15 votes

What is the idea behind the proof of the Isogeny theorem and Theorem III.7.9 (Serre) in Silverman's book?

Here is a rough idea of the proof of 1. I'll highlight (with italics) concepts that you may not know now, but will be useful to learn in number theory, which hopefully will be motivated by this ...
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15 votes

Polynomial values are powers of two

I'll prove a stronger statement. Let $S$ be a finite set of primes. I claim there is a $c_{n,S}$ such that a polynomial $f$ with rational coefficients cannot take only values that are $S$-units on $\{...
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13 votes

Polynomial values are powers of two

Yes, such $c_n$ is bounded by something effective. Below is a cubic bound, which probably may be improved. (Update: see $n^2\log n$ upper bound by Will in the comments.) Assume that $f(x)$ is a power ...
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11 votes
Accepted

If we have a nice formula for number of points on a curve over finite fields, can we get some geometric information of the curve from the formula?

This formula implies that the zeta function of $C$ is given by the formula $$\zeta_C( u) = e^{ \sum_{\alpha=1}^{\infty} | C(\mathbb F_{q^\alpha})| u^\alpha / \alpha } = e^{ \sum_{\alpha=1}^{\infty} (...
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8 votes

Weil height vs Moriwaki height

One can define a height with respect to a special metric so that some particular useful equation involving the height holds exactly, instead of approximately as for the Weil height. For example, one ...
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  • 115k
5 votes
Accepted

Rational points on a special class of surfaces

I am just posting my comments as an answer. Without a hypothesis that the geometric generic fiber of $\pi:S\to \mathbb{A}^1_t$ is irreducible, the result is false. For a smooth compactification of $...
3 votes
Accepted

On the Artin-Rees Lemma for non-commutative rings

There is some discussion of this in Rowen's "Ring Theory", volume I, Section 3.5, with additional references therein. Exercise 19 on p. 462 in op. cit. states that a polycentral ideal $I$ of ...
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2 votes

Why is the category of motives generated by varieties?

No, $\mathbb{Q}(X)$ is the presheaf "additively represented" by $X$; it is not constant. Now I will express my understanding of this matter; I did not check that it fits with Ayoub's ...
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1 vote

Singularities of arithmetic surface

There seems to be some confusion concerning "regularity" and "smoothness". First of all, if $X$ is a noetherian integral scheme and $x\in X$, then $X$ is regular at $x$ if $$\...
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