All Questions
2,494 questions
-1
votes
1
answer
342
views
Finite or polynomial number integral points clarification on Coppersmith's theorems (possibility of ellipse counter example?)
Coppersmith states if $f(x,y)$ is an irreducible bivariate with total degree $\delta$ then if he can list all roots $(X,Y)$ of the polynomial in $\mathsf{poly}(\log D,\delta)$ time if the roots ...
- 13.9k
3
votes
0
answers
87
views
Obstruction for points to be contained in smooth hypersurfaces in tterms of inseparability degree of residue field
Let $k$ be an imperfect field of char $p>0$ and
$x \in \mathbb{P}^n_k$ be closed point of projective space.
In this discussion Qing Liu wrote that
Over an imperfect field, a reduced point can not ...
- 619
4
votes
0
answers
179
views
Reference for Iwahori-Hecke algebras
I recently came across the notion of an Iwahori-Hecke algebra. I would like to learn the basics about this type of algebras (mainly to get an intuition about them, as they seem to be related to some ...
- 541
14
votes
2
answers
568
views
Does every non-isotrivial 1-parameter family of elliptic curves have a positive rank specialization?
Let $\mathcal{E}/\mathbb{Q}(t)$ be given by $$y^2=x^3+A(T)x+B(T)$$ for some $A(T),B(T)\in\mathbb{Q}[T]$ and assume $\mathcal{E}$ is non-isotrivial (the $j$-invariant $\frac{6912 A(T)^3}{4A(T)^3 + 27B(...
- 1,403
4
votes
0
answers
226
views
The coarse moduli schemes of the "Shimura stacks" are the canonical models of the corresponding Shimura varieties
Let $F$ be a number field, $B$ a central simple algebra over $F$, $*$ a positive involution on $B$ which fixes $F$, and
$O_B$ a maximal $O_F$-order of $B$ which is stable under $*$.
Assume that $(B, *)...
- 1,364
7
votes
1
answer
627
views
Weil height vs Moriwaki height
Let $X$ be a projective veriety over a number field. After fixing an embedding into $\mathbb P^n$ (i.e. a very ample line bundle $L$), one can define the Weil height $\hat h_{L}$ by restriction of the ...
- 1,237
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\...
- 541
21
votes
1
answer
2k
views
Cohomology of Shimura varieties and coherent sheaves on the stack of Langlands parameters
In Zhu's Coherent sheaves on the stack of Langlands parameters theorem 4.7.1 relates the cohomology of the moduli stack of shtukas to global sections of a certain sheaf on the stack of global ...
- 3,309
13
votes
2
answers
950
views
Is there an $\mathbb{R}$-valued cohomology theory for varieties over $\mathbb{F}_p$?
If $E$ is a supersingular elliptic curve over $\mathbb{F}_{p^m}$ with $m\geq 2$ its endomorphism ring is a maximal order in a quaternion algebra ramified at $p$ and $\infty$ so there can't be a Weil ...
user158636
3
votes
1
answer
443
views
The Weil restriction of a simple algebraic group
Let $F$ be a number field, $G$ an $F$-simple affine algebraic group.
Then is the Weil restriction $\operatorname{Res}_{F/\mathbb{Q}} G$ $\mathbb{Q}$-simple?
I couldn’t find any references.
- 185
3
votes
0
answers
145
views
A Galois equivariant Weil cohomology theory with coefficients in the rational numbers and a variation of the Tate/Hodge conjecture
A well-known example of Serre shows that there can be no Weil cohomology theory with $\mathbb Q$ coefficients for schemes over $\mathbb F_{p^2}$. However, this example is no obstruction to a Weil ...
- 7,746
4
votes
2
answers
500
views
Smoothness of fibers over finite fields
Let $f:X\rightarrow Y$ be a morphism of smooth projective varieties over a finite field of characteristic different from $2$. Is there any result on the existence of a point $y\in Y$ such that $X_y = ...
user58018
7
votes
2
answers
788
views
Reference request: the geometry of vanishing cycle
I’m currently studying basics on étale cohomology, by Fu’s and Milne’s book. The formalization of vanishing cycle and nearby cycle particularly interests me. I realized it may relate with reduction ...
- 375
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
88
views
Generic finite subgroups, associated to small finite fields, of reductive algebraic groups
Theorem 1 of [LS] Liebeck and Seitz - On the subgroup structure of exceptional groups says:
Let $X = X(q)$ be a quasisimple group of Lie type in characteristic $p$, and suppose that $X < G$, where ...
- 12.9k
1
vote
1
answer
141
views
Power series corresponding to $[a]\in \operatorname{End}(E)$ ($a \in R_K$) can be expressed as $[a](t)=at+\text{(term higher than degree $2$)}$?
Let $K$ be an imaginary quadratic field and $E/K$ be an elliptic curve which has complex multiplication on $K$.
Let $R_K$ be ring of integers of $K$.
Let $ \hat{E}$ be its formal group of $E$.
Take $...
- 1,541
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
8
votes
3
answers
1k
views
Further reading in algebraic geometry
I recently finished reading W. Fulton's "Algebraic Curves" and also attended a lecture series on moduli spaces and am interested in learning about them as well. I looked for a few books to ...
-4
votes
2
answers
405
views
Do these irrationals exist?
