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### Vanishing of motivic cohomology with finite coefficients in negative degrees

I wonder whether the "finite coefficient version" of Beilinson-Soule conjecture i.e. the following statement holds or not.
STATEMENT:
Let $X$ be a smooth and projective scheme over a finite field ...

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### Special values of zeta functions and extensions of base fields.

Let $X$ be a scheme of finite type over a finite field $k=\mathbb{F}_{q}$ of $q$ elements.
Then, one can define the zeta function $Z_{X/k}(T)$ of $X$ ovet $k$ as $\prod_{x\in ...

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### Classification of quasi-split unitary groups

Let $U$ be a unitary group defined with respect to an extension $E/F$ of non-archimedean local fields, and assume it is realised with respect to a pair $(V,q)$, where $V$ is an $n$-dimensional vector ...

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### How do you compute the primes of bad reduction?

Suppose that I am given a subscheme $Y$ of $\mathbf{P}^n_{\mathbf{Z}}$, flat over $\operatorname{Spec}\mathbf{Z}$ and with smooth generic fiber $Y_{\mathbf{Q}}$, defined by the vanishing of some ...

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### Torsion of elliptic curves is finite

Let $S$ be an integral 1-dimensional scheme with function field $K$.
Let $E$ be an elliptic curve over $K$. The torsion of $E$ over $K$ is not necessarily finite. As an example consider an elliptic ...

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### SAT and Arithmetic Geometry

This is an agglomeration of several questions, linked by a single observation: SAT is equivalent to determining the existence of roots for a system of polynomial equations over $\mathbb{F}_2$ (note ...

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### $\sigma$-conjugate iff $p$-adically close

First some notations. Let $p$ be a prime, $k$ a perfect field of characteristic $p$, $W=W(k)$ the ring of Witt vectors over $k$, $\sigma : W \rightarrow W$ the Frobenius, $R$ a commutative ...

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### Equation for simple Jacobian of a genus two curve

Let $X$ be a curve of genus two over a field $k$ with a $k$-rational point. Let $J$ be the Jacobian of $X$.
Can we write down an explicit equation for the abelian surface $J$?
I know $X$ can be ...

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### Detecting sections on an arithmetic variety

Let $S$ be Spec $O_K$ with $O_K$ the ring of integers of a number field $K$.
Let $X\to S $ be an arithmetic variety, i.e., an integral smooth quasi-projective $S$-scheme with generic fibre $X_\eta$ ...

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### Properties of subvarieties of a simple abelian variety

Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.)
Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension.
Suppose ...

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### An example of almost etale extension

In the paper of Faltings' "p-adic Hodge theory", Faltings showed an example of almost etale extension before he proved the almost purity theorem. The example is following:
Let $k$ be a perfect field ...

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### global section of vector bundle and reduction

Let $k$ be an algebraically closed field of char $p\neq 0$, $W_2(k)$ the witt vector of length 2. $C_1$ a smooth projective curve over $W_2(k)$, and $H_1$ a vector bundle over $C_1$. We denote $C_0$ ...

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### Representability of sheaf of Ext^1 of a Néron model by $\mathbb{G}_m$

Let's work over a trait $S=\mathrm{Spec}R$, where $R$ is a dvr with fraction field $K$, residue field $k$. Given an abelian variety $A_K$ with semi-stable reduction, let $A$ over $S$ be its Néron ...

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### Learning path for the proof of the Weil Conjectures

Assume you are an algebraic geometry advanced student who has mastered Hartshorne's book supplemented on the arithmetic side by the introduction of Lorenzini - "An Invitation to Arithmetic Geometry" ...

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### Are ranks of Jacobians over number fields unbounded?

Fix a number field $K$.
Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$?
Does there exist an integer $g$ such ...

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### Does the moduli space of genus three curves contain a complete genus two curve

Inspired by the question
Does the moduli space of smooth curves of genus g contain an elliptic curve
and its amazing answers, I ask (pure out of curiosity) whether the moduli space $M_3$ of (smooth ...

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### Grothendieck's section conjecture and base change: restricting sections

Let $X$ be a smooth projective geometrically connected curve over $\mathbf{Q}$ of genus at least two. Fix an algebraic closure $\overline{\mathbf{Q}}$ of $\mathbf{Q}$ and let $G_{\mathbf{Q}}$ be the ...

