33,780 reputation
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bio website math.harvard.edu/~elkies
location Harvard University, Cambridge, MA 02138
age 48
visits member for 3 years, 7 months
seen 8 hours ago
[Not sure yet what to use this space for...]

Dec
15
awarded  Good Answer
Dec
13
comment Fourier transform of $sin(\frac{1}{x})$ for $x > 0 (x > 1)$
Since $\int_0^\infty \exp(-ax-b/x) \, dx$ is basically a Bessel function I'd expect that there's a formula for $\int_0^\infty \sin(1/x) \, e^{ixy} \, dx$ in terms of Bessel functions too. But it looks like this question will be closed before I can find this formula... Try Gradshteyn & Ryzhik.
Dec
12
awarded  Nice Answer
Dec
12
comment Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?
Indeed $y^2 = x^3 + x^2 - t$ has discriminant $t$ and $j$-invariant $1/t$ in characteristic $3$. Likewise $y^2 + xy = x^3 + t$ in characteristic $2$.
Dec
12
comment Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?
There might not be an entirely simple proof of this result. The nonconstant elliptic curves over ${\mathbb C}^*$ are quadratic twists $y^2 = x^3 + at x^2 + bt^2 x + ct^3$, cubic twists $y^2 = x^3 + at^2$ and $y^2 = x^3 + at^4$, quartic twists $y^2 = x^3 + at x$ and $y^2 = x^3 + at^3 x$, and sextic twists $y^2 = x^3 + at$ and $y^2 = x^3 + at^5$; any argument must account for all of these. The proof I gave is conceptual but advanced; the Szpiro path is elementary (Szpiro is basically Mason = polynomial ABC) but requires case analysis.
Dec
12
answered Are there nonisotrivial elliptic curves over $\mathbb{G}_m$?
Dec
11
comment How to prove that two univariate polynomials are always algebraically dependent?
M.Stoll's suggestion is computational. For example, in the univariate case the resultant is exactly the determinant you get when you first have as many equations as variables.
Dec
5
answered A question on how polynomials split over $\mathbb{F}_p$ for large primes $p$
Dec
5
comment Is the Jacobi theta function invertible?
How do you "have $\theta(z( P))$" without knowing $P$?
Dec
5
awarded  Enlightened
Dec
5
comment Euler's Triangular Number closure properties
Are we really to believe that Euler, of all people, didn't notice the obvious generalization? "Opera Postuma" = posthumous works; this could have been a note to himself where he saw the pattern and saw no need to write down any more.
Dec
4
awarded  Enlightened
Dec
4
revised Contest problems with connections to deeper mathematics.
1896, not 1986! Also, revert \mathbb to \mathbf (I'd used {\bf Z} which is equivalent)
Dec
4
awarded  Nice Answer
Dec
4
awarded  Nice Answer
Dec
4
awarded  Nice Answer
Dec
4
awarded  Necromancer
Dec
4
revised Hilbert's Theorem on $L_2$ norm of polynomials in $\mathbb{Z}[X]$ - Explicit construction and a converse?
add link to MO188807, give global minimum when $b-a \geq 4$, and fix a couple of typos
Dec
4
answered minimizing an integral over integer-coefficient polynomials $\displaystyle \inf_{f \in \mathbb{Z}[x]} \int_a^b f(x)^2 \, dx $
Dec
4
awarded  Revival