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

3
votes
0answers
69 views

Automorphy of families of motives

I have a couple of elementary questions regarding automorphy of Galois representations arising from geometric families. Suppose we have an algebraic family of varieties over a number field, and ...
3
votes
0answers
87 views

Visualization of hidden structures in numbers

In the general context of Numbers and Geometry I was playing around with geometric visualizations of structures in the natural numbers and came up with a type of function graphs I haven't seen before....
4
votes
0answers
112 views

Twistor projective line

Let $\mathbf{P}^1$ be the projective line over $\mathbf{C}$, and $c : \mathbf{P}^1\to \mathbf{P}^1$ the anti-holomorphic involution sending $z$ to $-\overline{z}$. The quotient $\widetilde{\mathbf{P}}...
3
votes
0answers
105 views

Extension of Galois representations from geometry

Let $K$ be a number field. Take two representations $A$, $B$ of $G_K=\mathrm{Gal}(\bar{K}/K)$ over some ring $\Lambda$ (in fact, I only consider the case $\Lambda$ is a field, usually of ...
5
votes
0answers
60 views

Frey-Mazur for abelian varieties

Let $K$ be a number field. The Frey-Mazur conjecture asserts the existence of a constant $N_K$ such that for all primes $p>N_K$, and all pairs of elliptic curves $E_1$, $E_2/K$, if $\overline{\rho}...
7
votes
1answer
184 views

Do arithmetic schemes have non-singular alterations?

Let $X$ be an integral normal flat finite type scheme over $\mathbb{Z}$. Does there exist a proper surjective generically finite morphism of schemes $Y\to X$ with $Y$ an integral regular ...
26
votes
2answers
2k views

How to visualize Dirichlet’s unit theorem?

As the question title asks for, how do others "visualize" Dirichlet’s unit theorem? I just think of it as a result in algebraic number theory and not one in algebraic geometry. Bonus points for ...
7
votes
3answers
245 views

Infinite Galois descent for finitely generated commutative algebras over a field

Let $k_0$ be a field of characteristic 0, and let $k$ is a fixed algebraic closure of $k_0$. Write $G={\rm Gal}(k/k_0)$. Let $A_0$ be a finitely generated commutative $k_0$-algebra with a unit. Then ...
2
votes
0answers
59 views

Log canonical thresholds of decomposable forms

This question is related to Comparisons of log canonical thresholds, but I restrict here to polynomials of the form $$F(x_1,\dots,x_m)=L_1(x_1,\dots,x_m)\dots L_n(x_1,\dots,x_m)$$ where $L_i(x_1,\dots,...
2
votes
0answers
43 views

Uniformity of the set of poles of Igusa local zeta functions

Let $Ω_p$ denote the set of the real parts of the poles of the Igusa zeta function of a polynomial $f∈\mathbb{Z}[X_1,…,X_m]$ (assume $f(0)=0$ so that $\Omega_p\ne \emptyset$) at the prime p. From ...
4
votes
1answer
147 views

Pairing on arithmetic surfaces

Let $f: X \to S$ be an arithmetic surface, where $S=\operatorname{Spec } O_K$ for a number field $K$. It is well known that if we want to introduce a reasonable intersection theory on $X$ we have to ...
2
votes
1answer
108 views

Comparisons of log canonical thresholds

Premise Let $K$ be a field of characteristic zero and $f\in K[X_1,\dots,X_m]$. By Hironaka's theorem, there exists a log resolution (over $K$) of the ideal $(f)$. Let $\{(N_i,\nu_i)\}_i$ be the ...
8
votes
1answer
625 views

How to visualize the Frobenius endomorphism?

As the question title asks for, how do others "visualize" the Frobenius endomorphism? I asked my advisor and he said he didn't know and that I could go and ask on MO and possibly get miseducated. So ...
7
votes
1answer
354 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 ...
2
votes
1answer
143 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
214 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}_{\...
7
votes
0answers
171 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 ...
8
votes
0answers
153 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 ...
5
votes
0answers
181 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
209 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
109 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
94 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
237 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
193 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
68 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
59 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
224 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
135 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
50 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
242 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
39 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
126 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 ...
9
votes
0answers
249 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
173 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
123 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(...
8
votes
0answers
299 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 ...
5
votes
0answers
175 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
150 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
83 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
89 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
93 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
56 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]\...
11
votes
0answers
227 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
138 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
160 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
197 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
53 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 ...
3
votes
1answer
214 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
111 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 ...