Unanswered 'characteristic-p+characteristic-p+or+arithmetic-geometry' Questions - Page 18

1 vote

0 answers

209 views

A question about a set of prime numbers

Let $n$ an integer sufficiently large. I'm looking for a "big" set $S$ of prime number $p$ satisfying $10.n \geq p > \sqrt{18n} $ and $p| C_{9.n}^{k-n}C_{8n+k}^{8.n}$ for all integer $k$ ...

mamiladi

417

asked Jun 18, 2019 at 0:15

1 vote

0 answers

90 views

Is there an analogue of theta cycles for more general mod p automorphic forms?

The theory of $\theta$-cycles (due to Tate, I think) and filtrations is to me a very beautiful and powerful tool in proving many statements about mod $p$ modular forms in more explicit and elementary ...

GTA

1,024

asked Jun 10, 2019 at 20:55

1 vote

0 answers

282 views

Smooth absolutely irreducible (?) genus 1 plane pointless curve over $\mathbb{F}_{13}$

We got a family of genus 1 plane curves that may violate a bound in a paper. Explicitly: Let $F(x,y)$ be the degree 39 polynomial with integer coefficients: ...

joro

25.4k

asked May 29, 2019 at 9:30

1 vote

0 answers

206 views

Moduli spaces of arithmetic varieties with isomorphic $l$-adic cohomology

Given a positive integer $d$, a rational prime $l$ and a number field $K$, is it sensible to consider the moduli stack of $d$-dimensional varieties over $K$ whose $l$-adic cohomology rings are ...

user138328

11

asked Apr 13, 2019 at 15:58

1 vote

0 answers

150 views

Tate module is canonically isomorphic to a $\mathbb Z_p$-lattice on Shimura Variety of Hodge Type

Let $(G, X)$ be a Shimura datum of Hodge type. Suppose that $K \le G(\mathbb A_f)$ is such a compact open subgroup that its $p$th component $K_p = \mathcal G(\mathbb Z_p)$ is a hyperspecial subgroup ...

Jędrzej Garnek

421

asked Apr 8, 2019 at 22:02

1 vote

0 answers

102 views

Maximum number of bounded primitive integer points in a zero-dimensional system

Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...

VS.

1,836

asked Mar 22, 2019 at 15:21

1 vote

0 answers

88 views

What is the probability of 'yes' to this likely $coNP$ problem?

Pick a set of primitive (gcd of coordinates is $1$) integer points $\mathcal T$ in $\mathbb Z^n$. Denote the set of $n$ many algebraically independent homogeneous system of polynomials (thus zero-...

VS.

1,836

asked Mar 22, 2019 at 13:57

1 vote

0 answers

245 views

p-adic Hodge theory for singular projective varieties

In p-adic Hodge theory, one has comparison theorems relating, for example, the crystalline cohomology of the special fiber of a smooth proper family with the etale cohomology of the rigid-analytic ...

paul

375

asked Feb 14, 2019 at 15:40

1 vote

0 answers

139 views

finite $p$ extensions on adjoining $p$-torsion points of an elliptic curve

Let $K$ be a fixed number field and $E$ be any elliptic curve over $K$. When we adjoin to $K$ the $p$-torsion points $E[p]$, we obtain an extension whose Galois group can be embedded in $GL(2, \mathbb{...

debanjana

1,283

asked Jan 13, 2019 at 16:39

1 vote

0 answers

127 views

Height variation of abelian varieties within an isogeny class

Let $A$ be an abelian variety defined over a number field $K$ of dimension $g \geq 2$, and put $h_F(A)$ for the (stable) Faltings height of $A$. It is well-known from the seminal paper of Faltings ...

Stanley Yao Xiao

27k

asked Jan 7, 2019 at 2:44

1 vote

0 answers

127 views

Universal elliptic curve over anticanonical tower

While I'm reading Scholze's paper "On torsion in the cohomology of locally symmetric varieties" he constructs the anticanonical tower passing through the construction of an integral model $X_{\infty}$ ...

rime

445

asked Dec 26, 2018 at 20:35

1 vote

0 answers

157 views

A family of crystalline representations

Let $K$ be a number field and let $v$ be a finite place of $K$. Further, let $g \geq 1$ be a positive integer. Consider the family $F(K,v,g)$ consisting of abelian varieties $A$ of dimension $2g$, ...

