奥の細道を超えて

[Header Image Credit]
The first expedition of Xu Fu 徐福 to the Mount Penglai 蓬萊山 in search of the Magical Herbs of Longevity (by Utagawa Kuniyoshi 歌川国芳)
MoFA, Boston via Wikimedia Commons

Preface

💫Highly inspired by the following superb articles⬇️

The Complex Plane is indeed so fundamental for classical mathematics.
Elliptic Curves 楕円曲線 are defined on the Complex Plane using Weierstrass ℘-function or using a lattice .
Hyperelliptic curves 超楕円曲線 are defined as ramified double cover of the projective line, which is a compactification of the Complex Plane.
The Upper Half Plane 上半平面 H := {z ∊ C | Im(z) >0}, known as the Poincaré half-plane model, represents the Hyperbolic Plane and is so fundamental for Non-Euclidean Geometry 非ユークリッド幾何学　(accurately speaking Hyperbolic Geometry 双曲幾何学), with which we can define Modular Curves モジュラー曲線 in general.
These fascinating mathematical objects do form Shangri-La.

Challenges

In reality, there are many things that we don't understand even though the Complex Plane is not Terra Incognita. In other words, exploration along the Narrow Road to the Complex Plane is still very much worse pursuing.

The three different paths I can think of are as follows.

① Walk on the line s = 1/2 + it (t ∊ R) on C¹

When you walk that path carefully, you'd find infinite gems engraved with the Greek letter ζ (zeta). An extremely intriguing thing is that that such gems might be found nowhere else.

② Scrutinize the area near 1 in the form of β > 1 - c/(log q) to find the Philosopher's stone (or the Elixir of life)

It has nothing to do with "Harry Potter and the Philosopher's Stone", unfortunately. 
Should you be able to debunk the myth and find a "zero", felicitations!

③ Zoom into the Mandelbrot Set to confirm it's locally connected

This is completely different from ① and ②.
The MLC Conjecture (Mandelbrot Local Connectivity) is stated as follows:

• The Mandelbrot set is locally connected.

This would appear more visible than ②😉

Prerequisite & Caveats

These are evidently exceedingly difficult, yet also unfathomably rewarding.
All the three paths expect challengers are Black Belts who can explore without relying solely on AI.
Those who coin inexplicable terms with mathematical nuances assisted by AI will not face the same fate as the failed suitors of Turandot😱, but will not be entertained either. From an environmental conservation standpoint, I humbly wish AI-generated nonsense would not trespass into this territory.

Annotations

① is unarguably famous, and an annotation is not deemed required, however, I had better make a remark that the Riemann's ζ-function should be defined carefully by analytic continuation, without which the "trivial zeros" don't make sense.

② is less famous, or perhaps not known at all to laypeople, but is a very important theme for mathematics.
It's called Siegel zero, a definition of which is as follows:

The following theorem by D. R. Heath-Brown (explained by Terrence Tao) is highly important, and it's extremely intriguing, so different from typical statements of theorems😲

③ is far less famous as a problem, however the Mandelbrot Set, one of the most iconic fractals, is widely known to people even without any training in mathematics.

Here 's a much "simpler" example called "Topologist's sine curve" defined as follows:

The remarkable property of the topologist's sine curve T is that it is connected but neither locally connected nor path connected.

The Mandelbrot Local Connectivity is considered one of the most important unsolved problems in complex dynamics and modern mathematics, despite not being included in the Millennium Prize Problems.

A different route

Should you prefer exploring wider spaces over walking on flat ground, Algebraic (Arithmetic) Geometry awaits you with so many enchanting themes.
The present author is still on the following highly steep route🏔️

References

Please kindly be advised that the present author not only does not hold a mountain guide qualification but also has no personal experience traversing the aforementioned paths.
In addition, the present author feels some obligation to come up with three (3) treasures (if not tortures) (tres tesoros para tres monos)😎
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