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Consider the elliptic surface $E$ with affine equation $$y^2 = x(x-1)(x-t^2)$$ over the base $\mathbf{P}^1$ with parameter $t$ (with complex scalar field). Then $E$ has four points of bad reductio …

This question is about the formal smoothness property for schemes. A morphism $X\to S$ is formally smooth if for every affine $S$-scheme $Y$ and every subscheme $Y_0\subset Y$ cut out by a nilpotent …

The two groups you want to compare are canonically isomorphic, so long as C is connected. See Example 11.3 of Milne's notes: http://www.jmilne.org/math/CourseNotes/lec.html

Since you asked about software, I'd just like to point out (if you don't know already) that SAGE (available at sagemath.org) can compute division polynomials easily. The commands R.<A,B> = Polynomia …

Let's say $\mathcal{O}$ is a complete DVR with fraction field $K$ and algebraically closed residue field $k$. (The case I had in mind here was with $\mathcal{O}$ of equicharacteristic $p$, so assume …

In number theory, base change can also refer to an operation on automorphic representations. If L/K is an extension of number fields, and pi is an automorphic representation of a reductive group G ov …

Let $R$ be a complete discrete valuation ring and let $K$ be its field of fractions. Suppose $X$ is a smooth rigid-anaytic space over $K$. Often it is convenient to have a model of $X$ whose reducti …

Let $F$ be a nonarchimedean local field, and let $D/F$ be the central division algebra of invariant $1/d$. Let $k$ be the algebraic closure of the residue field of $F$ and let $\pi$ be a uniformizer. …

The covers you seek are exactly the elliptic curves which admit cyclic $n$-isogenies into $E$ (or out of $E$, it doesn't matter). Proof: if $X\to E$ is unramified, then $X$ is a genus one curve (by …

Let $\mathbf{F}_q$ be a finite field of odd characteristic. Let $X_t$ be the hyperelliptic curve over $\mathbf{F}_{q^2}(t)$ with affine equation $$y^2 = \left((x^{(q+1)/2}-(x-1)^{(q+1)/2})^2 - t\righ …

I have typeset Deligne's letter, and placed the result here: http://www.math.ias.edu/~jaredw/DeligneLetterToPiatetskiShapiro.pdf I have made some minor edits so that the text reads more naturally to …

Ok, it's time for me to ask my first question on MO. Consider the affine curve $Y+Y^q=X^{q+1}$ over the finite field $\mathbf{F}_q$. It's interesting because it has the largest number of points over …

This question is a sequel to an earlier question, which asked about the zeta function of a certain affine variety over a finite field $k$. The unusual thing about this variety is that it had the maxi …

This question is about Tate's 1963 paper "Algebraic Cycles and Poles of Zeta Functions". Here he announces a conjecture (now known as "the Tate conjecture") which states that certain classes in the c …