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319 views

### Bounding failures of the integral Hodge and Tate conjectures

It is well know that the integral versions of the Hodge and Tate conjectures can fail. I once heard an off hand comment however that they should only fail by a "bounded amount". My question is what ...

**17**

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438 views

### function field analogy and global/absolute geometry

The "function field analogy" seems to be a topic that is considerably bigger than any one existing writeup conveys. There are several old question on MO and and MathSE that ask for details. One of the ...

**17**

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401 views

### On Determinants of Laplacians on Riemann Surfaces

History of the Formula: In their famous paper "On Determinants of Laplacians on Riemann Surfaces" (1986), D'Hoker and Phong computed the determinant of the Laplacian $\Delta_n^+$ on the space $T^n$ of ...

**16**

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559 views

### Smooth proper schemes over Z with points everywhere locally

This is a variation on Poonen's question, taking Buzzard's fabulous example into account. It was earlier a part of this other question.
Question. Is there a smooth proper scheme ...

**16**

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927 views

### Two conjectures by Gabber on Brauer and Picard groups

In a paper I need to make reference to 2 conjectures by Gabber
(see Conjectures 2 and 3, page 1975)
http://www.mfo.de/programme/schedule/2004/32/OWR_2004_37.pdf
1) Let $R$ be a strictly henselian ...

**12**

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239 views

### Artin L-function and Zeta function of twisted Dirac operator

If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation $\sigma$ as a flat connection on the curve, then the definition of the Artin ...

**10**

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396 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 ...

**10**

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369 views

### What makes the Cartier operator “tick”?

Let $C$ be a smooth curve over a finite field of characteristic $p$. Let $t$ be a local parameter at a point. If $f$ is a regular function on a neighbourhood of the point, one can write uniquely
$$f ...

**10**

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509 views

### Points of bounded height in a number field

Let $K$ be a number field of absolute degree $d$, let $B$ be a positive real number, and write $S(K, B) = \{x \in K : H(x) \leq B\}$. Here $H$ is the absolute multiplicative height of an algebraic ...

**10**

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804 views

### Effective proofs of Siegel's theorem using arithmetic geometry

This is a speculation and perhaps naive. The theorem of Siegel that
There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is ...

**9**

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144 views

### Totally real points on curves

Let $X$ be a smooth, projective (geometrically integral) curve defined over $\mathbb{Q}$ with genus $g \geq 3$. Suppose that $X(\mathbb{R}) \neq \emptyset$. Does $X$ have a point defined over a ...

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172 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 ...

**9**

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181 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**

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357 views

### Is there an algorithm which determines if a curve has good reduction outside a given set of primes

Fix a number field $K/\mathbf{Q}$, a finite set of places $S$ in $K$, an integer $g$ and a curve $X$ over $K$ of genus $g$.
Is there an algorithm which tells you if $X$ has good reduction outside ...

**9**

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438 views

### An analogue of Deligne-Lusztig theory for positive depth representations?

Deligne-Lusztig theory is an important tool in understanding the depth zero representations of $p$-adic groups. Is there an analogue of Deligne-Lusztig theory that helps in understanding positive ...

**8**

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193 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 ...

**8**

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347 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 ...

**8**

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174 views

### Corresponding notion of unramified for motives (or de Rham cohomology)

The etale cohomolgoy of a variety $X$ over a number field $K$ is a Galois representation of $\mathrm{Gal}(\overline K/K)$ with some properties coming from $X$, e.g., it is unramified outside $S$ if ...

**8**

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336 views

### Crystalline realization of mixed Tate motives

Deligne and Goncharov, in their article of 2005, mention that the crystalline realization functor has yet to be worked out. What's the current state of the literature on this? And how big of an issue ...

**7**

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184 views

### Inequality regarding sum of gaussian on lattices

When S is a subset of an inner product space, let d(S) denote ${\sum\limits_{s \in S} e^{- \langle s,s \rangle}}$
Suppose L is a discrete additive subgroup of $\mathbb{R^n}$, M is a subgroup of L, ...

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152 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 ...

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1k views

### What are “fractional motives”?

Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them?
Edit: Further cases of "fractional motives" as ...

**6**

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186 views

### The Rappoport-Zink spectral sequence vs. the one of the complement of a normal crossing divisor

As far as I understand these matters, for a regular $\mathfrak{X}$ that is proper flat of finite type over $\operatorname{Spec}\mathbb{Z}_p$, the Rappoport-Zink spectral sequence relates the etale ...

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93 views

### On the definition of LGP-monoids in IUT III

I have been trying to understand, without success, the definition of "LGP-monoids" on p. 80 of Mochizuki's IUT III and was wondering if anyone could provide some more explanation than what is given ...

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141 views

### simple proof of relation between H^1 crystalline and Dieudonne module?

Hi,
Let $k$ be a perfect field of characteristic $p > 0$. Let $A/k$ be an abelian variety. Then the first crystalline cohomology group of $A$ with respect to $W(k)$ (= Witt vectors) is canonically ...

