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7 votes
0 answers
207 views

Unicritical rational functions on curves in characteristic $p$

Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$. How precisely can one describe the ...
4 votes
0 answers
413 views

On Stickelberger's Theorem over function fields

Here is the setup to Stickelberger's theorem over number fields (following Washington's book Intro. to cyclotomic fields). Let $M/\mathbb{Q}$ be a finite abelian extension with galois group $G$. ...
0 votes
1 answer
472 views

surjectivity of rational points induced by surjective map from affine space

Let $k$ be a local field of char $0$ (which is the case I concern). Let $V$ be a variety defined over $k$ and let $f: \mathbb A^n\to V$ be a surjective map (over the algebraic closure of $k$) ...
3 votes
2 answers
662 views

Moduli Space of Abelian Varieties with a N-torsion point

Does there exists (as scheme, or as some sort of stack) the moduli space of principally polarized Abelian Varieties together with a point of order $N$, for $N>1$ an integer? In the case of ...
3 votes
1 answer
667 views

What is an automorphic representation of CM type ?

In a recent paper of BL-Gee-Geraghty: "Sato-Tate for Hilbert modular forms" (JAMS 2011), a theorem is proved for regular algebrai cuspidal automorphic representation of $GL_2(\mathbb A_F)$ with $F$ a ...
14 votes
1 answer
835 views

A remark in Swinnerton-Dyer's paper in Cassels-Frohlich

In Swinnerton-Dyer's charming paper "An application of computing to classfield theory", in Cassels-Frohlich, he discusses the genesis of the Birch/Swinnerton-Dyer conjecture and numerical tests of it ...
13 votes
2 answers
1k views

Families of curves for which the Belyi degree can be easily bounded

I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above. The modular curves $X(n)$. They are ...
9 votes
2 answers
1k views

On Grothendieck's period relations

Let $V$ be a smooth projective variety defined over $\mathbf{Q}$ and denote by $$ \omega: H_{dR}^*(V,\mathbf{Q})){\otimes_{\mathbf{Q}}}\mathbf{C}\rightarrow H_{B}^*(V,\mathbf{Q})\otimes_{\mathbf{Q}}\...
4 votes
1 answer
918 views

Heisenberg group in characteristic two

I have seen two constructions called the Heisenberg group. If $k$ is a field of characteristic not equal to $2$ and $V$ is a $2n$-dimensional vector space over $k$ with symplectic form $\omega : V \...
4 votes
2 answers
1k views

Is the absolute Galois group of the field of Laurent series in positive characteristic finitely generated?

If $K$ is an algebraically closed field of characteristic $p>0$, then $K((t))$, the field of Laurent series with coefficients in $K$, has infinitely many Galois extensions of degree $p$. Indeed, ...
13 votes
2 answers
572 views

Existence of points on varieties which avoid a given number field.

Let C be a geometrically integral curve over a number field K and let K' be a number field containing K. Does there exist a number field L containing K such that $L \cap K' = K$, and $C(L) \neq \...
2 votes
1 answer
455 views

good references in moduli stack and stable reduction

Hello,everyone. I'm looking for some good references in moduli stack and stable reduction, so I ask here for some advice. I knew the famous paper of Deligne-Mumford, but this paper is hard for me ...
1 vote
0 answers
238 views

Classification of fibres in pencils of curves of genus two

For the case of characteristic positive, there exist some clasification of families of curves of genus two over a elliptic curve?.
2 votes
0 answers
622 views

The cohomology of the relative dualizing sheaf of a relative curve

Let $X\to S$ be a curve over $S=\mathrm{Spec} \ \mathbf{Z}$. Let $\omega$ be the relative dualizing sheaf of $X\to S$. Let $g$ be the genus of the generic fibre. Assume that $g\geq 2$. I know that $\...
1 vote
0 answers
187 views

stack quotient question

Hi, I have the following question: let $k$ a field with $char(k)= p>0$, which we can assume to be perfect, $W(k)$ the ring of Witt vector, and $a,b$ positive integers. Consider the ring $R=W(k)[x,...
4 votes
0 answers
269 views

Tate's theorem about abelian variteies in case of abelian scheme

For $k$ a finite field , $A,A'$ an abelian varieties over $k$, $G$ the Galois group of $k$, $l$ a prime number different from the characteristic of $k$ . Tate has proved that: $Q_l\otimes Hom_k(A,A')...
2 votes
1 answer
223 views

density of conjugate of arithmetic subgroup

$K=Q(\sqrt{d} ) , d<0 $, $\Gamma $ an arithmetic subgroup of $G=SU(2,1)(K)$ . Is $\cup_{g\in G}(g^{-1}\Gamma g)$ dense in G for the complex topology?
17 votes
1 answer
1k views

On the Hasse-Weil L-function of $P^n$

So let us start with the "simplest" scheme over $Spec(\mathbf{Z})$ namely $X_0=Spec(\mathbf{Z})$. Then the (reciprocal) Weil zeta function of $X_0$ at a prime $p$ is given by $Z_p(T)=1-T$ (a ...
23 votes
1 answer
2k views

Does smooth and proper over $\mathbb Z$ imply rational?

