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

Apr
17
revised Advice for number theory library
Changed to separate by commas, instead of each name on a separate line
Apr
16
comment How to prove that every polynomial in an infinite family is irreducible over Q?
I assume that there is a typo in (3) andyou mean that it is "reducible" for at most a finite number of $y$, not "irreducible" for at most a finite number of $y$.
Apr
12
comment Elliptic curves with square conductor
@Conder You're right, I misremembered. I was probably thinking of the exponent of the wild part, but even for that, the right numbers are 6 and 3, not 5 and 3.
Apr
12
answered Elliptic curves with square conductor
Apr
11
awarded  Nice Question
Apr
11
revised Do most degree $d$ morphisms of $P^n$ have smooth critical locus?
Fixed the exponents on the example function $f$
Apr
11
comment Do most degree $d$ morphisms of $P^n$ have smooth critical locus?
@KevinVentullo Yes, thanks, I'll fix it. (The exponents aren't quite right.)
Apr
11
asked Do most degree $d$ morphisms of $P^n$ have smooth critical locus?
Apr
10
comment Integer roots of a polynomial
With the edits that I just made, I think that this is a legitimate MO question, so please don't close it unless you feel that this version is not MO worthy.
Apr
10
revised Integer roots of a polynomial
Fixed exposition so that it is understandable (using the OP's added remarks)
Apr
10
comment Integer roots of a polynomial
You ask for integer roots of "power series", but what you write are polynomials, not power series. I'm confused about what you're asking. Is it the following: Do there exist integers $j\ge1$ and $p\ge1$ and real numbers $a_1,\ldots,a_p$ such that $1-\sum a_iX^i$ has all its roots outside the unit circle and the equation $\sum a_iX^i=1-X^j$ has a root that's an integer?
Apr
10
awarded  Informed
Apr
6
revised Divisible torsion Z-modules
Changed math to LaTeX
Apr
6
comment How to find all integer points on an elliptic curve?
@KevinBuzzard Hi Kevin, Sorry to nit-pick (and on such an old post), but the "priveleged rational point" $O$ is definitely not an integral point. In essence, we are taking a model for $E$ over $\mathbb{Z}$, and when we say a point $Q$ is integral, we mean integral with respect to $O$, which by definition means that for every prime $p$, the points $\tilde Q \bmod p$ and $\tilde O\bmod p$ are distinct. Clearly $O$ itself fails to have that property in quite spectacular fashion!
Apr
5
comment A question about the Sylvester determinant
There's a nice exposition in van der Waerden's classic Algebra. It probably appears in standard algebra texts such as Dumit and Foote, although I don't have a copy handy.
Apr
4
comment Can i use Shamir's secret sharing scheme for multiplicative homomorphism for secure multiparty computation?
This question belongs on another site in the Stack Exchange network, namely the cryptography site, but that's not an option in the "off-topic" list.
Mar
31
comment Kernel of a 3-isogeny between two elliptic curves
@Andrew The Weil pairing for a cyclic $N$-isogeny gives a Galois equivariant perfect pairing between ker$(\phi)$ and ker$(\hat\phi)$ with values in $\mu_N$.
Mar
31
comment Kernel of a 3-isogeny between two elliptic curves
There are probably explicit formulas for 3-isogenies written down somewhere. Or you can use the method in Velu's paper, which works for cyclic $N$-isogenies (although of course it becomes infeasible in practice if $N$ gets too large). Or you can probably work it out yourself with a bit of work; I did the (admittedly easier) two isogeny case in my Arithmetic of Elliptic Curves book.
Mar
30
comment Kernel of a 3-isogeny between two elliptic curves
To expand on Chris's comment, the Weil pairing shows that $\operatorname{ker}(\phi)$ and $\operatorname{ker}(\hat\phi)$ are dual as Galois modules, so $\operatorname{ker}(\phi)=\mu_3$ if and only if $\operatorname{ker}(\hat\phi)=\mathbb{Z}/3\mathbb{Z}$, and the latter is equivalent to the points in the kernel of $\hat\phi$ being in $\hat E(\mathbb{Q})$.
Mar
30
answered Kernel of a 3-isogeny between two elliptic curves