All Questions
2,494 questions
2
votes
0
answers
119
views
Resolution of singularities of the resultant locus
We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...
- 7,746
0
votes
0
answers
179
views
Points at which a polynomial becomes reducible
Let $n \geq 10$ and set $\mathbf{y} = (y_1,\ldots,y_n)$. Let $Q_1(\mathbf{y}),\ldots,Q_5(\mathbf{y})$ be non-zero quadratic forms with integer coefficients such that the cubic form $x_1Q_1(\mathbf{y})+...
- 105
0
votes
1
answer
240
views
Can I calculate congruent zeta function of given hyperelliptic curve by hand?
How can I calculate the numerator of congruent zeta function of given hyperelliptic curve ?
For example, let $C:y^2=(x^2+1)(x^4-8x^3+2x^2+8x+1)$.
numerator of congruent zeta function mod$23$ of this ...
- 4,193
5
votes
1
answer
312
views
Conductor at 2 of abelian surfaces with real multiplication
Let $A/\mathbb{Q}$ be an abelian surface such that $\text{End}_{\mathbb{Q}}(A)\otimes \mathbb{Q}$ is a real quadratic field $E$. I am interested in bounding the conductor of $A$ at $2$.
Let $\mathfrak{...
- 984
4
votes
1
answer
259
views
On inverse limits of $\pi$-adically complete algebras
Consider the following situation, let $\mathcal{R}$ be a discrete valuation ring with uniformizer $\pi$ (say the valuation ring of a finite extension $K$ of $\mathbb{Q}_{p}$. Let $\{ A_{n}\}_{n\in\...
- 63.9k
2
votes
1
answer
438
views
Sheaf cohomology in number theory
I have read the first three chapters of Hartshorne and was wondering what are the applications of the notions presented in number theory or arithmetic geometry. I already know that the notion of ...
- 7,320
1
vote
1
answer
315
views
About simple motives
I'm reading through Jannsen's paper Motives, numerical equivalence, and semi-simplicity and I'd like to pose two questions.
Suppose all motives are $F$-linear, for some characteristic zero field $F$, ...
- 149k
2
votes
0
answers
240
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References to let me know about current directions of research in arithmetic geometry
I have knowledge of basic algebraic geometry and good deal of number theory. I have studied roth theorem and I am currently studying proof of Mordell-Weil theorem. These two topics come under ...
- 793
3
votes
1
answer
308
views
Projective dimension of group ring
Assume that $G$ is a group and $R$ is a p.i.d. What can we say about the projective dimension of $R[G]$? For example can we say that this dimension is at most $1$ for reductive groups? (I think if $...
CommunityBot
- 1
5
votes
1
answer
268
views
Torus gerbes over curves
Setup: Let $k$ be an algebraically closed field. Let $C$ be a smooth connected curve over $k$. Let $K(C)$ be the function field of $C$.
Tsen's Theorem implies that every $\mathbb{G}_m$-gerbe over $K(C)...
- 63.9k
1
vote
0
answers
108
views
Does an open immersion "cut out" points surviving finite descent?
Let $X$ be a smooth affine curve over a number field $k$, and let $C$ be its smooth compactification. Let $i:X(\mathbb{A}_k) \rightarrow C(\mathbb{A}_k)$ be the induced morphism of adelic points by ...
- 895
2
votes
0
answers
197
views
Mumford's computation of the determinant of cohomology of a relative curve
In Integral Grothendieck-Riemann-Roch theorem, Pappas mentions that Mumford computed the determinant of cohomology of $f:X\to S$ a relative curve integrally, and thus proved an integral version of GRR ...
- 2,054
1
vote
1
answer
94
views
Reference for auto-duality of nearly ordinary deformations associated to Hida families
suppose we have a $p$-stabilized newform $f$ of classical level $\Gamma_0(p)$; then there is a unique ordinary deformation into a Hida family $\mathbf{f}$ defined over a finite flat extension $R/\...