Does smooth and proper over $\mathbb Z$ imply rational? I think someone told me that this is a standard conjecture. Is it a widely held? held at all? Did someone in particular make this conjecture? ...
21 votes
3 answers
3k views

Stacks in modern number theory/arithmetic geometry

Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was ...
14 votes
2 answers
2k views

"Purely local" proof of local Langlands

As from this website http://math.uchicago.edu/~lxiao/workshop_site/ My question is: What does it mean by "purely local"? Also, I heard about this phrase "purely local" in other problems as well, ...
0 votes
1 answer
223 views

Smooth quadric over p-adic integers

Let $k$ be a $p$-adic field with ring of integers $\mathcal{O}_K$ and residue field $\mathbb{F}$. Say I have a (projective) quadric $Q$ which is smooth over $\mathcal{O}_K$, such that the reduction $\...
6 votes
1 answer
1k views

Albert classification of rational endomorphism rings of simple Abelian varieties over finite fields

Recall the Albert classification of rational endomorphism rings with involution of simple Abelian varieties over arbitrary fields: Type I: totally real, trivial involution Type II and III: quaternion ...
1 vote
1 answer
435 views

Elliptic subfields of a function field

Let $C$ be a curve and $K(C)$ be its function field of genus 2, where $K$ = $\mathbb{C}$. The number of essential elliptic subfields of $K(C)$ is 0 or 2 or $\infty$. Edit: I am looking for a proof. ...
3 votes
1 answer
586 views

Conductor of an elliptic curve

Given any elliptic curve over $\mathbb{Q}$ of conductor $N$, by modularity of elliptic curves, there exists a surjective morphism from $X_0(N)$ $\rightarrow$ $E$.There may be several such 'N' and ...
18 votes
1 answer
3k views

Field with one element example?

$$\frac{1}{\mu(B)}\int_B \vert x \vert d\mu(x) = \frac{1}{p+1}$$ This formula holds for the unit ball in $\mathbb{Q_p}$. This formula also holds for $\mathbb{R}$ when $p=1$. Should one expect $$\...
7 votes
1 answer
334 views

Rational points on smooth compactifications

Let $X$ be as smooth variety over a field $k$ of characteristic $0$. Consider the following statements: The variety $X$ has no $k((t))$-rational points. No smooth compactification of $X$ has a $k$-...
2 votes
0 answers
279 views

deRham cohomoloy of CM liftings of Jacobians

Let $k$ be field of characteristic $p>0$ and $W=W(k)$ the ring of Witt vectors of $k$. We call a smooth curve over $k$, ordinary, when the Jacobian of $J(X)$ of $X$ is an ordinary abelian variety. ...
8 votes
1 answer
2k views

Order of Ш (Sha)

To prove the BSD conjecture, one has to know about 'the finiteness of the Shafarevich Tate group'. But, an example of an elliptic curve of rank 2 (whose Sha group $Ш(E/\mathbb{Q})$ is finite) is not ...
2 votes
1 answer
1k views

What is Mordell-Weil lattice?

What is Mordell-Weil lattice?
3 votes
0 answers
742 views

Explicit description of O^{cris}_n in Fontaine/Messing

Let $k$ be a perfect field of characteristic $p$, $W(k)$ the Witt ring and $K$ its quotient field. In their article "$p$-adic periods and $p$-adic etale cohomology" Fontaine and Messing give in II.1.4 ...
2 votes
0 answers
131 views

why find a good field extension such that the curve has semi-stable model is important?

Hello everyone, I'd like to ask some question about semi-stable reduction of curves. The Deligne-Mumford theorem tell us "Let $A$ be an Dedekind domain, $K=K(A)$, for any smooth curve $X$ over $K$ ...
17 votes
2 answers
5k views

why we need rigid geometry?

Hello, everyone. I want to ask some questions about rigid geometry. 1.what is the motivation of rigid geometry? 2.what is the applications of rigid geometry for solving arithmetic problems, ...
6 votes
1 answer
2k views

understanding the main theorem of complex multiplication (of elliptic curves)

I want to understand the meaning of the main theorem of complex multiplication (of elliptic curves) as given in Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, II.8, or Shimura, ...
4 votes
1 answer
453 views

Why is the kernel of reduction a pro-p group?

