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Questions tagged [arithmetic-geometry]

Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

11
votes
1answer
932 views

How to visualize the Frobenius endomorphism?

As the question title asks for, how do others "visualize" the Frobenius endomorphism? I asked some people in real life and they said they didn't know and that I could go and ask on MO and possibly get ...
7
votes
1answer
381 views

Mordel's conjecture for function fields in positive characteristic

Manin proves Mordel's conjecture for function fields in characteristic zero.his proof has a gap but Coleman fill this gap and restate Manin proof in a more modern language.both of them work over ...
3
votes
1answer
285 views

An explicit correspondence for reductions of modular curves $Y(N)$

Let $Y(N)$ be the modular curve associated with the principal congruence subgroup $\Gamma(N) \subset \mathrm{SL}(2, \mathbb{Z})$ of level $N \in \mathbb{N}$. It is well known that this curve has a ...
1
vote
1answer
297 views

Points of infinite level modular curve

Let us consider the anticanonical adic tower of modular curves which is described in Peter Scholze's paper "On torsion in the cohomology of locally symmetric varieties", and let us call $\mathcal{X}_{\...
8
votes
0answers
305 views

Moduli interpretation of Fargues-Fontaine curve

The Fargues-Fontaine curve is, in his schematic version, a noetherian regular scheme $X$ of dimension 1 associated to a pair $(E,F)$, where $E$ is a local field (i.e. complete w.r.t. a discrete ...
10
votes
0answers
207 views

What are the possible bad reductions for an abelian variety of dimension $g$ and a maximal endomorphism ring?

Perhaps the most basic fact about abelian varieties with CM is they have an everywhere potential good reduction (Serre-Tate). On the face of it it might appear that there isn't much more to be added ...
6
votes
0answers
218 views

Smooth morphisms to the moduli stack of elliptic curves

Fix a prime $p$, and let $\overline{M}$ be the Deligne-Mumford compactification of the moduli stack $M$ of elliptic curves (which, concretely, is the open substack of the stack of cubic curves which ...
4
votes
0answers
236 views

Strengthened supercongruences for Ramanujan-type formulas for $1/\pi^k$

The question below is again a follow-up of an old question. Motivation: Zhi-Wei Sun listed a number of supercongruences attached to Ramanujan-type $1/\pi$ formulas in the arXiv paper which can be ...
2
votes
0answers
147 views

Specialization map on geometric points

Let $\mathcal{X}$ be a proper and smooth scheme over $\text{Spec}(\mathbf{Z}_p)$, and let’s call $X$ the geometric generic fiber of $\mathcal{X}$, and $X_0$ the geometric special fiber of $\mathcal{X}$...
1
vote
0answers
132 views

Lefschetz trace formula and Frobenius elements

Let $\mathcal{X}$ be a smooth and proper scheme over $\text{Spec}(\mathbf{Z}_p)$ (with $\mathbf{Z}_p$ the $p$-adic integers). Let’s call $X$ its geometric generic fiber $X := \mathcal{X}_{\overline{\...
6
votes
1answer
248 views

Binomial coefficients in discrete valuation rings

Let $V$ be a complete discrete valuation ring whose residue field is a finite field $k=\mathbf{F}_q$. Let $\pi\in V$ be a uniformizer. For any integer $d,n\ge 0$, define: $${\pi^d \choose n} := \...
5
votes
1answer
209 views

Is there infinitely many prime $p$ such that the normalized trace of Frobenius $\frac{a_p(E)}{2\sqrt{p}}$ is arbitrarily small (but not zero)?

I just encountered the following problem and failed to find proper reference, could anyone kindly help to give me some illustration? For an elliptic curve $E$ without complex multiplication (just ...
3
votes
0answers
82 views

What is the canonical divisor on an arithemetic curve?

Let an arithemetic curve be an integral scheme $X$ whose structure morphism $\pi:X\rightarrow B=Spec(O_{K})$ is projective, flat and of pure dimension $0$, and whose generic fiber is regular. I am ...
1
vote
0answers
67 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 ...
3
votes
1answer
266 views

1-dimensional p-divisible groups, level structures and Cartier divisors

I am confused about the 1-dimensionality of $p$-divisible groups and its role in defining level structures. Here's how I view/understand/not understand things: If a $p$-divisible group arises from a ...
4
votes
1answer
154 views

Isogeny of Drinfeld module

Let $\phi$ be a Drinfeld module over a function field $K$. It is known that if the rank of $\phi$ is 1, then it is isomorphic over $K$ to one defined over $O_K$. Is it true that if the rank of $\phi$ ...
2
votes
0answers
53 views

Product formula over arithmetic surface

Let $X$ be an arithmetic surface and $f\in K(X)$ be a function in the function field. Is there any analagous "product formula" showing $\deg(f)=0$? This is motivated by the number field case, where $X=...
7
votes
1answer
253 views

Is there a converse to the Brauer–Nesbitt theorem?