- 2,054
1
vote
0
answers
96
views
Are the irreducible components appearing in the resolution of singularities of a Hilbert modular surface defined over $\mathbb{Q}$?
It seems to me that this is claimed in van der Geer's "Hilbert modular surfaces" on p. 245 at the beginning of XI.2 (without justification).
My current state of belief/knowledge:
The ...
- 93
20
votes
3
answers
2k
views
Rational points on algebraic curves over $\mathbb Q^\text{ab}$
Motivation:
Let $\mathbb{Q}_{\infty,p}$ be the field obtained by adjoining to $\mathbb{Q}$ all $p$-power roots of unity for a prime $p$. The union of these fields for all primes is the maximal ...
- 2,821
0
votes
0
answers
182
views
Does a $p$-adic power series $F(x,y)=\sum_{i,j \geq 0}b_{ij}x^iy^j \in \mathbb Z_p[[x,y]]$ have finitely many zeros in $\mathfrak{m}_{\mathbb C_p}$?
Let us consider the $p$-adic field $\mathbb Q_p$ with ring of integers $\mathbb Z_p$ and maximal ideal $\mathfrak{m}$.
Then any $p$-adic power series $f(x)=\sum_{n>0}a_nx^n \in \mathbb Z_p[[x]]$ ...
- 930
3
votes
0
answers
123
views
points on non-hyperelliptic curves of genus 3
I have a non-hyperelliptic curve $C$ of genus 3 and I'm interested in finding the $K$-rational points on the curve with $K$ a fixed imaginary quadratic number field.
As $C$ is a non-hyperelliptic ...
- 31
4
votes
0
answers
77
views
Conjugacy of cocharacters from conjugacy of labelled diagrams
Everything to follow is over some fixed algebraically closed field $k$. Although all the definitions make sense regardless of characteristic, the meat of the question is about small positive ...
- 12.9k
2
votes
0
answers
220
views
Is the ring of power series with $p$-adic coefficients Huber?
I have been reading the Berkeley lectures and got stuck with this question. Let $\mathbb{Q}_p [[t]]$ denote the ring of power series with $p$-adic coefficients. Is there a natural topology (e.g. the ...
- 103
1
vote
0
answers
192
views
Vanishing of the local étale cohomology sheaf (?)
Let $X$ be a locally noetherian regular scheme, and let $Z$ be a closed subscheme of $X$ whose codimension $d > 0$ at every point.
Let $U$ be the complement of $Z$ in $X$.
For a sheaf $\mathscr{F}$ ...
- 12.9k
4
votes
1
answer
545
views
On the local properties of rigid analytic varieties
Let $K$ be a non-archimedean field complete with respect to a discrete valuation with ring of integers $\mathcal{R}$, uniformizer $\pi$ and residue field $k$. Consider an affinoid analytic $K$-variety ...
- 783
7
votes
0
answers
231
views
Field extensions that preserve given cohomology classes
Let $G$ be a connected reductive group over $\mathbb{Q}$ and let $\operatorname{Ker}^1(\mathbb{Q},G) \subset H^1(\mathbb{Q},G)$ be the subset of classes that are trivial at all places. I am trying to ...
- 537
4
votes
0
answers
174
views
Spencer complex and de Rham Complex
in those lectures notes written by Claude Sabbbah: https://perso.pages.math.cnrs.fr/users/claude.sabbah/livres/sabbah_nankai110705.pdf
there is the proposition 1.4.4 where he says that there is a ...
- 385
7
votes
1
answer
264
views
Upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$
Does there exist an upper bound of the analytic rank of the modular Jacobian varieties $J_1(N)$?
(Or more generally of $J_\Gamma$ for a congruence subgroup $\Gamma_0 \subseteq \Gamma \subseteq \...
- 37k
16
votes
0
answers
1k
views
Finiteness for motivic local systems
Let $X$ be a smooth proper algebraic curve over $\mathbb{C}$. Say a complex local system $\mathbb{V}$ on $X$ is motivic if there exists a dense Zariski-open subset $U\subset X$, and a smooth proper ...