Let $A$ be an abelian variety over a $p$-adic field $K_v$, i.e., $K_v$ is a finite field extension of $\mathbb Q_p$, for $p$ a prime number. Denote by $k_v$ the residue field of $K_v$ and let $\...
6 votes
4 answers
1k views

Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry

I hear that the nonstandard methods are applied to many problems in various fields of mathematics such as functional analysis, topology, probability theory and so on. So, I have become interested in ...
17 votes
2 answers
2k views

Geometric interpretation of Hida isomorphism

[EDIT]: After getting a very nice answer by Kevin Buzzard I realize that my question was a little bit too vague and I try to restate it more precisely. As the title says, I would like to understand ...
5 votes
0 answers
397 views

false elliptic curves and principal polarizations

Hi, Let $\Delta$ be a quaternion algebra over $\mathbf Q$ and let $\mathcal O_\Delta$ be a maximal order in $\Delta$. Recall that a false elliptic curve over a field $K$ is a pair $(A/K,i)$ ...
26 votes
1 answer
2k views

Why does the Section Conjecture exclude curves of genus 1?

Let $X$ be an integral proper normal curve over a (perfect) field $F$, of genus $\geq 2$. One variant of Grothendieck's "section conjecture" states that the sections $G_F \rightarrow \pi_1(X)$ of the ...
2 votes
1 answer
510 views

hyperalgebras (positive characteristic)

The question is about commutator in integral forms. Let $A$ an associative algebra over a field of characteristic zero, $x\in A$ and $k\in \mathbb Z$, we denote $x^{(k)}=\frac{x^{k}}{k!}$. How to ...
15 votes
1 answer
807 views

Has the Weil conjectures been proved using other (Weil) cohomology theory?

According to Weil, the Weil conjecture should follow once one has a sufficiently powerful cohomology machine. And it is proved using one of them, namely étale cohomology. My question is, has there ...
6 votes
2 answers
2k views

Q-factorial and rational singularities on surfaces

Let $X$ be a normal surface. Is any rational singularity $\mathbf{Q}$-factorial? I've seen this somewhere for surfaces over fields, but what about the general case of an integral 2-dimensional ...
14 votes
1 answer
2k views

A curve with bad reduction for which the jacobian has good reduction

Let $K$ be a number field. If $X$ is a curve over $K$ with good reduction at a place $v$ of $K$, then the Jacobian of $X$ also has good reduction at $v$. This follows from the functoriality of the ...
2 votes
0 answers
445 views

Are torsors over unipotent groups trivial

I might have misunderstood something I heard somewhere. Are torsors over unipotent groups trivial? I couldn't find this in some standard references.
4 votes
1 answer
1k views

Bound for the number of rational points on the modular curve

By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977), we know that the only rational points of X_0(N) for N any prime > 163 are the two cusps (o) and (oo) (|X_0(N)(Q)| = 2 )....
3 votes
1 answer
980 views

Construction of Kummer map for abelian variety

Let $A$ be an abelian variety over the rational numbers $\mathbf{Q}$. Let $V=T_p A \otimes \mathbf{Q}_p$ be the $\mathbf{Q}_p$-Tate module of $A$. Let $G$ be the absolute Galois group of $\mathbf{Q}$. ...
5 votes
1 answer
4k views

Trace of Frobenius over $F_q$

Let $q_0$ be a prime and $q$ = $q_0^n$. Let $a(F_q/F_{q_0})$ denote any integer which is trace of Frobenius over the field $F_q$ for some elliptic curve which can be defined over $F_{q_0}$. It is ...
1 vote
0 answers
457 views

Why do twists of an algebraic group over k correspond to k-torsors over G

Let $G$ be an algebraic group over a field $k$. Let $k^s$ be the separable closure of $k$. I can't seem to figure out why isomorphism classes of twists of $G$ correspond to $k$-torsors over $G$. It'...
5 votes
1 answer
819 views

Does the Mordell conjecture imply the Shafarevich conjecture

The base field is a number field. It is known that the Shafarevich conjecture implies the Mordell conjecture (Kodaira-Parshin). Is the converse also true? Note that both conjectures are now ...
4 votes
0 answers
409 views

Every curve is a Hurwitz space in infinitely many ways

Diaz, Donagi and Harbater proved that every curve over $\overline{\mathbf{Q}}$ is a Hurwitz space. A Hurwitz space is a connected component of the curve $H_n$. The curve $H_n$ is (the ...

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