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 | |
stats | profile views | 21,772 |
[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 |