The arithmetic-geometry tag has no wiki summary.

**3**

votes

**2**answers

507 views

### Faltings height of a CM abelian variety

Let A be a CM abelian variety, say simple of dimension g, with $End(A) = O_K$, where $K$ is a CM field
of degree $2g$.
Is there an upper bound for the Faltings height $h(A)$ in terms of the ...

**3**

votes

**1**answer

141 views

### How do ideal sheaves behave on the special fibers of the projective line over the integers?

Let $X=\mathbb{P}^1_{\mathbb{Z}}$ and $Y\subset X$ be a local complete intersection of codimension two with Ideal sheaf $I_Y$.
(I'm mostly interested in the case where $Y$ is a single point $x$ ...

**2**

votes

**0**answers

90 views

### Vanishing theorems that work in positive characteristic

Let $X$ be a smooth projective variety over a field of characteristic $p>0$ of dimension at least $2$. I am looking for some examples when $H^2(\mathcal{O}_X)$ vanishes. Is there any standard way ...

**1**

vote

**0**answers

124 views

### algebraicity of Néron-Tate canonical height for Abelian varieties over global function fields

(transcendence of canonical heights)
Is the Néron-Tate canonical height for an Abelian variety $A$ over a global function field $K$, $\hat{h}: A(K) \times A^\vee(K) \to \mathbf{R}$ known to always ...

**3**

votes

**2**answers

274 views

### isogeny and congruence subgroup

Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$.
By ...

**3**

votes

**1**answer

183 views

### Etale covers of products of curves

Is a finite etale cover of a product of curves again a product of curves?
The answer is no in general. Here's one way to construct an example. Take the product $A$ of two elliptic curves and an ...

**4**

votes

**1**answer

219 views

### Is there a formula for a hyperelliptic curve over QQ, such that its Jacobian contains a rational torsion point of extact order n, for any given n?

Given a positive integer $n$, is there an algorithm (or even better a closed formula) that provides me with a hyperelliptic curve $C/\mathbb Q$, such that its Jacobian $J:=Jac(C)$ possesses a ...

**10**

votes

**1**answer

369 views

### Pro-algebraic versus continuous Galois cohomology, and schematic homotopy types

I've been thinking about Bertrand Toen's approach to studying the homotopy theory of schemes, and I've come across an inconsistency in my understanding of the subject that I was hoping somebody might ...

**7**

votes

**3**answers

905 views

### How many integer points does my favorite ellipse go through?

The equation of the ellipse interpolating the six lattice points $(0,0)$, $(1,0)$, $(0,1)$, $(d-1,d)$, $(d,d)$, $(d,d-1)$ in the plane for a fixed $d$ (at least 3) is
$$
x^2+y^2 - ...

**3**

votes

**2**answers

137 views

### Lattice-point-free buffers around circles

Let $C(r)$ be the origin-centered circle of radius $r$,
and let $\beta(r)$ be the exterior buffer around $C(r)$:
the distance from $C(r)$ to the closest lattice point exterior to $C(r)$:
...

**4**

votes

**1**answer

134 views

### Comparison of cycle maps

Let $X$ be an algebraic variety over $\bar{\mathbb{Q}}$ of dimension $d$, then there is the l-adic cycle map $\mathrm{cl}_{et}:\mathrm{CH}^i(X)\rightarrow\mathrm{H}^{2i}(X,\mathbb{Q}_\ell(i))$ from ...

**7**

votes

**0**answers

171 views

### Lattice radial-step (ratchet) spirals

(30Oct13: Now solved; see Addendum.)
Define a curve, a ratchet spiral, $S(r_0,\epsilon)$ as follows, where $r_0 > 0$ and $\epsilon < 1$.
$S(r_0,\epsilon)$ begins with the arc ...

**6**

votes

**0**answers

185 views

### Kernels and cokernels for morphisms of abelian schemes up to isogenies

For $S$ a noetherian scheme, let $\mathcal{A}(S)$ be the additive category of abelian schemes over $S$ and $\mathcal{A}_{\mathbb{Q}}(S)$ be the category of abelian schemes up to isogenies, i.e. ...

**5**

votes

**1**answer

300 views

### How many combinations of intersections of n hyperplanes there are in which a hyperplane is not crossed more than k times?

Consider you have $n$ hyperplanes with dimension $k$ that are not coplanar. Each hyperplane intersects the others $n-1$ times. Any intersection of $k$ such hyperplanes produces a vector. The number of ...

**10**

votes

**0**answers

195 views

### Purity for abelian schemes up to $p$-isogenies

Let $S$ be a noetherian excellent regular scheme and $U\subset S$ an everywhere dense open of codimension $\geq 2$. For some fibered categories of geometric objects, it makes sense to ask whether the ...

**14**

votes

**2**answers

1k views

### Deligne Weil II

Deligne's Weil I has been published under the title "La conjecture de Weil: I" in 1974, and Weil II in 1980. So did Deligne know in 1974 that there would be a Weil II, and can one explain the period ...

