I'm trying to figure out some facts about the pullback of Cartier divisors, but honestly I'm having a hard time describing the pullback of the sheaf associated to divisor. Let $\phi\colon X \to Y$ be a dominant morphism of schemes. Suppose $X, Y$ are both integral, noetherian, separated. If...

It is a well known fact that the Galois group $G$ of a Galois extension $K\subseteq L$ is a profinite group, as $G$ is equal to the inverse limit of the Galois groups of the finite subextensions of $K\subseteq L$. Therefore, $G$ gets a "natural" topology that turns it into a compact group. Why...

Recently we encountered a difficult problem in computer science, but since it is very closely related to mathematics, I was unsure which board would be more appropriate. In the end I posted it here on the mathematics board. To make the problem easier to understand, I will give both a...

Is the following statement true or false?Let $\mathbb{N}$ be the set of positive integers. For any $z > 2$, there always exist $x, y < z$ such that:$$f(x) + f(y) = f(z)$$Where the arithmetic function $f(n)$ is defined as:$$f(n) = \prod_{p^k \parallel n} \left( \frac{p^{k+1}-1}{p-1} \right) =...

As I mentioned in my previous post, I have been reading (Talpo and Vistoli)'s paper Infinite root stacks and quasi-coherent sheaves on logarithmic schemes. While editing that post, a second question came up, which I thought was sufficiently independent to deserve its own post. So here is the...

The discovery by Andrew Booker of an integer solution to, $$N=x^3 + y^3 +z^3=33$$ $$8866128975287528^3 - 8778405442862239^3 -2736111468807040^3=33$$ got some press and Youtube mileage back in 2019. As mentioned in Booker's July 2019 article, for $0<N<1000$, there used to be $13$ unsolved $N$,...

I am looking for a reference—either a book or a specific paper—that contains the actual proof of the result that no three consecutive positive integers are perfect powers. While reading Wacław Sierpiński's 250 Problems in Elementary Number Theory, I came across a remark stating that A. Mąkowski...

I ran a simulation where for each 2-digit combination with 30 symbols (so 0 to T), it checked, from base 2 to base 10,000, in how many bases that specific symbol combination resulted in a prime number. The top 10 were 65,6B,6H,6T,6N,61,6D,67,6J, and 6P. All starting with 6. Anyone have any idea...

This problem is from the book Álgebra comutativa em quatro movimentos by Borges and Tengan. Let $k$ be a field of characteristic not equal to $2$ or $3$. Let $f(x, y) = y^2 - x^3 - 10$ and consider the ring $R = k[x, y]/(f(x, y))$. Show that $R$ is a Dedekind domain. Edit. We also have to assume...

I am reading the paper Oscillatory Integrals with Uniformity in Parameters by Kiral, Petrow, and Young, and I am having trouble understanding the notion of an inert function introduced in Definition $2.1$. These are my confusions Since $X=X_T \in [1,\infty]$.Then if we consider a family of...

I'm trying to solve the following problem from a commutative algebra book (Álgebra comutativa em quatro movimentos by Borges and Tengan). The question has 30 concrete examples of rings (mostly quotients), but I will restrict to two. Compute the Krull dimension of the following rings. $R =...

$\newcommand{\e}[1]{\exp\left(#1\right)}$ $\newcommand{\le}{\leqslant}$ In Terry Tao's Math 254A Notes 5 (Bounding exponential sums and the zeta function), Exercise 5 outlines the derivation of the Vinogradov-Korobov bound for Dirichlet $L$-functions. Let $\chi$ be a non-principal character of...

Let P(n) and Q(n) be two non-trivial polynomials in n with rational coefficients and z[P, Q] is the value of infinite sum of P(n)/Q(n) from n=1 to +∞ (only when it converges, in which the degree of Q should be larger than or equal to the degree of P plus 2). Claim: It is impossible for z[P,Q] to...

Let $C$ be a plane curve and suppose, for simplicity, it has a singularity at the origin. Let $\tilde{C}$ be it's strict transform after blowing up $(0,0).$ Is it possible for $\tilde{C}$ to have a singular point not at the origin?

These days I've been reading (Talpo and Vistoli)'s paper Infinite root stacks and quasi-coherent sheaves on infinite root stacks. At the moment I'm looking at their proposition 4.9....

I mean, if a function $F(s)$ is in Selberg class, and $F(s)$ has a pole of order m at $s=1$, is it true that there exists a function $G(s)$ in Selberg class such that $F(s)=\zeta^m(s)G(s)$, and $G$ is entire?

Suppose $R$ is a reduced Noetherian ring. We know that $f \in R$ is a zero divisor if and only if $V(f)$ contains an irreducible component of $\mathrm{Spec} R$. I would like to know if one could use the irreducible components to say something about the annihilator of $f$ in $R$? (Perhaps it...

I found this problem where I think there is a mistake but am not 100% sure: Let $Z$ be a closed subset of a topological space $X$. If $Z$ is irreducible we call codim($Z,X$) as the supremum of lengths of the chains of irreducible closed subsets of $X$ which contains $Z$: $$Z\subset Z_0\subsetneq...

One of the axioms for Selberg class is that for some natural number $m$, $(s-1)^mF(s)$ extends to an entire function of finite order. What is the use of "finite order"? If we allow it to be of infinite order, what will happen?

Elliptic curves have L-functions that correspond to modular forms. Elliptic curves are degree 3 algebraic curves. I want to know if quadratic curves have L-functions. If they do, are these L-functions related to modular forms?