Stanley Yao Xiao

27k

asked Dec 7, 2018 at 17:38

1 vote

0 answers

54 views

Finite generation for a restricted ramification idele module

Let $k$ be a number field, let $\bar k \subseteq \mathbb{C}$ be a fixed algebraic closure of $k$, and let $S$ be the set of infinite primes of $k$. Denote by $k_S$ the maximal extension of $k$ inside $...

Pablo

11.3k

asked Oct 23, 2018 at 0:45

1 vote

0 answers

242 views

Hardy-Littlewood vs heuristics on the zeta zeros

The first Hardy-Littlewood Conjecture asserts: Conjecture 1: Fix integers $0< a_1 < \ldots < a_k$. The density of those primes $p\le x$ such that $p + 2a_1,\ldots, p+2a_k$ are also primes, is:...

user129745

asked Oct 5, 2018 at 10:10

1 vote

0 answers

111 views

Reference request: Number of elliptic and hyperbolic quadratic forms of a given rank over a finite field

My question is over the finite field $\mathbf{F}_q$ of $q$ elements. It is well known that a symmetric matrix of odd rank corresponds to a parabolic quadratic form but even rank symmetric matrices ...

Singh

179

asked Jun 19, 2018 at 13:14

1 vote

0 answers

99 views

Special formal lifts of smooth algebras

Let $A$ be a smooth algebra over $k$ a finite field. Say $B$ is a $p$-adically complete smooth algebra over the Witt ring $W(k)$, lifting $A$. Assume $B$ is of the form $W(k)\langle t_1,\ldots, t_n\...

user124171

asked May 22, 2018 at 14:50

1 vote

0 answers

147 views

The specific elliptic fibration on the Kummer surface of the superspecial abelian surface

Consider two copies $E_1$, $E_2$ of the supersingular elliptic curve $$ y^2 = x^3 - 1\qquad (y^2 = x^4 - 1) $$ over a finite field $\mathbb{F}_{p^2}$ of odd characteristics $p$ such that $$p \...

Dimitri Koshelev

1,833

asked May 21, 2018 at 11:30

1 vote

0 answers

607 views

Push-forward along closed immersion

Let $X$ be a scheme, $p : Z\to X$ a closed immersion, $\mathcal{F}$ a locally free sheaf of modules on $Z$ of finite rank. Assume both $\mathcal{O}_Z$ and $\mathcal{O}_X$ are coherent sheaves of $\...

user120812

asked May 9, 2018 at 13:30

1 vote

0 answers

68 views

For the geometric meaning of this value for complex curve with model over $\mathbb{Q}$

Let $X$ be a smooth projective algebraic curve defined over $\mathbb{Q}$ with genus $g$, we have an isomorphism $H^1_{dR}(X/\mathbb{Q})\otimes_\mathbb{Q}\mathbb{C}\cong H^1_{sing}(X,\mathbb{Z})\...

Bonbon

806

asked May 8, 2018 at 19:18

1 vote

0 answers

34 views

Hodge classes generated in degree $1$

Let $X$ be a smooth projective variety over the complex numbers, and $\text{Hdg}^p(X)_{\mathbf{Q}}$ the abelian group of Hodge classes in $H^p(X,\mathbf{Q}(p))$. Denote by $\text{Hdg}^*(X)$ the ...

user95222

asked Mar 14, 2018 at 18:58

1 vote

0 answers

290 views

Coniveau in étale motivic cohomology

Let $X$ be a smooth variety over a field. Is there a spectral sequence: $$E_1^{p,q} := \bigoplus_{x\in X^{(p)}}H^{q-p}(\kappa(x)_{\rm ét},\mathbf{Z}(n))\Rightarrow H^{q-p}(X_{\rm ét},\mathbf{Z}(n))$$...

user92332

asked Mar 13, 2018 at 1:31

1 vote

0 answers

118 views

Torsion homologically trivial cycles

Is there an example of a smooth projective variety $X$ over the complex numbers, such that $$\ker(\text{CH}^2(X)\to H^4(X,\mathbf{Z}(2))$$ is not torsion?

