I am working on a sentence of Yves Hellegouarch in his book "Invitation aux mathématiques de Fermat-Wiles" . In the Fermat section, page 38, he tells that Fermat probably associated the equation $z^p=x^p+y^p$ to the form $X^2+(-1)^{(p+1)/2}pY^2$. I don't really understand the reason he thinks...

I am curious if there is a notion of how to find a Gröbner basis for any ideal $I$ of a finitely generated algebra $R\cong \mathbb{K}[x_1,\dots,x_k]/J$. I know that Gröbner bases are generaly developed as a tool for polynomial rings, but I wonder what fails in this case or in which cases it's...

I'm struggling to understand an argument in the book "Weil Conjectures, Perverse Sheaves, and $l$-adic Fourier Transform" by Kiehl and Weissauer. In Theorem I.6.1, they prove (in specific cases) that the $i$-th cohomology of a pure sheaf of weight $w$ has weight $w+i$. I'm confused by their...

I am trying to prove that the prime ideals above 5 in $K=\mathbb Q(\sqrt{359})$ are non-principal. I calculated the splitting to be $5O_K = (5,\sqrt{359} + 2)(5,\sqrt{359} + 3)$. If either of the ideals were principal their generator would have norm $\pm 5$. Showing that $a^2-359b^2 = -5$ has no...

Let $\mathcal P$ denote the set of prime numbers, and consider $$ N =\frac{1}{1-\sum_{p\in\mathcal P}2^{-p}}. $$ Numerically, the binary expansion of $N$ appears to behave like that of a base-$2$ normal number. For example, among the first $10^6$ binary digits, the frequencies of $0$ and $1$,...

Let $N \geq 5$ be an integer so that the $\Gamma_1(N)$-moduli problem is representable over $\mathbb{Z}[1/N]$ (both in terms of elliptic curves/generalized elliptic curves). I am interested in Katz modular forms of this level and their $q$-expansions. From my modest understanding, there are two...

As a soon-to-be senior undergraduate planning to pursue research in Schubert calculus under a supervisor specializing in this field, I have struggled to locate accessible introductory textbooks or lecture notes for this subject, as well as more advanced reference materials to save for my future...

There is a philosophy attributed to Grothendieck that it is better to have a good category (e.g. mapping objects, abelian category, etc) with bad objects than a bad category with nice objects. What are some examples of this? Please also describe some ways these good categories have been helpful.

Let $p \geq 3$ be an odd prime. I would like to know whether the following conjecture is true. Conjecture For every odd prime $p \geq 3$, there exist integers $a$ and $b$ such that: $a+b+3$ is prime; $4a+2b+3=p$ $\gcd(a,b,3)=1$ $b^2\leq 4a$ $a\geq 1$. Here, $\mathbb{P}$ denotes the set of prime...

In the rather new MSE post Recursive sequence $a_{k+1} = a_k - \gcd(a_k, (n+k)^2 - 1)$ generating twin primes, a sequence of integers $(a_k)_{k \geq 1}$ is associated to each natural number $n$, namely $a_0 = n^2$ and $a_{k+1} = a_k - \gcd(a_k,(n+k)^2-1)$. What is interesting, as pointed out by...

Let $X$ be a smooth algebraic variety over a field $k$, and let $K/k$ be any field extension. I want to prove that $$ X_K:=X\times_{\operatorname{Spec}k}\operatorname{Spec}K $$ is smooth over $K$. I want to use only the following facts: Jacobian criterion (rational points): If $$...

I fail to understand the highlighted statement in my screenshot below. If $U \subset Y$ is a non empty open subset, then the natural map $g: \mathscr{O}_Y(U) \to k(Y)$ given by $(U,f) \mapsto [U,f]$ is naturally injective. Indeed, if $g((U,f_1)) = g((U,f_2))$, i.e $[U, f_1] = [U,f_2]$, then...

I find it easiest to explain the motivation as a "game" or task: you are given an arbitrary (but guaranteded to be valid) subsequence of the characteristic function of primes, so just '1's and '0's, one after another, and your task is to identify the numbers they represent. To simplify, let's...

I am an independent researcher investigating the arithmetic and structural properties of prime distributions formulated within the twin-track lattice of $6n \pm 1$. I would like to inquire about a potential algebraic and geometric invariant regarding Goldbach pairs. Consider the following model...

I read little bit about Special Relativity and there was one moment that I can't understand. It was about that there was two irreducible polynomials that have common roots: I was confused because each of these polynomials have different variables. My question is: is it possible that two...

Determine all monic polynomials $P(x)$ with integer coefficients for which there exists a monic polynomial $Q(x)$ with integer coefficients such that, for every pair of positive integers $m,n$, $P(m^2+mn+n^2)\ne 0$ and $$ P(m^2+mn+n^2)\mid Q(m^4+m^2n^2+n^4). $$ Let $$ A=m^2+mn+n^2,\qquad...

(I'm sorry for my English.) Hello. I have been reading B.Mazur's article "An introduction to the deformation theory of Galois representations". I'm at the proof of proposition1 in §30, where he discusses I-ordinary deformation. Let me write down the settings : $A$ : a Noetherian local ring which...

I am self learning Algebraic Geometry from Daniel Perrin's Algebraic Geometry textbook. I have a question on last paragraph of page $41$. Let $D(f)$ be the set of points where the function doesn't vanish and $\Gamma(V)= k[X_1,...,X_n]/I(V)$ where $k$ is a commutative field. Let $r$ denote the...

I am reading up on some algebraic geometry and came across the above definition of sheaves. Some things confuse me. When the author says in 2) of Definition 4.1, "For each inclusion of open sets $V \subset U$..", does he mean that $V$ is an open set in $X$ or in the induced topology on $U$? In...

Problem Statement Let $f: X \to Y$ be a surjective morphism of finite type between affine Noetherian schemes, where $X = \operatorname{Spec} B$ and $Y = \operatorname{Spec} A$. Suppose that for every closed point $y \in Y$ and for every $x \in X_y$ that is closed in $X_y$, the stalk map...