Title: Primes as $X-Y$ with $\gcd(X,Y)=1$ and $\operatorname{rad}(XY)$ equal to the product of all primes below $p$ — is this known? While experimenting with multiplicative decompositions of primes, I arrived at the following question. I would like to know whether it is already in the...

Consider a non-cuspidal point $x\in X_0(p)(K)$ representing an elliptic curve $E/K$ over a quadratic field $K$, and suppose that $E$ has potentially multiplicative reduction over a prime $\mathfrak{q}$ of $K$ over the rational prime $q$. Let $0,\infty$ be the cusps of $X_0(p)$. Using the...

This is about the umpteenth discrepancy of definitions in algebraic geometry. It's about recipes to define properties of morphisms of algebraic spaces out of properties of morphisms of schemes. When the morphism of algebraic spaces in question is representable the recipe is clear [SP, 025V;...

We say that a topological space $X$ is irreducible if it can not be written as the union of two proper closed subsets. My question is: Do we consider this definition when the topological space is defined in terms of open subsets or closed subsets? When working with the zariski topology, the...

I am looking for a topological or geometric interpretation of the Albert–Brauer–Hasse–Noether (ABHN) theorem, which establishes the local-global principle for central simple algebras over a global field $K$. The classical exact sequence is given by: $$0\rightarrow \text{Br}(K)\rightarrow...

We consider the $L$-function associated with a nonzero Maass cusp form $f$ of weight $0$, level $N$, and Laplace eigenvalue $1/4+r^2$. Let $t(n)$ be the normalized Fourier coefficient corresponding to the Maass cusp form $f$. I need the estimate of $$\sum_{n\le T}t(n)e^{2\pi i n x},$$ where $x...

One question from a past qualifying exam is to find the Galois group of $x^6+3$ over the finite field $\mathbb{F}_5$. With some creativity, we can write: $$x^6+3=x^6+8=(x^2)^3+2^3=(x^2+2)(x^4-2x^2+4)=(x^2+2)(x^4+4x^2+4-16x^2)\\ =(x^2+2)((x^2+2)^2-(4x)^2)=(x^2+2)(x^2-2x+2)(x^2+2x+2).$$ Since...

One question in a past qualifying exam asked us to find the Galois group of $x^6+3$ over $\mathbb{F}_5$, $\mathbb{F}_7$, and $\mathbb{Q}$. Each part of this question has been answered individually on this website: over $\mathbb{F}_5$, $\mathbb{F}_7$, and $\mathbb{Q}$. I asked this question again...

I am reading Arithmetic Moduli of Elliptic Curves by Katz and Mazur, and I have a question about the beginning of Chapter 2, §2. Let $f:E\to S$ be an elliptic curve. Since the sheaf of relative diffrentials $\Omega^1_{E/S}$ is an invertible sheaf on $E$, one defines...

I found the following infinite continued fraction: $$ \frac{1}{G} = \frac1{(2\pi)^{-3/2}\,\Gamma^2(1/4)} = \cfrac{4}{1+\cfrac{5}{1+\cfrac{6}{1+\cfrac{9}{1+\cfrac{8}{1+\cfrac{13}{\ddots}}}}}} $$ where $G$ denotes Gauss's constant, and the partial numerators are defined by interweaving sequences...

I've been experimenting with a self-similar construction for the Collatz map (the $n \to n/2$ / $n \to 3n+1$ function) and found a pattern I can partially — but not fully — explain. I'd appreciate a sanity check and any pointers to relevant literature. Construction. Fix an odd integer $x$ with...

Let $A$ be a commutative ring and $f, g \in A[S, T]$ be two homogeneous polynomials in two variables of homogeneous degrees $d$, resp. $e$. Their resultant $\newcommand{\Res}{\mathrm{Res}}\Res(f, g)$ is defined to be the determinant of the linear map of free modules of rank $d + e$ $$ (f, g)...

So recently the Jacobian conjecture has been disproven. The counterexample had a determinant of $-2$. See for instance : https://www.newscientist.com/article/2580374-ais-solution-to-87-year-old-riddle-takes-mathematicians-by-surprise/ or Wikipedia. Now I wonder if this polynomial can lead to an...

Let $E/K$ be a finite field extension, $G=\operatorname{Aut}_K(E)$ be the group of automorphisms fixing elements of $K$, and $E^G$ be the fixed field of $E$ under $G$. We know $E/E^G/K$ and so $$[E:K]=[E:E^G][E^G:K]=|G|\,[E^G:K].$$ Additionally, $[E:K]=[E:K]_s[E:K]_i,$ where $[E:K]_s$ denote the...

A field extension $F/K$ is Galois if $F^{\operatorname{Aut}_K(F)}=K$ in Hungerford's Algebra, and it is Galois if it is normal and separable in Lang's Algebra. For the finite extension, these two definitions are the same. For infinite algebraic extensions or for transcendental extensions, which...

I notice that the recently-publicized counterexample(s) to the Jacobian Conjecture are not surjective. Is that necessarily the case? That is, (Q) If $F:\mathbb C^n \to\mathbb C^n$ is algebraic and everywhere locally injective, and also surjective, must it be injective? Maybe the relevant setting...

Let $X,Y$ be two algebraic surfaces (=smooth, proper $2$-dim schemes over fixed base field $k$) and let $E \subset X$ exceptional curve, ie $E \cong \Bbb P^1$ with self intersection $E^2=-1$. By Castelnuovo's contraction theorem $E$ can be contracted to a smooth point of a smooth surface leaving...

Let $F/K$ be a finite field extension, $G=\text{Aut}_K(F)$ be the group of automorphisms of $F$ that fix elements of $K$, and $F^G$ be the fixed field of $G$. We then have $$[F:F^G]\le |G|.$$ This is proven in Hungerford Chapter V, Lemma 2.9. Hungerford used this lemma to prove "$F^G=K$ iff...

While creating quantitative aptitude problems for management entrance exam preparation, I noticed that the Euclidean algorithm is almost always preferred over prime factorization for computing the greatest common divisor.

I am in the middle of a problem which needs showing that the following system of congruences has finite number of solutions. I have verified up to some extent through sage that this has only two solutions for $(q, r)$ (with $q<r$) namely $(11, 17)$ and $(23, 103)$. The system of congruences is...