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### Varieties with infinitely many etale covers and rational points

Let $X$ be a (smooth projective geometrically connected) variety over a field $k$.
Consider the set Et$(X,k)$ of finite etale covers $Y\to X$ over $k$, with $Y$ geometrically connected over $k$.
...

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### “Good reduction” for singular varieties

A projective nonsingular variety $X$ over a number field $K$ has the notion of good reduction at places $p$ of $K$. Informally, $X$ has good reduction modulo $p$ if $X$ remains nonsingular when ...

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### Do all curves have Néron models

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$.
Does there exist a Néron model $\mathcal X$ for $X$ over $O_K$?
By a Néron model, I mean ...

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### Can one ignore primes lying over $l$ in the Fontaine-Mazur conjecture? Counterexamples?

The Fontaine-Mazur conjecture predicts that an $l$-adic Galois representation of a number field is 'geometric' if it is unramified outside a finite set of primes and is De Rham for primes lying over ...

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### Dieudonné modules over rings of charateristic zero

Dear Colleagues,
would appreciate if you could recommend references, if such a theory exits, for the following question.
Let $A$ be an Abelian scheme over $\text{Spec}(R)$, where $R$ is a subring of ...

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### Does a curve over a number field have a finite etale cover of given degree

Let $X$ be a (smooth projective geometrically connected) curve over a number field $K$ of genus $g\geq 2$. Let $d\geq 2$ be an integer.
Does there exist a curve $Y$ over $K$ with a finite etale ...

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### What is the reduction of this hyperelliptic curve

Let $K$ be a number field and $E/K$ an elliptic curve with equation $Y^2Z = X^3 +AXZ^2+BZ^3$ in $\mathbf{P}^2_K$, where $A,B\in K$.
Let $S$ be non-empty finite set of finite places of $K$ and suppose ...

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### Defining isogenies over smaller fields

I'm having some issues with abelian varieties and fields of definition. This already became clear in my previous question on Jacobians. Here's another question. If somebody can explain some nice facts ...

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### Jacobians defined over smaller fields

Let $L/K$ be an extension of number fields.
Let $X$ be a curve over $L$ which can not be defined over $K$. Let $J(X)$ be the Jacobian of $X$ over $L$.
In general, the Jacobian $J(X)$ probably ...

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### Modularity of higher dimensional abelian varieties

In another question I asked about strategies for giving an effective version of the Shafarevich conjecture for abelian varieties over $\mathbb{Q}$.
For elliptic curves, one can give a proof using ...

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### The significance of modularity for all Galois representations

On pg. 1 of the slides of a talk, Henri Darmon wrote:
Question: What is an interesting Diophantine equation?
A “working definition”. A Diophantine equation is interesting
if it reveals or ...

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### Is the geometry of a variety determined by the counts of rational points?

In Diophantine Geometry: An introduction, Hindry and Silverman write "Geometry Determines Arithmetic" (pg. 2) and "Geometry Governs Arithmetic" (pg. 474).
On pg. 211 of the same book, the authors ...

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### Is There a Mayer-Vietoris Spectral Sequence of Motivic Cohomology for Closed Coverings?

For etale cohomology, there is a spectral sequence of the following form ("Mayer-Vietories spectral sequence for closed covers"):
$E_{1}^{p,q}=\oplus_{i_{0}< \cdots < i_{p}} H_{ Y_{i_{0} \cdots ...

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### Over which fields does the Mordell-Weil theorem hold?

According to a well-known theorem of Mordell, the group of rational points $E(\mathbf{Q})$ of an elliptic curve $E/\mathbf{Q}$ is finitely generated. Weil generalized this theorem to abelian varieties ...

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### Fields over which cubic hypersurfaces are rational

All cubic hypersurfaces having at least one double point are birational to some $P^n$ over an algebraically closed field. How does the statement change as I pass to non alg closed fields? Does it hold ...

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### is Hasse principle a birational invariant?

...it is probably a very trivial question, but I am a beginner in arithmetics.

