$k$ is an algebraically closed field. $\Gamma(V)$ is image of $r: k[X_1,...,X_n]\to F(V,k)$ and $V$ is affine algebraic set. I am self studying Algebraic Geometry from Daniel Perrin's Algebraic Geometry and have a question in proof of Proposition $4.8 $ of Chapter $1$ on Page $17$. Proposition...

Let $$ t_n = {16 \over (2n+1)5^{2n+1}} - {4 \over (2n+1)239^{2n+1}}. $$ Define $$ S_N = \sum_{n=0}^N (-1)^n t_n. $$ For each integer $m \geq 0$, define $$ I_m = [S_{2m+1}, S_{2m}]. $$ By Machin's formula, $$ \pi = 16\arctan(1/5) - 4\arctan(1/239), $$ so $$ S_{2m+1} \leq \pi \leq S_{2m}. $$ Also,...

This is extended computational evidence for the Conjecture from this question, focusing on the $n=1$ case. Conjecture A for $n=1$ states: for all sufficiently large primes $p$ with $q \mid p-1$, $$v_p\!\left(\prod_{k=1}^{p} \Phi_q(k)\right) = q-1.$$ Reduction to $v_p(k^{p-1}-1)$. By the...

Let $N_n=\binom{2n}{n}^3(42n+5)$. Define $L_m=\sum_{n=0}^m \frac{N_n}{2^{12n+4}}$ and $U_m=L_m+\frac{4N_{m+1}}{3\cdot 2^{12m+16}}$. These are rational intervals coming from a Ramanujan series for $1/\pi$. For this question, I only want to study rational points inside these explicit intervals....

In Kedlayas lecture notes (https://kskedlaya.org/prismatic/sec_overview.html) about prismatic cohomology he introduces the ring of Witt vectors using $\delta$-rings via the fact that the Witt vector functor $W$ is the right adjoint to the forgetful functor $\mathbf{Ring}_{\delta}\to...

Let $G$ be a group object in an $\infty$-topos $\mathcal T$, and let $P \to X$ be a $G$-torsor as in the accepted answer by Daniël Apol to this question; equivalently, as in the answer, such a $G$-torsor is given by a map $\varphi \colon X \to BG$, and $P$ is the pullback of $\varphi$ along the...

Let us consider a prime number $p$ and three distinct positive integers $n_1<n_2<n_3$ less than $p$. Let us assume that the triple $(n_1, n_2, n_3)$ satisfies the following condition $$k+[kn_1]+[kn_2]+[kn_3]=2p, \ \ for \ all \ 1\leq k\leq p-1$$ Where $[kn_i]$ denotes the rest of the division of...

For a semiprime $N = p×q$, consider the sequence $f(k) = k^2 + k + N$, for $k = 0, 1, 2, ...$ Since $f(k) ≡ k(k+1) \mod N$, we have gcd($f(k), N$) = gcd($k(k+1), N$). This means a factor of $N$ is revealed at position $k$ whenever $p$ divides $k$ or $k+1$. For $N = 77 = 7×11, f(k) = k^2 + k +...

While trying to prove FLT for $n=7$, I came to find that there could be a certain class of solutions (counterfactual) for the counterfactual case that FLT is false for $n=7$: $(d^7+x^7+y^7)^7 = (d^7+x^7-y^7)^7 + (d^7-x^7+y^7)^7$ where $x, y$ and $d$ are coprime and all of them are coprime to...

For an elliptic curve, an L-function is associated and has an Euler product. Is the Euler factor at prime $p$ the same as the zeta function of this elliptic curve over $\mathbb{F}_p$ at $p^{-s}$? That is $\exp(\sum_{n=1}^{\infty}\frac{N_n}{n}t^n)$ with $t=p^{-s}$, where $N_n$ is the number of...

Let $\varphi: X\to Y$ be a dominant morphism of affine varieties over algebraically closed field $k$, $\varphi*: k[Y]\to k[X]$ be the induced $k$-algebra monomorphism. Let $\mathfrak{m}_y$ be the ideal of a point $y\in Y$. I believe the ideal $I(\varphi^{-1}(y))$ of the fiber $\varphi^{-1}(y)$...

Let $G$ be a finite group, let $k$ be a field of characteristic $p$, and let $S_p$ be a Sylow $p$-subgroup of $G$. It is a well-known fact from modular representation theory that every irreducible representation factors through any normal $p$-subgroup of $G$. In particular, it factors through...

Let $\mathcal{A}, \mathcal{B}, \mathcal{C}, \mathcal{D}, \mathcal{E}, \mathcal{F}$ be infinite arithmetic progressions of integers. We consider the set of all second-order quadratic recurrence relations: $$s_{n+1} = a s_n^2 + b s_n s_{n-1} + c s_{n-1}^2 + d s_n + e s_{n-1} + f$$ where $(a, b, c,...

I started from a modular experiment for integers of the form $$ N=2k+11, $$ where I iterated $$ U_0=k+1,\qquad U_{n+1}=U_n^2-k\pmod N, $$ and called an integer $N$ captured if the orbit reached a solution of $$ x^2\equiv k\pmod N. $$ After rewriting the iteration symbolically using $$...

I'm working through Vakil and encountered the following exercise. 5.1.B. EXERCISE. Exercise 3.7.F showed that there is a bijection between irreducible closed subsets and points for affine schemes (the map sending a point p to the closed subset $\overline{\{p\}}$ is a bijection). Show that this...

Let $A$ be a (associative with unit) $\mathbb{Q}_p$- and $\mathbb{Q}_q$-algebra where $p,q \in \mathbb{P}\cup \{\infty\}$. Does it follow that $p=q$? If $A$ would be a Hausdorff locally compact skew-field and topological $\mathbb{Q}_p$- and $\mathbb{Q}_q$-algebra then it would be necessarily...

In the proof of Theorem 8.12 in Morandi's Field and Galois Theory, the author writes: Because $KN$ is the composite of a Galois extension of $S$ with a purely inseparable (hence normal) extension, $KN/S$ is normal. Thus, $\sigma_j(K)\subseteq KN$ by Proposition 3.28. I do not understand this...

Here is a quite interesting problem I've come up with: Let $\mathbb{N}^+ = \{1, 2, 3, \dots\}$. Does there exist a sequence of sets $A_1, A_2, \dots$ such that: $1$. $A_k \subset \mathbb{N}^+$ is nonempty and finite. $2$. $A_i \cap A_j = \varnothing$ for $i \neq j$. $3$. $\bigcup_{k=1}^{\infty}...

How to prove that for any faithfully flat quasi-compact (or, more generally, fpqc) morphisms are epimorphisms? My attempt: Let $f:X\to Y$ be faithfully flat quasi-compact (or, more generally, fpqc) and suppose that $g_1,g_2: Y\to Z$ are two morphisms such that $g_1\circ f=g_2\circ f$. Since $f$...

I am working on moments of the Riemann zeta function, and want to get a numerical lower bound for the $k$th moment of $\zeta(s)$ of the form $$\int_1^T|\zeta(1/2+it)|^{2k}dt>C(k)T(log T)^{k^2}.$$ Soundararajan (https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300011438)...