- 23k
5
votes
2
answers
314
views
Generalization of $j(E) \in \overline { \Bbb{Z}}$ to abelian varieties of arbitrary dimension
Let $E/ \Bbb{C}$ be an elliptic curve which has complex multiplication over a number field $K$.
Then it is widely known that $j(E) \in \overline { \Bbb{Z}}$.
What is the known generalization of this ...
- 47.4k
1
vote
0
answers
83
views
Are there known situations where this weaker form of the section conjecture holds?
Let $k$ be a number field. The section conjecture predicts that for a (smooth geometrically connected) hyperbolic curve over $k$, the profinite Kummer map $\kappa :X(k) \rightarrow \mathscr{J}_{\pi_1(...
- 895
3
votes
1
answer
171
views
On the stability of having a normal formal model under finite extensions of the base field
Let $K$ be a finite extension of the $p$-adic numbers with valuation ring $\mathcal{R}$ and uniformizer $\pi$. Consider a smooth and connected rigid $K$-variety $X=Sp(A)$ and assume that the affine ...
- 125
4
votes
3
answers
756
views
Reference for Skinner-Urban on the Iwasawa main conjecture for $\mathrm{GL}_2$
Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban
"The Iwasawa main conjecture for $\mathrm{GL}_2$"?
I am interested in partucular in the ...
- 63.9k
5
votes
1
answer
340
views
The $\mathbb{Q}$-rational cuspidal group of $J_0(N)$
Let $N$ be a positive integer and consider the modular curve $X_0(N)$ over $\mathbb{Q}$. Also, consider the Jacobian variety $J_0(N)$ of $X_0(N)$, which is an abelian variety defined over $\mathbb{Q}$....
- 2,224
2
votes
0
answers
90
views
Reconciling two notions of finite descent obstructions
Let $k$ be a number field and $X$ a smooth geometrically connected variety over $k$. We denote by $H(k,X)$ the set of sections $G_k \rightarrow \pi_1(X)$, where $G_k$ is the absolute Galois group of $...
- 895
1
vote
0
answers
180
views
Maximal unramified extension and algebraic closure of $\operatorname{Frac}(\widehat{A_{\mathfrak{m}_A}})$
$\DeclareMathOperator\trdeg{trdeg} \DeclareMathOperator\Frac{Frac} $
Let $k$ be an algebraically closed field of characteristic $0$ and $K$ a function field over $k$. let $(A, \mathfrak{m}_A)$ a ...
- 6,006
2
votes
0
answers
145
views
How to compute the character of the Steinberg module for the group $\mathrm{SL}_n$ over a field of characteristic $p$?
It is known that the Steinberg representation $V$ of the group $\mathrm{SL}_n$ over a field $k$ of characteristic $p$ (maybe one needs to assume that $k$ is perfect, I am not sure) is the irreducible ...
- 63.9k
6
votes
2
answers
2k
views
"Bad" reduction of Shimura curves via dual graphs
I have the following naive (and inexpert) question about the
reduction of Shimura curves at primes dividing the discriminant
of the underlying quaternion algebra. It requires some background
to state. ...
- 1,446
3
votes
0
answers
127
views
Isogeny of elliptic curve over positive characteristic $p$ which does not come from characteristic $0$
Let $K$ be quadratic imaginary field. Let $E$ be an elliptic curve which has CM over $R_K$
($R_K$ is ring of integers of $K$).
According to SIlverman's ''ADvanced topics in the arithmetic of elliptic ...
- 1,541
1
vote
1
answer
431
views
Is it easy to define weights for $Q_l$-sheaves over finite type $Z[1/l]$-schemes?
In her paper "Mixed perverse sheaves for schemes over number fields" A. Huber defines certain weights for certain categories of $\mathbb{Q}_l$-sheaves over a finite type $\mathbb{Q}$-scheme $...
- 1,446
2
votes
0
answers
47
views
Characters of simple $\mathfrak{sl}_n$-modules in positive characteristic with subregular nilpotent central character
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$ (i.e. $\chi$ is a nilpotent matrix whose Jordan normal form has two blocks ...
