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14
votes
2answers
825 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
1answer
127 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) ...
1
vote
1answer
475 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 ?
2
votes
0answers
164 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
0answers
169 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
1answer
912 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 ...
0
votes
0answers
100 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
1answer
249 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
0answers
122 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
0answers
180 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 ...
1
vote
1answer
144 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
1answer
391 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
1answer
326 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
0answers
96 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
0answers
86 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
0answers
374 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
2answers
299 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
1answer
172 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
2answers
168 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
1answer
150 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
1answer
238 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
0answers
118 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 $C$ ...
7
votes
1answer
217 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
1answer
240 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 ...
18
votes
2answers
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
0answers
310 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
2answers
221 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
0answers
118 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
0answers
126 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
1answer
182 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
0answers
178 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
0answers
153 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
1answer
95 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
1answer
369 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 ...
0
votes
0answers
80 views

How to construct a reduction type that we need from a smooth curve

Let $X_{k}$ be a stable curve over a algebraically closed field $k$. We can find a complete DVR $R$ and deform $X_{k}$. Then we can obtain a stable curve $X$ over $R$ whose generic fiber $X_{\overline ...
4
votes
1answer
190 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 ...
10
votes
1answer
790 views

Are overlaps among {algebraic geometry, arithmetic geometry, algebraic number theory} growing?

From a naive outsider's viewpoint, just watching the MO postings in those three fields scroll by, and hearing of breakthroughs in the news, it appears there might be increasing overlap among the ...
5
votes
0answers
86 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 ...
4
votes
1answer
262 views

$R^2f_{\operatorname{et},*}\mathbb{G}_m$ vs $R^2f_{\operatorname{Zar},*}\mathbb{G}_m$

Let $S$ be the spectrum of a discrete valuation ring and $f:X\rightarrow S$ be a relative projective curve with generic fiber smooth and special fiber semistable. How much differ the sheaf ...
0
votes
0answers
97 views

How does one see level structure in the ell-adic Galois representation

This is probably a very easy question for those familiar with the arithmetic theory of abelian varieties because it is either possible or impossible for "trivial reasons". Let $A$ be an abelian ...
0
votes
1answer
172 views

Algebraic varieties in “mixed” affine spaces

Let $K\subset L$ be a field extension and let $K\subset F_1,F_2,...,F_n\subset L$ be proper intermediate fields. Consider the "mixed" affine space $\mathbb{A}_{(F_i)}:=\prod_{i=1}^n F_i$ instead of ...
15
votes
1answer
439 views

Is there a known example of a curve X of genus > 1 over Q such that we know the number of points of X over the n-th cyclotomic field, for every n?

By Falting's theorem, these numbers are of course finite. Is there an example where we can explicitly compute them for every $n$? Thank you!
9
votes
0answers
274 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 ...
1
vote
1answer
192 views

Comparison between Etale and Zariski topology on schemes

Let $Sch_{Zar}, Sch_{et}$ denote scheme with Zariski and Etale topology respectively. Is there a functor from $Sch_{et}$ to $Sch_{Zar}$ (or from $Sch_{Zar}$ to $Sch_{et}$) which preserves fiber ...
6
votes
1answer
235 views

Vanishing cohomology of de-Rham Witt complex

Let $X$ be a smooth scheme over $\mathbb{F}_{p}$ for a prime number $p$. As far as I understand, there is a surjective morphism from $\Omega^\bullet_{W\mathcal{O}_X} \to W \Omega_{X}^\bullet$ which ...
3
votes
1answer
174 views

Why is the base change functor faithful

Let $L/k$ be a field extension of algebraically closed fields of characteristic zero. Let $U$ be a smooth quasi-projective variety over $k$. I am trying to understand why the base-change functor from ...
7
votes
1answer
352 views

Potentially good, semi-stable reduction => good reduction ?

Does a smooth proper variety having semi-stable reduction as well as potentially good reduction have good reduction ? Note that over a $p$-adic field, this is true for the Galois representations in ...
0
votes
0answers
107 views

Density of integral (rational) points on affine varieties

In a series of papers due to Browning, Heath-Brown, and Salberger (some of which are joint work, some individual), they established that for any projective variety $X \subset \mathbb{P}^n$, they ...
3
votes
1answer
105 views

Rational points on the curve y^p=f(x) in characteristic p

Let $K$ be a finite extension of $\mathbb{F}_q(t)$ and define the curve $C$ by the equation $y^p=f(x)$ where $p=\mathbf{char} K$ and $f\in K[x]$. What is the genus of $C$? When does it have infinitely ...
8
votes
1answer
242 views

Belyi's theorem for function fields

Belyi's theorem states that every smooth projective algebraic curve $C$ defined over $\bar{\mathbb{Q}}$ admits a map $C\to\mathbb{P}^1$ ramified only over $0,1,\infty$. Is there an analogue of this ...