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4
votes
2answers
115 views

Bound for number of points on surface over $\mathbb{F}_p$

I know of the bound for the number of points on an elliptic curve over a finite field: $$|\# E(\mathbb{F}_q) - q - 1| < 2\sqrt{q}$$ where this includes the point at infinity. I have been told that ...
8
votes
2answers
372 views

How to Compute Genus

How to compute the genus of $ \{X^4+Y^4+Z^4=0\} \cap \{X^3+Y^3+(Z-tW)^3=0\} \subset \mathbb{P}^3$? We know that the genus of $ \{X^4+Y^4+Z^4=0\} \subset \mathbb{P}^3$ is 3 because the degree is 4. ...
0
votes
1answer
99 views

multiplicity of schemes in positive characteristic

let $K$ a field and $X$ a scheme of finite type over $K$. let $X_1,\dots,X_r$ the irreducible components of $X$ with generic points $\eta_i$. Let $\bar{X}=X\times Spec(\bar{K})$ where $\bar{K}$ is an ...
4
votes
0answers
112 views

On local rings of a normalization

Let X an irreducible singular curve with a singular point $x$. Consider $A$, the normalization of $\mathcal{O}_{X,x}$, and $x_1,\dots,x_r$ the points over $x$ in the normalization of $X$. Why $A\cong ...
1
vote
0answers
178 views

Normalization of a node

How do you describe the normalization of $\mathrm{Spec}(k[[x,y]]/(xy))$? In particular, call this $N$. Why can $N$ be written as the intersection of 2 local rings and the same holds for the maximal ...
3
votes
2answers
415 views

Why a smooth surjective morphism of schemes admits a section etale-locally?

Why a smooth surjective morphism of schemes admits a section etale-locally?
4
votes
0answers
92 views

find valuations

consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are ...
6
votes
1answer
197 views

places and primes

what does it means that a place divide a prime on an algebraic number field?
3
votes
0answers
112 views

places valuations and so on

I am looking for complete references on places, valuations and so on. In particular I need to understand the meaning of "a finite place does not divide a prime number $\ell$" and what is the Frobenius ...
2
votes
0answers
119 views

Primitive nth roots of unity and moduli of elliptic curves

In various descriptions of the moduli space of elliptic curves with level structures, such as the description of $X_0(N)$ being defined over $Spec\ \mathbb{Z}[1/N]$, the primitive $N^{th}$ roots of ...
15
votes
2answers
1k views

Theories of $p$-adic integration

What is the compelling need for introducing a theory of $p$-adic integration? Do the existing theories of $p$-adic integration use some kind of analogues of Lebesgue measures? That is, do we put a ...
11
votes
3answers
605 views

Importance of determining whether a number is squarefree, using geometry

Despite appearances, this is not a question on computational aspects of number theory. The background is as follows. I once asked a number theorist about what he considered to be the most important ...