Comment by R.P. on Is the set of points on an abelian surface which project...
You should understand the set of $k$-rational points on $X$ as the disjoint union of the set of $k$-rational points on all quadratic twists of $J$ (where $k$ is any field extension of the base field)....
View ArticleComment by R.P. on Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the...
+1. I think any question that gets a follow-up question by a 100k rep user deserves more upvotes.
View ArticleComment by R.P. on Most elementary proof showing that exponential growth wins...
@MaartenHavinga I don't think that would work so easily. The desired result is the existence of a real number $n_0$ such that for all $n>n_0$ we have $2^n>n^k$. But each value $x$ from your...
View ArticleComment by R.P. on What are some ways to stay engaged with the mathematical...
@MichaelHardy Well, maybe I was being too negative. I guess my sense that mathematicians can't really be said to form a community really says more about me than about them. But since this is exactly...
View ArticleComment by R.P. on Is there a regular pentagon with a rational point on each...
@Gro-Tsen That seems to me to be pretty trivial. You can start with any three points, say $(0,0)$, $(0,1)$, and $(1,0)$, and draw three lines through them in such a way that they are (i) all at 60...
View ArticleComment by R.P. on Is there an elliptic curve analogue to the 4-term exact...
$X=E(K)$, $Y=\textrm{Sha}(E/K)$. Doesn't get more natural than that, seems to me. :)
View ArticleComment by R.P. on Is this an instance of the snake lemma?
This kind of inquiry seems to beg the question: can every diagram chase lemma be formally derived from the "canonical" ones (snake, five, nine, etc.)? I don't even know if this question can be made...
View ArticleAnswer by R.P. for Two queries on triangles, the sides of which have rational...
If for the equation derived by Chris Wuthrich, we introduce the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, we obtain the more symmetric relation $$pqr(p+q+r) = A^2.$$Now I knew I recognized...
View ArticleAnswer by R.P. for Possible $p$-torsion subgroup of $E(\mathbb{Q}_p)$, and if...
I was interested in this question myself a while back, particularly for the additive reduction case. I wrote up a little note about my results here. The main result was a nice looking numerical...
View ArticleAnswer by R.P. for Diophantine representation of the set of prime numbers of...
Call your polynomial $P$. I propose the following polynomial:$$P' = (\xi^2+1)(1 - (\xi^2+1-P)^2)$$Proof (that the positive values of $P'$ are exactly the primes of the form $N^2+1$): Let $P_0$ be one...
View ArticleAnswer by R.P. for Does the expression $x^4 +y^4$ take on all values in...
emtom has found the right reference, but there is a more explicit result in that book (Ireland and Rosen, A Classical Introduction to Modern Number Theory). In fact, Theorem 5 of Chapter 8 (on page...
View ArticleAnswer by R.P. for Is there a name for sum of increases of f(x) on ranges...
Not as far as I know, but one can write down a formula (supposing that $f$ is differentiable, with $f'$ Riemann-integrable):$$\int_a^b f'(x) H(f'(x)) dx $$where $H$ is the Heaviside step function.
View ArticleAnswer by R.P. for Results that are widely accepted but no proof has appeared
I think one example is given in this MO question of mine: a quartic in $\mathbb{P}^3$ with at worst Du Val singularities is a K3 surface (and similar statements for two types of complete intersections...
View ArticleSingular models of K3 surfaces
Let us work over a ground field of characteristic zero. As is well-known, a K3 surface is a smooth projective geometrically integral surface $X$ whose canonical class $\omega_X$ is trivial and for...
View ArticleAnswer by R.P. for Singular models of K3 surfaces
For what it's worth, I wrote up a proof (pretty detailed) following the hints in Francesco Polizzi's answer. It's in an unpublished preprint found here (p. 38 onwards). I am not a geometer, so the...
View ArticleAnswer by R.P. for Finding $q(x)$ such that $p(q(x))$ is reducible over...
Note: in this answer, I have inadvertently disregarded your requirement for $q$ to have integral coefficients. I do however prove that a $q$ with rational coefficients does exist, so I will just let...
View ArticleAnswer by R.P. for What is the state-of-the-art for solving polynomials...
For the reals, I particularly like the book by Sturmfels mentioned by Alexandre Eremenko. For the rational numbers, you can hardly do better than Bjorn Poonen's book Rational Points on Varieties, which...
View ArticleAnswer by R.P. for Conic sections are to cones as quadric surfaces are to what?
The thing that makes quadric surfaces "3D analogs of conic sections" is just that they are defined by a single equation of degree 2. It's not a particularly helpful characterization though, I would...
View ArticleAnswer by R.P. for Generalization of Weak Nullstellensatz?
