in K^2 \mid y^2=f(x)}=E(K). The first isomorphism is the contravariant equivalence between the category of affine schemes and the category of commutative rings, and the second is just ring theory. From the definition of the Néron model and the universality of fiber products, a natural injective from ε(R) to E(K) can be defined, and it follows from NMP that this is bijective. The injectivity is also immediately evident from the fact that the composition of ε(R) with r,s:SpecR→ε and the natural morphism SpecK→SpecR coincides as a morphism on a generated point scheme. This can also be seen from the valuation test for separability.

The group E(K)/E0(K) that we wish to show is finite can be shown to be isomorphic to the group obtained by dividing the special fiber ε~ of ε by its identity component, using the Tate algorithm. The fact that the quotient group is a finite group follows from the fact that for algebraic groups, the identity component is of finite index. Proof of the fact: Since G is an algebraic variety, its connected components are finite. Let the connected components of G be G_0,...G_n. Here, G_0 is a normal subgroup of G (because for any g \in G, gG_0g^-1\subset G_0 and gG_0 g^-1 is connected). Define a mapping Φ from G/G_0 to {all connected components of G} by gG_0 \to gG_o. For any connected component C of G, there exists g\in G such that C=gG_0. In fact, let g be any element of C. Clearly, C \subset g C_0. On the other hand, since g \in g C_0 is maximal as a connected component of C containing g, g C_0 \subset C. Hence, g C_0 \subset C. The injectivity of Φ is clear, and Φ is bijective. Therefore, [G:G_0] = (the number of connected components in G) = n < \infty.

References JHSilverman Advanced topis of the arithmetic of elliptic curves

If more than 62.5% of the pairs of elements are commutative, we can conclude that it is an Abelian group!

If there is even one pair of elements that is non-commutative, then it is a non-commutative group, but conversely, how many non-commutative pairs are needed to declare that the group is commutative? This question asks this question. A problem showing the above inequality was once asked at the Stanford University Graduate School. Experiments with S3 give 1/2, while experiments with D4 or Q8 give 5/8. If more than 5/8 pairs of elements are non-commutative, then it can be declared to be an Abelian group, which seems a bit strange. https://arxiv.org/pdf/1205.4757.pdf looks interesting, so I would like to read it next time I have some free time.

Infinite dimensional Galois theory

Galois groups contain a topology called the Krull topology, which is a topology in which U_σ in the image above is a fundamental neighborhood system of σ. In other words, it is saying that "if you want to know whether U ⊆ Gal(L/K) is an open set, you can take any σ∈U and say that you can find U_σ such that σ∈U_σ⊆U." This gives you all the open sets.
The important part in the definition of the Krull topology is that M over K is a finite extension (it is convenient to limit the Galois condition to Galois in order to write Gal(M/K) when taking M). When M can run up to L (when it is finite-dimensional), it becomes a discrete topology, and when it cannot run up to L (when it is infinite-dimensional), the p-adic topology or the topology of Z^ appears.
Two elements f and g of a Galois group are defined as "close" if f | M = g | M for a finite intermediate field M over K. Once this is recognized, it is natural that the discrete topology is equivalent to the finite order, and furthermore that Galois comes from the finite in the definition of the Krull topology.

Taking the Galois extension Q(√2,√3)/Q and its Galois group {1,σ,τ,στ} (σ:√2→-√2,τ:√3→-√3) Klein's quaternion group) as an example, U_1＝U_σ＝U_τ＝U_στ＝{1,σ,τ,στ}, and the union of the fundamental neighborhood system is a basis, so it is a discrete topology. In the infinite-dimensional case, a one-point set cannot be an open set, and it becomes a topological space with no isolated points.

We have put together some cool examples and exercises. I've arranged some playful examples and standard textbook problems.