**3**

votes

**1**answer

136 views

### Section on $\mathcal{X}(1)$ induced by a cusp of $X(1)$

I'm reading the article Generalized arithmetic intersection numbers of U. Kuehn. At the beginning of section 4.12 "Modular forms over $\mathbb{Z}$" we are in the following situation.
Let $\Gamma(1) ...

**0**

votes

**0**answers

647 views

### “Descending cohomology, geometrically” by Mazur:

(Exist texts of that talk or related texts: http://ttv.mit.edu/collections/harris60/videos/13881-problem-session-barry-mazur ?) Article: http://www.math.harvard.edu/~mazur/papers/page37.pdf

**2**

votes

**0**answers

233 views

### “Extended” Weil Cohomology Theories

According to Wikipedia, a Weil cohomology theory is a functor from the category of smooth projective varieties over a field $k$, to graded algebras over a field $K$ of characteristic zero, together ...

**9**

votes

**0**answers

199 views

### Relation between the arithmetic Frobenius and the Frobenius of the $\varphi$-module of an unramified representation

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$ with perfect residue field $k$. Suppose $V$ is an unramified representation with associated continuous homomorphism ...

**9**

votes

**1**answer

1k views

### Why A. Weil considered elimination theory to be eliminated?

It is well known that André Weil declared, in the 1940's, that elimination theory must be eliminated from algebraic geometry. I would like to understand his mathematical reasons to adopt such an ...

**1**

vote

**0**answers

106 views

### Rational solutions of equations of the form $y^2 x = f(x)$

Let $k$ be any number field, and suppose we want to study the $k$-rational points on
$$y^2 x = f(x),$$ where $f$ is a polynomial of degree greater or equal than 3. In other words, $y^2 x = f(x)$ is a ...

**4**

votes

**1**answer

344 views

### Cubic forms and Hasse Principle

It's well-known that quadratic forms over the rational numbers $\mathbf{Q}$ satisfy the Hasse-Minkowski theorem. This is to say that they are isotropic over $\mathbf{Q}$ if and only if they are ...

**4**

votes

**0**answers

138 views

### Shimura varieties and Maximal conditions

Working with Shimura varieties, I have been convinced to call them (or the families giving rise to them especially in $A_{g}$) somehow the "maximal" families. The motivation of this, has been for ...

**8**

votes

**0**answers

197 views

### congruences of level 1 and level p modular forms

I've been carrying out some experiments on the computer and I noticed the following congruence phenomenon: fixing a prime $p$, it seems that any modular form over $SL_2(\mathbb{Z})$ and of weight $k ...

**2**

votes

**1**answer

170 views

### Are Isom-schemes geometrically connected

This question is about properties of Isom-schemes that are well-known over algebraically closed fields.
Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically ...

**13**

votes

**1**answer

466 views

### How fast can we numerically calculate Kloosterman sums?

Define the usual Kloosterman sum by $$S(m,n;c) = \sum_{\substack{x \pmod{c} \\ (x,c) = 1}} e\Big(\frac{mx + n\overline{x}}{c}\Big),$$
where $x \overline{x} \equiv 1 \pmod{c}$, and $e(x) = e^{2 \pi i ...

**7**

votes

**1**answer

357 views

### Do all varieties have only finitely many etale covers of fixed degree

I've been wondering about the following "finiteness statement" concerning etale covers for a while.
Let $K$ be a field of characteristic zero, not necessarily algebraically closed. A variety over $K$ ...

**0**

votes

**0**answers

117 views

### Existence of a curve with no points over finite separable field extensions

Does there exist a field $K$, and a smooth projective geometrically connected curve $C$ over $K$ such that, for all finite separable field extensions $L/K$ the curve $C$ has no $L$-rational points?
I ...

**1**

vote

**0**answers

90 views

### topological invariance of direct image in the \'etale topology

Let $R$ be a complete local ring (even of dimension one if it helps) and write it as limit of artinian rings $R_n$. Let $X\rightarrow S=Spec(R)$ be proper, finite type even smooth outside the maximal ...

**10**

votes

**0**answers

438 views

### Why does $H^i(X_{ét},\mathbb{Q}_p)$ have a Hodge-Tate structure?

Let $X$ be a variety over a $p$-adic field $K$.
Is there a simple or intuitive explanation of why the $G_K$ representation $H^i(X_{ét},\mathbb{Q}_p)$ is Hodge-Tate? More precisely, why do the powers ...

**9**

votes

**2**answers

330 views

### Rational points on circular spirals

Is it the case that every unit-radius circular spiral,
$$x = \cos(t)$$
$$y = \sin(t)$$
$$z = c \cdot t$$
for $c \in \mathbb{R}^+$
is dense in rational-coordinate points
(i.e., all three coordinates ...

**2**

votes

**1**answer

202 views

### Purely additive reduction of Jacobian of Hyperelliptic curve

For general, let X be an abelian variety of dimension g.
We say that X has 'purely additive reduction' at prime p if the dimension of the unipotent radical of the special fiber of the Neron Model of ...

