bio | website | math.brown.edu/~jhs |
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location | Brown University Mathematics Department | |
age | ||
visits | member for | 3 years, 8 months |
seen | 14 hours ago | |
stats | profile views | 4,619 |
Professor of Mathematics, Brown University.
Primary interests: number theory, arithmetic geometry, elliptic curves, arithmetic dynamics, cryptography.
Sep 11 |
awarded | Enlightened |
Sep 11 |
awarded | Nice Answer |
Sep 11 |
revised |
How did height in algeb. number theory/elliptic curves started?
added 1153 characters in body |
Sep 11 |
answered | How did height in algeb. number theory/elliptic curves started? |
Sep 8 |
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Separation of lattice points on the Mordell elliptic curve
Ryan, why don't you send me an email so that we can discuss this offline. You can find my email address on the webpage that's linked on my MathOverflow home page. -- JS |
Sep 6 |
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Why are $S$-arithmetic groups interesting?
Nice answer. You might want to change the \R in the last paragraph to \mathbb{R}. (It is annoying that MO doesn't recognize our favorite self-defined macros!) |
Sep 6 |
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Multiplicity of a variety along a subvariety
Even if $X$ is a complete intersection, wouldn't you need to consider something like the order of vanishing of the $r$-by-$r$ minors of the Jacobian matrix of $F_1,\ldots,F_r$? |
Sep 2 |
answered | A question on degree 4 binary forms |
Aug 29 |
awarded | nt.number-theory |
Aug 28 |
answered | Ideal classes fixed by the Galois group |
Aug 27 |
comment |
Separation of lattice points on the Mordell elliptic curve
@RyanD'Mello Actually, if you assume that $(x,y)$ is a solution with $x$ reasonably large and $|x^3-y^2|\asymp\sqrt{|x|}$, and if you let $(x+\ell,y+k)$ be the next largest solution, then approximating $(x+\ell)^3\approx x^3+3\ell x^2$ and $(y+k)^2\approx y^2+2ky$ gives $k$ in terms of $\ell$ (more or less), and then looking at the next largest terms may well give a lower bound for $\ell$ in terms of $x$. I'll let you work out the details (and maybe this doesn't work at all). |
Aug 27 |
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Separation of lattice points on the Mordell elliptic curve
@RyanD'Mello Please don't misquote me. I didn't say that there is no simple answer using currently known (or even possibly quite elementary) methods. I simply said that at a quick glance, the results/methods in my paper don't seem to yield anything, and that offhand I don't recall seeing this problem addressed in the literature (which is a much weaker statement than saying that it hasn't been considered before!). |
Aug 26 |
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Separation of lattice points on the Mordell elliptic curve
@RyanD'Mello Offhand I don't know what lower bound on $x_2-x_1$ can be regarded as trivial and what would be significant. The best way to figure that out is to try to prove something, which I'll let you do, rather than my working more on the problem. |
Aug 26 |
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Composition of a transcendental function with a rational function
By assumption, $\mathbb{C}(y,f(y))$ is transcendental over $\mathbb{C}(y)$. Setting $y=\psi(x)$ is merely making an algebraic extension of each, so $\mathbb{C}(\psi(x),f(\psi(x)))$ is transcendental over $\mathbb{C}(\psi(x))$. Hence the tower $\mathbb{C}(\psi(x),f(\psi(x)))\supset \mathbb{C}(\psi(x))\supset \mathbb{C}(x)$ is transcendental, so $f(\psi(x))$ is transcendental over $\mathbb{C}(x)$. [This does seem a bit elementary for MO, so don't be surprised if it's closed.] |
Aug 25 |
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Asymptotics on prime divisors
I forgot to put this in my answer, so will leave it as a comment. You mention Zsigmondy's theorem. It is certainly a very interesting question to ask if all but finitely many $a(n)$ have a primitive prime divisor. I don't know the answer. But since your sequence is not a divisibility sequence, it's behavior may be somewhat different from the Zsigmondy-type results for classical and elliptic divisibility sequences. |
Aug 25 |
answered | Asymptotics on prime divisors |
Aug 25 |
revised |
Separation of lattice points on the Mordell elliptic curve
Changed a plus sign to a minus sign. Added some material. |
Aug 25 |
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Separation of lattice points on the Mordell elliptic curve
I'd forgotten about that Danilov article. Anyway, I thought about it for a little bit and don't see an immediate way to get a gap estimate using (local) canonical height estimates. Seems like an interesting problem. My article with an elliptic curve gap principle is: A quantitative version of Siegel's theorem: Integral points on elliptic curves and Catalan curves, J. Reine Angew. Math. 378 (1987), 60-100. For the curves $E_k:Y^2=X^3+k$ with $k$ 6'th power free, I prove that there is an absolute constant $C$ so that $|E_k(\mathbb{Z})|\le C^{1+rank~E_k(\mathbb{Q})}$`. |
Aug 24 |
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Separation of lattice points on the Mordell elliptic curve
@Lucia That would be a gap principle sort of statement. I'll have to think about it. I wrote a paper with a general gap principle for integral points on elliptic curves, but I'm not sure if it's relevant here. OTOH, if the specific question is about the $x\in\mathbb{N}$ such that there is a $y$ satisfying $0<|x^3-y^2|<\sqrt{x}$, it's not at all clear (at least to me) that the set of such $x$ is infinite. But maybe one could fix a small $\epsilon$ and take $x$ values admitting a solution to $0<|x^3-y^2|<x^{1/2+\epsilon}$, or use an upper bound of $x^{1/2}(\log x)^k$ for some fixed $k$. |
Aug 23 |
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Separation of lattice points on the Mordell elliptic curve
@KConrad Thanks for filling in $C$, I'd forgotten the exact value and didn't feel like rederiving, it. Actually, if you take $x$ and $y$ to be $S$-integers in a function field of genus $g$, the constant is something like $-(2g-2+\#S)$. As for the $x^3-y^2\ne0$ condition, I didn't think that it was necessary to repeat it, but I guess it doesn't hurt to put it in. |