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272 views

### Does a lower bound for models of finite group schemes exist?

Let $R$ be a discrete valuation ring (as beautiful as you like) and set $K:=Frac(R)$. Let $G_K$ be a finite $K$-group scheme, $G_1$ and $G_2$ two affine and flat models of $G_K$ of finite type, i.e. ...

**6**

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224 views

### Do Scharaschkin's results on Brauer-Manin obstructions on curves generalize to non-projective curves?

Theorem: Let X be a smooth projective curve over a number field K, and let $\delta$ be the index of X (i.e., the minimal degree of a K-rational divisor on X). Then V. Scharaschkin proved in this ...

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630 views

### Can you get Siegel's theorem “for free” from modularity and Mazur's Eisenstein Ideal paper?

There is a well-known theorem of Shafarevich that given a finite set $S$ of primes the number of isomorphism classes of elliptic curves over $\Bbb Q$ with everywhere good reduction outside $S$ is ...

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926 views

### An example computation of etale cohomology

(edited for clarity)
In a comment on a response to this question, moonface states the following: "Even if you tried to compute H^2 [etale with Z/5Z-coefficients] of a surface fibered in genus 2 ...

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91 views

### Can the hyperbolic core of a curve over $\mathbb Q$ be defined over $\mathbb Q$ as an algebraic stack

Here is a question I've been wondering about for a while. Currently it is mere curiosity and I do not have any direct applications in mind.
Let $X$ be a smooth quasi-projective geometrically ...

**5**

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150 views

### When are Galois representations with open image attached to elliptic curves?

Let $K$ be a number field with absolute Galois group $G_K$.
Let $\rho:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$ be a Galois representation such that the image of $\rho$ is open in ...

**5**

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145 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**

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203 views

### Isogenous elliptic curves have same conductor

Let $E/K, E'/K$ be elliptic curves defined over a number field $K$. Let $\phi: E \to E'$ be a non-constant isogeny defined over $K$. Why must the conductors be equal?
I know that this is an ...

**5**

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437 views

### Grothendieck monodromy theorem for l-adic sheaves

Hi,
Suppose that $F$ is a local field, $G_F$ its Galois group, $I$ the inertia subgroup, $k$ its residue field.
Let $X$ be a finite type scheme over $k$. Let $C$ be a constructible $l$-adic sheaf on ...

**5**

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674 views

### Motivic Galois group and Shimura varieties

Hi,
Suppose that one has a Shimura variety $Sh(G,X)$ where $(G,X)$ is the corresponding Shimura datum and suppose that it can be interpreted as a moduli space of motives (e.g. PEL type Shimura ...

**5**

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424 views

### a naive question about p-adic local monodromy theorem

The question is about whether one can view the p-adic local monodromy theorem as the quasi-unipotence of some monodromy operator.
it is known that the classical local monodromy theorem (i.e. for ...

**5**

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**0**answers

414 views

### Formal groups in the supersingular reduction case

Dear MO,
Let $E/\mathbb{Q}$ be an elliptic curve with potential good supersingular reduction at $p$. Thus, there is a finite extension $K/\mathbb{Q}_p$ such that $E/K$ has good supersingular ...

**5**

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504 views

### Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?

This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases?
Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...

**4**

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84 views

### minimal conductors among elliptic curves with a fixed CM type

Let $K$ be a quadratic imaginary field. To simplify my life, let us assume
that $K$ has class number one.
Consider the following infinite set:
$S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an ...

**4**

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**0**answers

98 views

### Complexes of arithmetic $\mathcal{D}$-modules with Frobenius structure

This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth ...

**4**

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128 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 ...

**4**

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148 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 ...

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301 views

### Does the Riemann hypothesis for liftable varieties over a finite field imply the Riemann hypothesis for all varieties over a finite field

The Riemann hypothesis for varieties over a finite field has been proven by Deligne. Still I would like to ask the following question.
A variety $X$ over a finite field $k$ is liftable if there ...

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**0**answers

168 views

### Do regular noetherian schemes of dimension one only have finitely many etale covers of bounded degree

Let $X$ be a regular noetherian scheme of dimension one. Let $d$ be an integer.
Question. Are there only finitely many finite etale morphisms $Y\to X$ of degree $d$?
I want to exclude finite etale ...

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164 views

### Is the moduli space of genus three smooth quartics affine?

Non-hyperelliptic curves of genus three are smooth quartics. Is the moduli space of such curves affine?
I think this follows from a more general result on smooth complete intersections, but I'm ...

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250 views

### 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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255 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 $G$. ...

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230 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 ...

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228 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 $(A/K,i)$ ...

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115 views

### Detecting linear dependence on multiplicative groups

Let G = $\mathbb{G}_m^2/\mathbb{Q}$ and let $\Gamma \subseteq G(\mathbb{Q})$ be a free abelian group of rank 2. Assume that the set of primes $p$ for which $\Gamma \mod p$ is cyclic has positive ...