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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 15 hours ago
Professor of Mathematics, Brown University. Primary interests: number theory, arithmetic geometry, elliptic curves, arithmetic dynamics, cryptography.

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awarded  Fanatic
2d
awarded  Critic
Dec
21
answered Can the pre-image of the real points in the complex upper-half plane of a modular elliptic curve under the modular parametrization be identified?
Dec
19
answered Simple Isogeny Question
Dec
17
comment Graduate program applications that require questionnaires and other non-letter 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
comment Graduate program applications that require questionnaires and other non-letter 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
comment 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 $h(E)=\max\bigl\{1,\log|j(E)|\bigr\}$. 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
comment Adjoining torsion points from abelian varieties
The following paper might help: MR1361754 Masser, D. W.(CH-BASL); Wüstholz, G.(CH-ETHZ) 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
comment 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
comment Well-known or prolific mathematicians that have never written a sole-author article?
@IgorRivin We must be reading different letters. I've generally found that co-author letters from good mathematicians are quite candid and specific as to what contributions each of the co-authors made to the paper. This isn't universally true, of course, but in my experience, it's true more often than not.
Dec
1
comment computing height on elliptic curve of the form $y^2=x^3-nx$
@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 2-torsion 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 $y^2=x^3-nx$
Nov
22
awarded  Custodian
Nov
22
reviewed Leave Open Is elliptic curve point division defined over the field of real numbers?
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.)