bio  website  math.brown.edu/~jhs 

location  Brown University Mathematics Department  
age  
visits  member for  3 years, 11 months 
seen  15 hours ago  
stats  profile views  4,981 
Professor of Mathematics, Brown University.
Primary interests: number theory, arithmetic geometry, elliptic curves, arithmetic dynamics, cryptography.
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awarded  Fanatic 
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awarded  Critic 
Dec 21 
answered  Can the preimage of the real points in the complex upperhalf plane of a modular elliptic curve under the modular parametrization be identified? 
Dec 19 
answered  Simple Isogeny Question 
Dec 17 
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Graduate program applications that require questionnaires and other nonletter material
Ah, but in a year or two when you have to use UM's system again, will they force you to remember your password (or force you to request a new one be sent to your email address)! 
Dec 17 
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Graduate program applications that require questionnaires and other nonletter material
@KConrad Are you referring to questions such as "what is the candidates potential to be a leader" and vague stuff like that, or do you also mean questions such as "rank in class" and check boxes with "top 1%, top 5%, ..."? Both types are annoying, and generally useless except maybe for initial screening at schools that get many hundreds of applications. OTOH, my experience is that almost all math grad programs pose such questions to letter writers, so you may get a more useful (and shorter) list if you ask for those schools that don't! 
Dec 16 
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Weil height of an Abelian Variety with everywhere (potentially) good reduction
If $j(E)=0$, your formula $h(E)=log j(E)$ is problematic! Indeed, ditto if $j(E)<0$. So you really do need to write it as $h(E)=\max\bigl\{1,\logj(E)\bigr\}$. And also, although you must realize this, your height ignores twisting, so $y^2=x^3+1$ and $y^2=x^3+1234567890$ have the same height, although arithmetically they may be quite different. 
Dec 5 
answered  Good lecture notes/books on Jacobian of hyperelliptic curve 
Dec 4 
answered  Why there are two point at infinity on certain elliptic curve 
Dec 4 
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Adjoining torsion points from abelian varieties
The following paper might help: MR1361754 Masser, D. W.(CHBASL); Wüstholz, G.(CHETHZ) Factorization estimates for abelian varieties. Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 5–24. They give effective estimates for the degree of the isogeny and fields of definition for the factorization of $A$ into simple factors. So one might be able to use that to prove that almost all of the Jacobians of $y^2=PQ$, counted by height, are simple. It wouldn't follow directly, but might be the right tool to apply. (Just a thought.) 
Dec 3 
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Adjoining torsion points from abelian varieties
Excellent. Now, to answer the further question, can you modify this so that the associated Jacobians are geometrically simple? Most of the ones you construct will be, but not all, I think. Maybe use $y^2=P(x)Q(x)$ for a generic $Q(x)$ so that $\deg(PQ)$ is odd? 
Dec 3 
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Wellknown or prolific mathematicians that have never written a soleauthor article?
@IgorRivin We must be reading different letters. I've generally found that coauthor letters from good mathematicians are quite candid and specific as to what contributions each of the coauthors made to the paper. This isn't universally true, of course, but in my experience, it's true more often than not. 
Dec 1 
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computing height on elliptic curve of the form $y^2=x^3nx$
@somayehdidari Up to possible small adjustments(and assuming that your $\theta$ is the function that I think it is), the function $\theta(2P)/\theta(P)^4$ is an elliptic function that vanishes at the 2torsion points and has a triple pole at 0, so it is a multiple of $\wp'(z)$, the derivative of the Weierstrass $\wp$function. 
Nov 30 
answered  computing height on elliptic curve of the form $y^2=x^3nx$ 
Nov 22 
awarded  Custodian 
Nov 22 
reviewed  Leave Open Is elliptic curve point division defined over the field of real numbers? 
Nov 20 
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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 
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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 
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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 nontrivial automorphism, namely $P\toP$, 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.) 