user119470

asked Mar 11, 2018 at 19:12

1 vote

0 answers

118 views

Filtrations and the Betti cycle map

Let $X$ be a smooth projective complex variety. Calling $f : X_{\rm an}\to X_{\rm Zar}$ the "change-of-topology" morphism of sites induced by sending a Zariski open $U$ of $X$ to its analytification, ...

user119470

asked Mar 11, 2018 at 0:05

1 vote

0 answers

585 views

Blow-ups in étale cohomology

If $f:X\to\text{Spec}(k)$ is a smooth projective variety over a separably closed field, with $i:Z\subset X$ a closed smooth sub variety, with complement $j: U\to X$, and $f':X'\to \text{Spec}(k)$ is ...

user92332

asked Feb 27, 2018 at 16:27

1 vote

0 answers

106 views

Supersingular isogenies of elliptic curves preserving divisibility of points

I hope my question is clear. In summary, If $\phi:E\to E'$ is an isogeny and $P\in E$ is not divisible by $2$, under which conditions $\phi(P)\in E'$ is also not divisble by $2$. Here is the detail ...

Eduardo R. Duarte

795

asked Jun 7, 2017 at 21:08

1 vote

0 answers

348 views

rigid analytic geometry positive characteristic

I am a beginning graduate student. I have the following basic question I am very confused about: Suppose $C$ is a smooth geometrically irreducible curve over a finite field $\mathbb{F}_q$, $q=p^m$, $...

Sam Taylor

147

asked May 12, 2017 at 11:21

1 vote

0 answers

67 views

Basis of homomorpshims of abelian varieties with minimal degree

Let $A, B$ be simple abelian varieties of dimension $g$ defined over a finite field $k$. We know that $Hom_k(A, B)$ is a free $\mathbb{Z}$-module of dimension $2g$. Is it always possible to have a $\...

A. GM

389

asked Mar 13, 2017 at 14:02

1 vote

0 answers

187 views

Unitary dual of the Heisenberg group over non-archimedean local fields of characteristic two

What is the unitary dual of the Heisenberg group over non-archimedean local fields k of characteristic two? This is well-known for the real Heisenberg group and in fact, when local fields have ...

m07kl

1,702

asked Feb 5, 2017 at 1:41

1 vote

0 answers

149 views

Smoothability of stable curves in mixed characteristic

Let $R$ be a complete DVR with residue field $k$ algebraically closed of characteristic $p$ and fraction field $K$ of characteristic zero. Let $C$ be a stable curve (in the sense of Mumford-Deligne) ...

user45397

2,323

asked Nov 7, 2016 at 22:18

1 vote

0 answers

136 views

Is there a reference for boundedness of smooth canonically polarized varieties over Z (No...)

In Kollár's paper Quotient spaces modulo algebraic groups, Kollár mentions right above Theorem 1.8 that the stack $\mathcal M_P$ of smooth canonically polarized varieties over Spec $\mathbb Z$ with ...

Canu

11

asked Jun 10, 2016 at 15:23

1 vote

0 answers

109 views

Intersection of modular polynomial roots

Let $l,l'$ and $p$ be three distinct prime numbers and $\Phi_k(X,Y)$ is $k$-th modular polynomial defined over $GF(p)$. Suppose that we know $\Phi_l(X,j)$ and $\Phi_{l'}(X,j)$ have two roots. Is this ...

Meysam Ghahramani

507

asked Apr 11, 2016 at 14:10

1 vote

0 answers

169 views

Weil restriction of fiber products

Let $X,Y,Z$ be smooth geometrically integral proper varieties over a field $K$ where $K/k$ is a finite extension of a number field $k$. Let $R|_{K/k}$ denote the Weil restriction. Suppose we have $K$-...

Karim

11

asked Apr 1, 2016 at 23:00

1 vote

0 answers

189 views

A definition of arithmetic divisor with conic singularities?

I have a question related to the preprint "Heights and metrics with logarithmic singularities" by G. Freixas i Montplet. Let $X$ be an arithmetic variety with arithmetic divisor $D$ how can we ...

user21574

asked Feb 16, 2016 at 9:56

1 vote

0 answers

96 views

Shortest paths stepping on rational points of height $h$

Q. Do shortest paths walking between rational points of height $h$ ever properly cross themselves? Explaining this question takes a bit of definitional exposition. First, I copy definitions from ...

