Tagged Questions

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Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and let $E(\mathbb{Q}_p) \supset E^0(\mathbb{Q}_p) \supset E^1(\mathbb{Q}_p) \supset \cdots$ be its $p$-adic filtration, where $E^n(\mathbb{Q}_p) = \{P ... 0answers 19 views inflexion point on elliptic curve Let$C: y^2 = x^3 + ax^2 + bx + c$be an elliptic curve over$\mathbb{Q}$($a,b,c \in \mathbb{Z}$) and let$\overline{C}$be its reduction over$\mathbb{F}_p$. Then does$\overline{C}$necessarily ... 1answer 48 views $p$-adic numbers and projective coordinates Let$E/\mathbb{Q}_p$be an elliptic curve and let$E^0(\mathbb{Q}_p)$denote its nonsingular points. We accept that$E^0(\mathbb{Q}_p)$is a subgroup of$E(\mathbb{Q}_p)$. Then let ... 1answer 49 views elliptic curves over$\mathbb{Q}_p$Let$E: Y^2 = X^3 + AX + B$be an elliptic curve over$\mathbb{Q}_p$, i.e.$A,B \in \mathbb{Q}_p$and$4A^3 + 27B^2 \neq 0$. Then, according to page 47 of Cassels' Lectures on Elliptic Curves, if ... 0answers 50 views How can we compute the order of 1-form on Riemann surfaces Let X be a hyperellictic curve defined by$y^2=h(x)$. Let$\pi:X\rightarrow\mathbb{P}^1$be the double covering map seding$(x,y)$to$x$. Let$\omega=\pi^*(dx/h(x))$. How can we compute the orders of ... 0answers 52 views Finding the prime element of a place of a function field of a elliptic curve. Let$p$be an elliptic curve in$\mathbb{C}[X,Y] $. Consider the quotient ring$A = \mathbb{C}[X,Y]/(p) $and its field of fractions$F = frac(A) $. For all$f + (p) \in A$, define$deg_A(f + (p))= ...
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Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of ...
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Isomorphic Elliptic Curves

I want to solve the following exercise: Show that the two elliptic curves $E/ \mathbb{Q}$ and $E'/ \mathbb{Q}$ are isomorphic. $E: y^2 = x^3+x-2$ and $E': y'^2 = x'^3-\frac{1}{3}x' - \frac{52}{27}$. ...
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Could a hyper elliptic curve of degree 5 admit only three ordinary double points?

The definition for hyper elliptic curves is those curves which admit a ramified double cover of $P^1$. The given degree five is for some homogeneous equations of degree five that defines the curve, ...
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What are some applications of the Weil conjectures for algebraic curves?

I have been interested in the Weil conjectures for some time, and the easiest place to start has been in studying them for elliptic curves. I've been able to see some of their applications and ...
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Is $y^2 =$ quartic in $x$ smooth at infinity?

Let $q(x)\in K(x)$ be a quartic polynomial in x with distinct roots over the algebraically closed field $K$. Consider the curve $C\subset \Bbb P^2$ given by $y^2-q(x)$. Is $C$ smooth? Well, at least ...
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Genus of Edwards curve

Let us work over a field $\Bbbk$ of characteristic not equal to two. Let $d\in\Bbbk\setminus\{0,1\}$. It is said in the wikipedia article about Edwards curves that the plane quartic defined by the ...
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Dimension of family of hyperelliptic curves

Suppose we have an elliptic curve E with a point $P$ of order $5$ over a field of characteristic $0$. Denote $E'$ the curve $E/\langle P\rangle$. Now let $x$ (resp. $x'$) be a function on $E$ (resp. ...