**3**

votes

**2**answers

190 views

### Cohen-Macaulay and descent

Let $X$ be a Cohen-Macaulay scheme and $f:X\rightarrow Y$ a morphism. Under which "non trivial conditions" on $f$ can we conclude that $Y$ is also Cohen-Macaulay? By "trivial conditions" I mean ...

**1**

vote

**1**answer

159 views

### Normality and descent

Let $R$ be a normal noetherian domain. Write it as intersection of discrete valued domains $\bigcap_p R_p$. Assume I have schemes $X_p\rightarrow \operatorname{Spec}(R_p)$. Via the inclusion ...

**1**

vote

**1**answer

272 views

### curves with good reduction everywhere

It seems to be a folklore that for any genus $g$, there is a number field $K$ and a curve $X$ over $K$, such that $X$ has good reduction at all the places of $K$. Are any simple proofs of this?

**3**

votes

**0**answers

130 views

### Effective Lang-Weil bounds for del Pezzo surfaces

Let $X$ be variety in $\mathbb{P}^N$ over $\mathbb{F}_q$ of dimension $n$ and degree $d$.
By the Lang-Weil bounds
$ |\# X(\mathbb{F}_q) - q^n| \le (d-1)(d-2)q^{n-1/2} + Cq^{n-1}$for a constant $C$ ...

**8**

votes

**1**answer

241 views

### Shimura surfaces that do not contain a Shimura curve

Let $S$ be a Shimura surface i.e. a Shimura variety with $dimS=2$. Does $S$ necessarily contain a Shimura curve? I know that probably the answer is No, but do not have an explicit example. What is the ...

**4**

votes

**1**answer

292 views

### Weil pairing, fixed field of a $p$-adic Galois representation

Let $A$ be an abelian variety over a $p$-adic field $K$. If $K(A_{p^\infty})$ is the field extension of $K$ obtained by adjoining the coordinates of all $p$-power division points of $A$. By the Weil ...

**19**

votes

**2**answers

1k views

### Are most curves over Q pointless?

Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points. Don Zagier suggests the paraphrase : Most ...

**8**

votes

**0**answers

400 views

### Torelli-like theorem for K3 surfaces on terms of its étale cohomology

Is there a proof of a Torelli-like Theorem for a K3-surface over any field (non complex) in terms of its etale or crystalline cohomology?
For example: If $K\ne \mathbb{C} $ and $X\rightarrow ...

**4**

votes

**2**answers

254 views

### Restricting the composition factors of subgroups of GL_m(Z/nZ)

For a positive integer $m$, let $\mathcal{A}(m)$ be the set of all integers $k \geq 5$ such that: there is a positive integer $n$ and a subgroup $G \subset \operatorname{GL}_m(\mathbb{Z}/n\mathbb{Z})$ ...

**3**

votes

**0**answers

140 views

### Notion of good supersingular reduction for proper smooth variety over a $p$-adic field

Let $X$ be a proper smooth variety over a $p$-adic field $K$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $k$, its residue field. We say that $X$ has good ordinary reduction if there is a ...

**4**

votes

**0**answers

155 views

### Semiabelian actions appearing in the toroidal campactification of a degenearting abelian varieties

Given a totally degenerated abelian variety $A_K$ (to make it easier) over a complete discrete valuation field $K$ with $R$, $\pi$ and $k$ the corresponding discrete valuation ring, uniformiser and ...

**1**

vote

**1**answer

215 views

### Unexplained techniques in Demjanenko's “Sums of 4 Cubes”:

A "not well understood" proof, do you know if one knows by now the conceptual background?: http://www.math.u-bordeaux1.fr/~cohen/sum4cub.ps

**1**

vote

**0**answers

225 views

### Katz's paper on Serre Tate local moduli

In katz's paper "Serre-Tate local moduli" chaper 3 has the following construction:
Let $A$ be a fixed ordinary elliptic curve defined over $k$ of char $p>0$. Consider the deformation of $A$ to ...

**1**

vote

**0**answers

157 views

### Compactifications of group schemes

Let $G$ be a group scheme over a scheme $S$ which is the spectrum of a discrete valuation ring. Let $\eta$ (resp. $s$) be the generic (resp. closed) point. Assume that the generic fiber $G_{\eta}$ is ...

**3**

votes

**1**answer

120 views

### Elliptic curves over global function fields and independence of l-adic representations

Serre has shown that the family of $\ell$-adic Galois representations of an elliptic curve defined over a number field $K$ is almost independent. More explicitly:
let $E/K$ be an elliptic curve and ...

**6**

votes

**1**answer

428 views

### Serre-Tate 1964 Woods Hole notes

I am not sure if this is the right venue to ask this. Apologies in advance.
I would like to clarify the following. When people give as reference:
J.-P. SERRE and J. TATE.-Mimeographed notes from ...

**4**

votes

**1**answer

224 views

### Relation between Lee and Yang' s “circle theorem”, zeta functions and Weil conjectures?

Ruelle mentions ( http://www.ihes.fr/~ruelle/PUBLICATIONS/%5B94%5D.pdf ) Lee and Yang' s "circle theorem", which comes from statistical mechanics and shall have not yet explored connections with zeta ...