$\DeclareMathOperator\Tr{Tr}$Say that we have an algebra $R$ over $\mathbb{C}$. If, for two finitely generated (edit: and semisimple) $R$-modules $M, N$ we know that $\Tr_M(r)=\Tr_N(r)$ for all $r\in ...
2
votes
0answers
43 views

Counting intersections of rectilinear lattices

The following proposition Let $g>0$ be an integer and let $\Lambda \subset \mathbb{R}^g$ be a rectilinear lattice (possibly shifted) with mesh $d$ at most $D$. Then we have $$ \left| \#(\Lambda \...
4
votes
1answer
133 views

Can an algebraic variety over a field $k$ be the union of proper closed subsets $(S_i)_{i\in I}$ with $I < k$

Let $k$ be an algebraically closed field (of characteristic zero, if it helps). Let $X$ be an algebraic variety over $k$. Let $I$ be an index set such that the cardinality of $I$ is smaller than the ...
11
votes
0answers
366 views

Reference request: Newton above Hodge

Let $K$ be a p-adic field, and let $\mathcal{O}$ be the ring of integers inside $K$ with residue field $k$. Let $\mathcal{X}$ be a smooth proper formal scheme over $\mathcal{O}$ (with topology given ...
4
votes
2answers
188 views

Singular abelian surfaces that can be defined over $\mathbb Q$

An abelian surface $A$ is called singular if it has maximal Picard number $\rho(A) = 4$. By work of Shioda-Mitani, any singular abelian surface $A$ is the product $A = E_1 \times E_2$ of two ...
2
votes
0answers
136 views

Lefschetz trace formula over truncated Witt ring

Let $k$ be a finite field, $W_n(k)$ be its $n$-th truncated Witt ring. We have a Frobenius on $W_n(\bar{k})$ whose fixed point is exactly $W_n(k)$. Let $X$ be a finite type separated scheme over $W_n(...
9
votes
0answers
387 views

On Topological Hochschild Homology

Nowdays, I hear talking about Topological Hochschild Homology more and more often, and I was wondering if someone could point out references to explain why it's important and interesting, and what ...
6
votes
0answers
226 views

Etale cohomology of the variety of matrices with given characterestic polynomial

A continuation of Number of points of the nilpotent cone over a finite field and its cohomology. Let $k=\Bbb F_q$ be a finite field, $p=\text{char}(k),$ $P \in k[\lambda]$ be a monic polynomial of ...
7
votes
1answer
351 views

Arithmetic symplectic geometry via mirror symmetry?

Homological mirror symmetry in the classical setting relates the bounded derived category of coherent sheaves on a Calabi-Yau manifold to the split-closure of the derived Fukaya category of the mirror ...
4
votes
0answers
165 views

Canonical differential on K3 surface

On an elliptic curve over $\mathbb{Q}$, we can associate a canonical Neron model and with it a Neron differential, whose coefficients in some natural coordinates yield the Dirichlet coefficients of ...
1
vote
0answers
90 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\...
1
vote
0answers
106 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 \...
2
votes
0answers
140 views

Elliptic fibrations on the Fermat quartic surface

Consider the Fermat quartic surface $$ x^4 + y^4 + z^4 + t^4 = 0 $$ over an algebraically closed field $k$ of characteristics $p$, where $p \equiv 3$ ($\mathrm{mod}$ $4$). Is there the full ...
0
votes
0answers
72 views

Twisted sheaves on tower of $\mathbb{P}^n$

Take the projective space $\mathbb{P}^n$ over a ring $W$. We call $\mathcal{O}(q)$ the usual twisted line bundle. Now take the map $f: \mathbb{P}^n\to\mathbb{P}^n$ defined by $$[x_0,\ldots, x_n]\...
13
votes
0answers
404 views

Topological finiteness of etale fundamental group for arithmetic schemes

By comparison theorem and finite CW complex structure we know every complex variety's etale fundamental group is topological finitely generated (see here). It seems natural to ask similiar things for ...
3
votes
0answers
166 views

Non algebraizable formal abelian schemes

I'd like to collect a good number of examples of formal schemes over $\text{Spf}(\mathbf{Z}_p)$, whose special fiber is a projective variety over $\mathbf{F}_p$, but that are not algebraizable. If ...
2
votes
0answers
176 views

Is the Fermat quartic surface a generalized Zariski surface?