- 63.9k
4
votes
0
answers
64
views
An analog of a BGG resolution in subregular case in positive characteristic
Consider representations of $\mathfrak{sl}_n$ in positive characteristic with a subregular nilpotent central character $\chi$. For every regular weight $\lambda$ of $\mathfrak{sl}_n$, we have the ...
8
votes
1
answer
339
views
On actions of finite groups on adic spaces
Let $K$ be an algebraically closed complete non-archimedean field and consider the unit ball $\mathbb{B}^{1}_{K}=Sp(K\langle t\rangle)$. We have an action of $\mathbb{Z}/2\mathbb{Z}$ on $\mathbb{B}^{1}...
- 28.2k
11
votes
1
answer
1k
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 ...
- 328
5
votes
0
answers
546
views
Perfect algebraic spaces on a paper of Xinwen Zhu
I have problem reading Xinwen Zhu's paper Affine Grassmannians and the geometric Satake in mixed characteristic about perfect algebraic spaces in Section A.1.
Let $k$ be a perfect field of ...
- 151
3
votes
0
answers
120
views
Resolving the "wild" singularities of $\mathbb A^n/C_n$
Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by ...
- 7,746
27
votes
6
answers
4k
views
Does the moduli space of smooth curves of genus g contain an elliptic curve
Let $M_g$ be the moduli space of smooth projective geometrically connected curves over a field $k$ with $g\geq 2$. Note that $M_g$ is not complete.
Does $M_g$ contain an elliptic curve?
The answer ...
- 2,821
0
votes
0
answers
101
views
Identity component of $\mathrm{Ker}(E^n→E^m)$ in the advanced topics in the arithmetic of elliptic curves
$\DeclareMathOperator\Ker{Ker}$Silverman's "Advanced topics in the arithmetic of elliptic curves", p.115 reads $$0\to\mathfrak{a}^{-1}Λ\to\Bbb{C}\to\Ker(E^n\to E^m)\to Λ^n/A^tΛ^m\quad (1)$$ ...
- 1,541
17
votes
1
answer
2k
views
How does the cohomology of the Lubin-Tate/Drinfeld tower fit into categorical p-adic local Langlands?
In conjecture 6.1.14 of this article, Emerton-Gee-Hellmann formulate the p-adic local Langlands conjecture, which posits the existence of a fully faithful functor from (the appropriate derived ...
- 21.3k
3
votes
0
answers
189
views
Resolutions of configuration space of the projective line where the complement is of "Tate type"
I would like to find a nice compactification $X_n$ of $F(\mathbb P^1,n)$ (considered as a scheme over $\mathbb Z$), the $n$-fold configuration space of the projective line with the property that the $...
- 7,746
2
votes
0
answers
147
views
Can we say anything about the zeros and Galois group of the polynomial $(x^p-a)^{p^2}-p^{p^2+1}x+p^{p^2} a=0$?
Let $p$ be an odd prime number and $\mathbb Q_p$ be the $p$-adic number field. Let $K=\mathbb Q_p(a)$ be the extension by $a=p^{\frac{p^2+1}{p^3-1}}$.
Consider the polynomial $f(x)=(x^p-a)^{p^2}-p^{p^...
- 875
1
vote
1
answer
149
views
When is $R$ a direct summand of Frobenius pushforwards?
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module ...
- 20.5k
2
votes
0
answers
177
views
How do characters of representations in cohomology depend on the (positive-characteristic) field?
The following sentence appears in Jantzen - Representations of algebraic groups, 2nd edition, p. x, where $G$ is a reductive group over an algebraically closed field $k$, $B$ is a Borel subgroup, $T$ ...
- 12.9k
41
votes
2
answers
9k
views
What should I read before reading about Arakelov theory?
I tried reading about Arakelov theory before, but I could never get very far. It seems that this theory draws its motivation from geometric ideas that I'm not very familiar with.
What should I read ...
- 760