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### Honda-Tate in families

Let $k$ be a finite field, say with $q=p^a$ elements. Honda-Tate theory states that there is a bijection between isogeny classes of simple abelian varieties over $k$ and ...

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### Heuristic for the Fermat-Catalan conjecture

[Edit: I've since realized that my question is confused: in particular, the minimum value of k that you need to sum from increases with the largest exponent under consideration so that the sum over ...

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### Counting higher dimensional abelian varieties of a given conductor

This question is a follow up to an earlier question of mine on enumerating elliptic curves of a given conductor.
I've heard people say that studying higher dimensional varieties via explicit ...

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### $K$-groups and dual graphs of special fibers

Let $p$ be a prime number, let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $\mathcal{E}_p$ be the special fiber of the Néron model of $E$ over $\mathbb{Z}_p$ and let ...

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### Isogeny classes of elliptic curves

Let $E \subset \mathbb{P}_\mathbb{C}^2$ be an elliptic curve. If $E$ has complex multiplication (by anything) then the theory of complex multiplication in particular tells us that if $\sigma \in ...

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### Finiteness of elliptic curves of a given conductor

It follows from the modularity theorem for elliptic curves over $\mathbb{Q}$ that there are finitely many elliptic curves of a given conductor $N$. Moreover, one can algorithmically enumerate them. ...

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### local galois representation with higher coefficient

Suppose K is a local field , G is its galois group, V a fine dimensional Vector space over F, which is a sub field of K, and totally ramified over $Q_p$. Consdider the linear action of G on V (V is ...

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### Algebraicity of power series over the rationals from the algebraicity over Fp

Van der Poorten conjectures [in "Power series representing algebraic functions," Sem. Th. Nombres Paris 1990-91] that if a power series over the rationals is the [complete] diagonal [of a rational ...

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### Special value of $L$-function

Let $p$ be a prime number. Let $f$ be a newform of weight 2 on $Γ_0(p)$, and $E_f$ denote the associated newform quotient of $J_0 (N)$ over $\mathbb{Q}$. Is there a way to express the
algebraic part ...

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### Examples of finiteness of rational points for hypersurfaces in $\mathbb P^3_{\mathbb Q}$ of degree $>4$.

Given an homogeneous polynomial $F(X,Y,Z,T)\in \mathbb Q[X,Y,Z,T]$ of degree $>4$, the surface it defines is well-known to be of general type. Suppose, moreover, that this surface doesn't contain ...

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### Is there a semisimple $\mathbf{Q}_\ell$-representation of $G_F$ ramified at an infinite set of places?

See http://math.uni.lu/~wiese/galois/Boeckle-Luxemburg-Notes.pdf, Theorem 1.4(a): Is there an example of a semisimple $\mathbf{Q}_\ell$-representation $V$ of $G_F$ ($F$ a global field) ramified at a ...

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### Question about witt vector of some ring

Suppose $R=Z_p[t]$ , and $\hat{R}$ its p-adic completion, suppose we have Endormorphism $\Phi$ of $\hat{R}$, whose redution mop p is just the absolute Frobenius of $\hat{R}/p\hat{R}$. And ...

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### Does semi-stable reduction behave well with Weil restriction of scalars

Let $A$ be an abelian variety over a number field $K$ with semi-stable reduction over $O_K$.
Does the Weil restriction $\textrm{Res}_{K/\mathbf{Q}}A$ of $A$ to $\mathbf{Q}$ have semi-stable reduction ...

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### Mordell-Weil group of the universal abelian scheme

Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$
...

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### Does the self-product of a $g$-dimensional abelian variety contain an abelian variety of dimension smaller than $g$ at some point

Let me be more precise than the title. (This will be my last attempt to do something with abelian varieties. Sorry for all the basic questions. The answers have been great!)
Let $A$ be a simple ...

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### Is any simple abelian variety covered by a non-simple abelian variety

Let $A/k$ be a simple abelian variety.
Does there exist a non-simple abelian variety $B/k$ and a finite homomorphism $f:B\to A$ over $k$?
I don't need $f:B\to A$ to be etale.

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### Are abelian varieties degree two covers of some projective space

Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$.
There exists a finite morphism $A\to \mathbf{P}^g_k$. Here's the question.
Does there exist a finite morphism $A\to ...