Consider the Fermat quartic surface $$F\!: x^4 + y^4 + z^4 + t^4 = 0$$ over an algebraically closed field $k$ of odd characterstics $p$. Shioda proved that for $p=3$ this surface is a generalized ...
6
votes
1answer
219 views

Smooth algebras always lift

Let $k$ be a finite field, $A$ a smooth $k$-algebra. Does there exists a smooth algebra $B$ over the Witt vectors $W(k)$, such that $B/p\simeq A$? How is it constructed?
3
votes
1answer
113 views

Liftings and closed immersions

Let $A$ be a flat $\mathbf{Z}_p$-algebra, $\overline{I}\subset A/p$ an ideal. Can we find an ideal $I\subset A$ such that $I$ mod $p$ = $\overline{I}$ $I$ does not contain $p$. It's harder than it ...
2
votes
0answers
59 views

Under what conditions are superspecial abelian surfaces isomorphic over a finite field?

Let $E_1$, $E_2$, $E_3$, $E_4$ be supersingular elliptic curves over a finite field $\mathbb{F}_{p^2}$, where $p$ is an odd prime. There is a well known theorem stating that over the algebraic closure ...
4
votes
1answer
238 views

Coherent modules over complete adic rings: counterexamples

Let $A$ be a coherent ring, complete with respect to the adic topology generated by a finitely generated ideal $I$. Define the category $Coh(A,I)$ whose objects are inverse systems $\{M_n\}$ of $A$-...
2
votes
0answers
124 views

Absolute approximation of formal schemes

Let $\mathfrak{X}_j$ be an inverse system of qcqs $p$-adic formal scheme, flat over $\mathbf{Z}_p$, with affine transition maps, and assume $\mathcal{O}_{\mathfrak{X}_j}$ is a coherent sheaf of ...
2
votes
0answers
163 views

Liftability of varieties, after fpqc base change

Let $X$ be a smooth projective variety over a finite field, with a closed immersion to some other smooth projective variety $S$, with $S$ liftable. Suppose there exists an fpqc cover $S'\to S$, such ...
2
votes
0answers
70 views

affine Lefschetz and Poincaré duality for syntomic cohomology

Let $X$ be a smooth variety over an algebraically closed field $k$ of characteristic $p > 0$. Is there an affine Lefschetz theorem and Poincaré duality for sheaves represented by finite flat ...
2
votes
0answers
129 views

Is there any generalization of Weil conjecture for non-smooth variety?

Is there any generalization of Weil conjecture for any non-smooth geometric-connected variety? For example, for more general curve, or at least some numerical evindence (example)?
5
votes
0answers
123 views

Do non-constant maps specialize to non-constant maps?

Let $R$ be a dvr with fraction field $K$ and residue field $k$. Let $\mathcal{X}\to \mathcal{Y}$ be a morphism of $R$-schemes such that $\mathcal{X}_K\to \mathcal{Y}_K$ is non-constant. Is the ...
1
vote
1answer
157 views

Approximation of constructible abelian étale sheaves

Let $F$ be a constructible abelian étale sheaf of modules over a finite ring $\Lambda$ on a scheme $X$ over a field $k$, with the size of $\Lambda$ invertible on $X$. Suppose $X = \varprojlim X_j$, ...
3
votes
1answer
172 views

Projective embeddings and quasi-compactness

Let $X$ be a projective scheme over a ring $R$, and $p : X\to\mathbf{P}^n_R$ a projective embedding. Does there exist $n$ large enough so that the complement $U\subset \mathbf{P}^n_R$ of $p(X)$ in ...
1
vote
0answers
80 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 $\...
3
votes
1answer
120 views

Étale torsors and equivariant structures

Let $X$ be a scheme over a separably closed field, equipped with an action of a constant group scheme $G$. Let $H$ be a finite group whose size is invertibile on $X$, and $Y\to X$ an $H$-torsor with ...
1
vote
0answers
66 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})\...
3
votes
1answer
441 views

Descent of étale torsors

Let $X$ be a scheme over a field $k$, $G$ a finite abelian group of size invertible on $X$. Suppose $K/k$ is a Galois field extension and let $Y\to X_K$ be an étale $G$-torsor. For what field ...
5
votes
0answers
204 views

Is there a relationship between the zeta function of a Laplacian and the Selberg Zeta Function?

Let us define the zeta function of an elliptic differential operator $H$ with eigenvalues $\lambda_n$ like so: \begin{aligned} \zeta_H(s) := tr( H^{-s} ) \\ := \sum^\infty_{n=1} ...