Or see Proposition 2.4.6 in Bjorn Poonen's book Rational Points on Varieties (link). This is almost exactly the result you conjectured, just a bit more general:Let $X$ be a $k$-variety. Then the map...
View ArticleAnswer by R.P. for Reference request: Diophantine equations
This may be a good choice for someone who (like yourself) is already superficially acquainted with some of the definitions and methods of Diophantine geometry:Marc Hindry, Joseph H. Silverman --...
View ArticleThe exponent of Ш of $y^2 = x^3 + px$, where $p$ is a Fermat prime
For $d$ a non-zero integer, let $E_d$ be the elliptic curve$$E_d : y^2 = x^3+dx.$$When we let $d$ be $p = 2^{2^k}+1$, for $k \in \{1,2,3,4\}$, sage tells us that, conditionally on BSD,$$\# Ш(E_p) =...
View ArticleAnswer by R.P. for Diophantine approximation on spheres
dodd is right. Every point over $\mathbb{Z}[1/2]$ must have coordinates in $\frac{1}{2} \mathbb{Z}$, since by clearing denominators we get four squares of integers, not all even, summing to a power of...
View ArticleAnswer by R.P. for Diophantine approximation on spheres
Here is a proof that $S(\mathbb{Z}[\frac{1}{p}])$ lies dense in $S^3$ for all primes $p \equiv 1 \pmod{4}$.Since this is a wholly algebraic/arithmetical question, it is easier to switch to...
View ArticleAnswer by R.P. for A road map through group cohomology
Chapter 2 of these notes by Milne have been helpful to me.
View ArticleBrauer-Manin obstruction on an open subset of an elliptic curve
First a disclaimer. This is an old question that I considered years ago and that I recently remembered. Since I am no longer in active research it may be considered as 'idle curiosity', although I feel...
View ArticleAnswer by R.P. for Set of primes $p$ such that $\mathrm{Hom}(A,...
Here's a sketch of an answer. I think the answer is that you can get three types of sets: (i) finite sets, (ii) co-finite sets, and (iii) sets of the form$$S_f = \{ p : f(x) ~ \textrm{has a root in...
View ArticleAnswer by R.P. for Is pure mathematics useful outside of mathematics itself?
I think you're right to have these doubts, that is to say, I have had similar misgivings about the "usefulness" of my work in pure mathematics, and not only that, but the "usefulness" of pure...
View ArticleAnswer by R.P. for How to show an invariant subfield of rational function...
The answer as to the surjectivity of $\alpha$ is no. As in algebraic number theory, the simplest way to prove that an element is not a norm is by local considerations. Let us...
View ArticleAnswer by R.P. for Square root in number field
This answer is meant to answer only your second question.Claim. Let $K=\mathbb{Q}(\sqrt[3]{2})$ and $\alpha = \sqrt[3]{2}-\sqrt[3]{4} \in K$. Then there does not exist $\beta \in K$ such that...
View ArticleAnswer by R.P. for Rational solutions to $P(x,y)=0$ for $P$ reducible over...
This also follows from Prop. 2.3.26(i) in Bjorn Poonen's Rational Points on Varieties, where it is stated that if for a finite type $k$-scheme $X$ the set of rational points $X(k)$ is Zariski dense,...
View ArticleAnswer by R.P. for What to do after a pure math academic path?
I sympathize with you, because I have been in a similar situation. I was in mathematics because it fascinated me, although sometimes more than others. The reason I left mathematics research was not so...
View ArticleAnswer by R.P. for Why do we make such big deal about the 'unsolvability' of...
I like this question because I agree with its sentiment. Let me give an additional reason why the insolubility of the quintic is an overrated result in my opinion. I believe that we shouldn't even be...
View ArticleAnswer by R.P. for Irreducibility of polynomials over some number fields
Here is a different approach, which is arguably a bit more elementary. If $f=X^n-p$ splits in $K$, and $g$ is one of its factors, then the constant term of $g$, being a product of zeros of $f$, must be...
View ArticleAnswer by R.P. for How can we solve the following number theory problem?
This is a variant of IMO 1981/3 and can be solved by the exact same method.
View ArticleAnswer by R.P. for Most elementary proof showing that exponential growth wins...
I would suggest considering instead the stronger statement that $n^k/2^n \to 0$. (This would of course necessitate an explanation of what it means for a function to converge to zero as the argument...
View ArticleComment by R.P. on Daunting papers/books and how to finally read them
It is not always obvious when it's time to move on. For me the emphasis should be on "go slower" rather than "keep moving". But sometimes you should just forge ahead, even when there are still some...
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