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bio website math.rutgers.edu/~ttyrrell
location New Jersey
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visits member for 1 year, 6 months
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Mar
22
comment Zeta Function of a Curve
You may find these lecture notes interesting: math.lsa.umich.edu/~mmustata/lecture3.pdf. Have you read Silverman's book on elliptic curves? It contains a section on zeta functions.
Mar
14
comment $a^4 = x^2 + y^2 (6+8b)$ solutions $(x,y)$?
The lines that pass through $(a^2,0)$ are of the form $x = my + a^2$, where $m$ is a rational number. Substituting for $x$ into the main equation, we can solve for $y$ in terms of $m$, and likewise for $x$. For what values of $m$ are the resulting values of $x,y$ integral?
Aug
26
comment Kähler differentials of a hyperelliptic curve.
The ds and dZ play a role in the restriction/transition maps.
Aug
26
comment Kähler differentials of a hyperelliptic curve.
With the restriction maps you can explicitly take a section on U' and a germ at $(0,0)$, and produce a family of sections on neighborhoods of $(0,0)$. The map from $\Omega^1_{V'/V}$ to the stalk at $(0,0)$ sends a polynomial to its constant term. So, we can extend germs (constants) to open nbhds as the constant terms of polynomials in $s$ with the higher order terms coming from the section on $U'$ restricted to $U' \cap V'$.
Aug
26
comment Kähler differentials of a hyperelliptic curve.
If you work out the restriction maps it might make the isomorphism more clear. In the odd case, with the decomposition $X = U' \cup \{(s,z) = (0,0)\}$, we can give global sections by gluing sections from the open cover of U' plus a basis around $(0,0)$. A global section is then a relative differential on $U'$, and a collection of differentials on open nbhds of $(0,0)$ - i.e. a germ. Perhaps the "as" is misleading, but there should be a slick way to see the direct sum decomposition.
Aug
26
comment Kähler differentials of a hyperelliptic curve.
What is the restriction map from $\Omega^1_{V'/V}$ to $U' \cap V'$? When the degree of $P$ is odd $\Omega^1_{U'/U}$ and $\Omega^1_{V'/V}$ have different dimensions over k (d versus d+1), so when you glue $U'$ and $V'$ to construct a global relative differential, this is where that extra dimension comes from - the single point over $s=0$.
Aug
26
comment Prove $X^2+Y^2-1$ is irreducible using geometrical tools.
Indeed. I've read your answer and I will edit mine.
Aug
26
comment Prove $X^2+Y^2-1$ is irreducible using geometrical tools.
In terms of style, the author may simply want to give a very general proof (one that applies when $k$ is not the real numbers). We also don't need to consider the whole infinite set of solutions to finish the proof, so the author might just be trying to keep things as simple as possible. It is not as important here that our field is infinite as that its characteristic is not 2.
Aug
22
comment a diophantine equation from Stewart and Tall
Have you tried reduction modulo a prime?
Jul
23
comment Visualize a projective curve $X^3+Y^3=Z^3$ in $P_2(C)$ as a torus
I would first replace y with y+1 and do some algebra to get the equation into Weierstrass form. Then, you can describe the torus with two patches.
Jun
17
comment Gluing schemes Hartshorne example
$\phi$ is the isomorphism we use to glue the schemes $X_1$ and $X_2$ together. The equality you are unable to interpret is the statement that the sections of our glued scheme X on an open set V are pairs $(s_1, s_2)$ which are equal when restricted to $U_1$ and $U_2$, respectively; we use $\phi$ to identify $U_1$ with $U_2$ in $X$.
Feb
14
comment A Function Meromorphic with Poles at the Primes
Can you construct a function with poles at the integers?
Jan
30
comment Proving the condition for two elliptic curves given in Weierstrass form to be isomorphic
There is a chapter devoted to it in Silverman's book.
Jan
29
comment Proving the condition for two elliptic curves given in Weierstrass form to be isomorphic
If you don't like this Riemann-Roch, function field business, have you read the main theorems concerning elliptic curves over the complex numbers? You can prove the same statement - all elliptic curves have a Weierstrass equation - using tools from analysis (eg. comparing coefficients of Laurent expansions).
Jan
16
comment Properties of the Mandelbrot set
Why is this tagged as algebraic geometry?
Jan
5
comment Question about the definition of weekly modular of weight k
A lot is known about non-holomorphic modular forms (search for Maass Forms). Note that there is no trouble with the definition; if f has a pole at $\tau$, then $\gamma(\tau)$ must also be a pole as the factor $(cx+ d)^k$ is certainly holomorphic at $\tau$.
Dec
19
comment Computing the degree of an isogeny
One way to show separability is by looking at the associated extension of function fields. What is the function field of $E/\langle P \rangle$ (considered as a subfield of $K(E)$ via $\phi$) and how is it constructed?
Dec
18
comment Integer values of difference between cube and sqaure
This problem can be thought of in terms of integral points on elliptic curves. This article by Keith Conrad covers some examples (in particular k=4).
Dec
18
comment Let $\phi$ be a Euclidean function, prove that if $a|b$ and $\phi (a) = \phi (b)$, then $a\sim b$
How does $\phi$ behave when applied to the product of 2 integers?
Dec
8
comment Self intersection of a curve?
Another reference would be Qing Liu's "Algebraic Geometry and Arithmetic Curves".