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6

This may only narrowly qualify for the requirements of the question, but they do teach quadratic reciprocity at Ross and PROMYS, so it can be pre-university mathematics. (of course, the factorization over the Gaussian integers could be considered elementary as well—the construction of a Euclidean function on $\mathbb{Z}[i]$ is very much elementary, and the ...


5

Not sure about the second question, but here is my take on the first. Let $p = (p_1, p_2)$. Suppose $p_2^2 \neq p_1^3$, then $p$ lies on $y^2 = x^3 + (p_2^2 - p_1^3)$; note, here $\Delta = -432(p_2^2 - p_1^3)^2 \neq 0$, so $y^2 = x^3 + (p_2^2 - p_1^3)$ is non-singular. If $p_2^2 = p_1^3$, then $p$ lies on $y^2 = x^3 + p_1x - p_1^2$, but $\Delta = ...


4

The short Weierstrass form for an elliptic curve $E$ over a field $K$ of characteristic not $2$ or $3$ is given by $y^2=x^3+ax+b$, such that the discriminant $\Delta=-16(4a^3+27b^2)$ is nonzero, see http://en.wikipedia.org/wiki/Elliptic_curve. However, the general definition of an elliptic curve is that $E$ is a smooth curve of degree $3$ over $K$, which ...


3

I think that taking the quotient amounts to the following. I think about this at the level of function fields. So let $k$ be the field of constants. Then the function field of your curve is $F=k(x,y)$, which is a quadratic extension of the field of rational functions $K=k(x)$, where $y^2=(x^3-1)(x^3-a)$. The symmetry that you talk about can be viewed as an ...


2

Aha, this isn't what I thought at first: I thought they'd be talking about half the elliptic curves $E$, but they're really talking about half the derivatives $L^{(k)}(E,1)$ as $k$ varies, for a fixed $E$. (1.3.3) reads $$ (-1)^k \Lambda^{(k)}(E,2-s) = \epsilon\Lambda^{(k)}(E,s). $$ (I've corrected a typo on the right-hand side). Here $\epsilon\in\{1,-1\}$ ...


1

The area is easy, just a generalization of the area of a circle, $A = \pi a b$ where $a$ and $b$ are the major and minor radii. The perimeter, as noted by others above is difficult and needs an infinite series. According to Wikipedia a good approximation is $P \approx \pi \left[3(a+b)-\sqrt{10ab+3(a^2+b^2)} \right]$. Hope this helps.


1

Assume an elliptic curve $E$ has a rational parametrization by a holomorphic map $$f: \mathbb P^1 \longrightarrow E.$$ The formula of Riemann-Hurwitz implies $$g(\mathbb P^1) = b/2 + (deg \ f)(g(E) -1) + 1$$ with genus $g(\mathbb P^1) = 0$, $g(E) = 1$ and $b \ge 0$ the total branching order of $f$, a contradiction, q.e.d


1

Silverman's book is advancing at a relatively quick pace early on if you have zero experience with local parameters of curves and projective spaces in general. The material prior to this is to some extent only a quick review. Spend more time with chapter I and its exercises. A general useful fact about all algebraic plane curves is: Fact. Assume that the ...



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