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bio website math.brown.edu/~jhs
location Brown University Mathematics Department
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visits member for 3 years, 11 months
seen 11 mins ago
Professor of Mathematics, Brown University. Primary interests: number theory, arithmetic geometry, elliptic curves, arithmetic dynamics, cryptography.

Nov
22
awarded  Custodian
Nov
22
reviewed Leave Open Is elliptic curve point division defined over the field of real numbers?
Nov
22
reviewed Reviewed Problem regarding heat equation partial differential equation
Nov
20
comment Is elliptic curve point division defined over the field of real numbers?
@RichApodaca The process is very feasible, in the following sense. I'll describe one case. Often $E(\mathbb{R})$ is isomorphic as a group to $\mathbb{R}^*/q^{\mathbb Z}$ for a real $q$ that's easy to compute to lots of decimal places. Further, the isomorphism is given by an explicit, rapidly converging series. So we have an isomorphism $f:\mathbb{R}^*/q^{\mathbb Z}\to E(\mathbb{R})$. You're taking a point $P\in E(\mathbb{R})$ and asking to solve $f(nt)=P$. So it's really a problem of numerical analysis: how easily can one solve such equations for an explicit power series $f(t)$?
Nov
20
comment degree of polynomials in nullstellensatz
To expand on Felipe's comment, the link is to the article: JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume I, Number 4, October 1988: SHARP EFFECTIVE NULLSTELLENSATZ by JANOS KOLLAR
Nov
20
answered Is elliptic curve point division defined over the field of real numbers?
Nov
14
comment Questions about the “universal elliptic curve” over the affine $j$-line punctured at 0 and 1728
You presumably threw out $i$ and $\rho$ because those curves have extra automorphisms. But every elliptic curve has a non-trivial automorphism, namely $P\to-P$, so there's a problem at every point. Roughly speaking, if you "loop around infinity," your points become negated, which is why the construction doesn't work. (This comment is meant only to help with the intuition. What Tom Church wrote is the formal mathematical formulation.)
Nov
13
comment Disjoint images of polynomials
You beat me to it. I was about to post "A silly observation is you can take $f(x)=1$ and $g(x)=2$. So presumably you want $f$ and $g$ non-constant," and then follow that with the observation that the OP is looking at points on curves defined over $\mathbb{Q}^{\text{ab}}$, for which it shouldn't be hard to produce examples with no points (as you just did).
Nov
12
comment Two rings…are they isomorphic?
You might want to say what $t$ is. (You mentioned it in your comment to Bjorn's answer, but it really needs to be in the statement of your question. I originally assumed it was a new variable.)
Oct
25
awarded  Enlightened
Oct
25
awarded  Nice Answer
Oct
24
answered $j$-invariants of elliptic curves over finite fields
Oct
20
comment Who first defined quantum integers?
In the first displayed equation, presumably $m$ is supposed to be $n$.
Oct
14
revised The linear projection of projective spaces
Fixed TeX and punctuation
Oct
10
comment Why do roots of polynomials tend to have absolute value close to 1?
@IanMorris Indeed, the roots will be uniformly distributed on the unit circle, as per the result of Erdos and Turan that I described in my answer (so sorry, but this fact was not unremarked). But as you say, the heuristic of Kostlan gives a very nice, albeit rough, explanation of why it should be true. The Erdos-Turan theorem, which is not at all easy prove, gives a precise quantitative formulation.
Oct
5
awarded  Good Answer
Oct
3
awarded  Nice Answer
Oct
3
answered Why do roots of polynomials tend to have absolute value close to 1?
Oct
1
comment What is the most useful non-existing object of your field?
@ToddTrimble In fairness, it should be noted that Hellegouarch defined these curves associated to Fermat solutions and used them to prove some interesting things relating Fermat solutions to rational torsion points on elliptic curves. I believe that this was before Frey started looking at them. Not to take anything away from Frey, who made the key observation that these curves provide a link between Fermat and the modularity conjecture. Anyway, although it makes for a longer name, these non-existent curves might better called Frey-Hellegouarch curves.
Sep
30
awarded  Explainer