Joseph O'Rourke

151k

asked Nov 1, 2015 at 2:29

1 vote

0 answers

235 views

On the Weil Chatelet Group

Let $A$ be a abelian curve over a number field $K$. The Weil Chatelet group parametrizes the twists of $A$, modulo the twists with a $K$ rational point. We can assume that $A$ is a plane curve. My ...

camilo

527

asked Sep 6, 2015 at 21:54

1 vote

0 answers

122 views

Statements generalizing representability of integers by binary quadratic forms to $n$-variable higher homogeneous forms?

Representing integers through the theory of binary quadratic forms is a well studied topic. We know that given $a,b,c\in\Bbb N$, based on discrimant $b^2-4ac$, we can study the representability of ...

user76479

asked Aug 31, 2015 at 22:55

1 vote

0 answers

101 views

Points on the intersection of an affine quadric and cubic over a finite field

Are there absolute constants $N$ and $B$ such that the following is true? Let $p>B$ be a prime. Let $q(x_0,\dotsc,x_n)$ and $c(x_0,\dotsc,x_n)$ be homogeneous of degree $2$ and $3$ with ...

Siksek

3,142

asked Aug 16, 2014 at 10:28

1 vote

0 answers

183 views

Interpretation of the Gross-Zagier formula for Green function

I am reading the paper of Gross and Zagier on heights of Heegner points and would like to check with the experts whether the following (meta?)mathematical statement makes sense. In the calculation of ...

Lev Borisov

5,186

asked Jul 21, 2014 at 21:53

1 vote

0 answers

132 views

Points with minimal height

Let $K$ be an algebraically number field and $$\phi : \mathbb P^n (K) \to \mathbb P^m (K)$$ a polynomial map, such that $\forall \alpha \in \mathbb P^n$, where $\alpha = [\alpha_0, \dots , \alpha_n]$, ...

user24815

asked Jul 7, 2014 at 18:08

1 vote

0 answers

246 views

Uniqueness of lifting of very ample line bundle on smooth proper surfaces over DVR

Let $R$ be a complete Henselian discrete valuation ring of characteristic 0, $X_R$ be a surface smooth, proper and flat over $R$. Assume that the residue field $k$ of $R$ is algebraically closed of ...

Kali

503

asked Jul 4, 2014 at 18:05

1 vote

0 answers

103 views

degree of isogenies between Jacobians and Abelian Varieties

Let $K$ be a local field of characteristic zero and positive residual characteristic. Let $A$ be a simple abelian variety and assume we have an isogeny $f:Jac_C\rightarrow A$ with $C$ a smooth curve ...

meffi

11

asked Apr 7, 2014 at 15:07

1 vote

0 answers

149 views

Covers of modular curves

I'm interested in covers of modular curves (especially cyclic covers) and I'm sure there's a lot of information out there available on this topic. However, I'm unable to locate any literature (on ...

Daniel Larsson

1,841

asked Jan 23, 2014 at 10:11

1 vote

0 answers

120 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 ...

user43198

1,981

asked Dec 5, 2013 at 0:41

1 vote

0 answers

118 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 ...

user40810

asked Oct 3, 2013 at 6:55

1 vote

0 answers

193 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 ...

Keesjan

51

asked Sep 11, 2013 at 9:58

1 vote

0 answers

111 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 ...

lillo

11

asked Sep 11, 2013 at 9:41

1 vote

0 answers

572 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 $W(k)...

Lan

699

asked Jul 23, 2013 at 12:13

1 vote

0 answers

190 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 ...

mton

41

asked Jul 17, 2013 at 15:53

1 vote

0 answers

301 views

How to Construct a ''Nice'' Birational Model in Characteristic $p>0$?

Let $X={\rm Spec} A$ be a normal affine variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Also, assume that $S\subset X$ is a prime Weil-divisor on $X$. Now,...

Omprokash

165

asked Mar 22, 2013 at 21:00

1 vote

0 answers

411 views

a question about Beauville-Laszlo

Hi, let $V$ be a complete DVR with uniformizer $\pi$. Let $m$ be a NON zero integer, $a\in V[[u,v]]/(uv-\pi)^{\times}$ and $f=\pi^{m}a$. Consider $F$ as the kernel of the diagram $$ V[[u,v]]/(uv-\pi)...

questio

11

asked Feb 25, 2013 at 9:05

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