An irrational $a$ verifies : $\{a\times n+k;(n,k)\in\mathbb Z^2 \}$ is dense in $\mathbb R$.
If you take $a$ universe then : $\forall b\in \mathbb N^*, \{a\times n^{b}+k;(n,k)\in\mathbb Z^2\}=A(a,b)$ ...
- 4,073
5
votes
2
answers
572
views
Birational geometry over finite fields
I apologize in advance since probably my questions are very naive. I would like to understand some central notions in birational geometry, that are clear to me over the complex numbers, for varieties ...
user114666
3
votes
0
answers
110
views
Local global principle over infinite extension of $\Bbb{Q}$ which is not algebraically closed
Let $A$ be an algebraic variety over a field $K$, which is finite extension of $ \Bbb{Q}$.
We say local global principle holds if $A(K_v) \neq \emptyset$ implies $A(K) \neq \emptyset$, where $K_v$ is ...
- 1,541
14
votes
3
answers
2k
views
Motivation for the Jacobian Variety
I've been to many talks in Number Theory and for some reason I've yet to fully grasp, we all seem to like Jacobian Varieties a lot. I know that they are Abelian varieties, which give information about ...
- 901
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
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
4
votes
1
answer
227
views
Compute de Rham-Witt sheaves
I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction.
It seems to be really difficult to compute anything with those beasts: like I cannot find any ...
- 309
0
votes
0
answers
97
views
Relation between divisibility problem of Shafarevich group and group structure of $Ш(E/K)$
For abelian variety $A/K$,
divisibility problem (i.e. $\forall n≧1$, $Ш(A/K)⊂p^nH^1(G_K,A)$ holds for fixed prime $p$?) was asked by Cassels in 1962 and even now discussed.
On the other hand, once ...
- 1,541
28
votes
1
answer
984
views
Relation between Schanuel's theorem and class number equation
(Crossposted on math stack exchange: https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation)
It was recently brought to my attention that there ...
- 595
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
2
votes
1
answer
189
views
On the upper-bound for a type of quintuple Kloosterman sums
Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound.
My question is, for any $x,y,z,w,\delta \in \mathbb{Z}$ ...
- 563
3
votes
1
answer
153
views
Effective Mordell for (twists of) hyperelliptic curves in $$\mathbb{P}(1,1,g+1)$$
We consider the following model for a hyperelliptic curve:
$$\displaystyle C_F : z^2 = F(x,y), \deg F = 2g+2, F \in \mathbb{Z}[x,y]$$
with $F$ homogeneous and non-singular. We also consider twists of ...
- 27k
35
votes
1
answer
2k
views
The modularity theorem as a special case of the Bloch-Kato conjecture
In the homepage for the CRM's special semester this year, I found the interesting statement that the modularity theorem (formerly the Taniyama-Shimura-Weil conjecture) is a special case of the Bloch-...
- 3,309
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 $\mathscr{F}$ ...
- 185
4
votes
1
answer
219
views
Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$?
Suppose that $\pi:X\to S$ is a smooth projective morphism of relative dimension 1. If $S$ is the spectrum of an algebraically closed field, then it is known that $X$ embeds into $\mathbb{P}^3_S$.
...
- 1,142
6
votes
1
answer
395
views
On the Artin-Rees Lemma for non-commutative rings
Consider a commutative noetherian ring $A$ with an ideal $I\subset A$. The Artin-Rees lemma implies that for f.g. modules $N\subset M$, the $I$-adic topology on $N$ agrees with the subspace topology ...
- 541
5
votes
0
answers
481
views
What does Colmez's conjecture tell us?
There is the well known conjecture of Colmez, which describes the logarithmic derivative of the $L$-function of a character via the periods of CM-abelian varieties. Equivalently it describes the ...
- 4,428
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 ...
- 413
11
votes
3
answers
1k
views
Are "large enough" finite etale covers arithmetic?
Let $X$ be a variety over a number field $K$. Then it is known that for any topological covering $X' \to X(\mathbb{C})$, the topological space $X'$ can be given the structure of a $\overline{K}$-...
- 437
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
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 ...
- 541
3
votes
1
answer
359
views
Description of a Shimura variety
Let $(G, X)$ be a Shimura datum and let $U \subseteq G(\mathbb A_f)$ be an open compact subgroup. By the general theory of Shimura varieties, we get a corresponding algebraic variety $Y(U)$ defined ...
- 133
43
votes
1
answer
4k
views
A mysterious connection between Ramanujan-type formulas for $1/\pi^k$ and hypergeometric motives
The question below is the follow-up of this question on MathOverflow.
Motivation: As is stated in the former question, those identities(formula (35)-(44)) of $1/\pi$ attributed to Ramanujan are ...
- 3,337
5
votes
1
answer
483
views
Does p-adic etale cohomology know the variety has ordinary reduction or not?
For a smooth proper variety $X$ over discrete valuation ring $\mathcal{O}$ of mixed characteristic $(0,p)$, let $X_K$ be the generic fibre over a generic point $\textbf{Spec} K$ and let $X_k$ be the ...
- 349
3
votes
1
answer
150
views
Selmer groups and fppf cohomology
Let $\mathcal{O}$ be a Dedekind domain and $K = \mathrm{Frac}(\mathcal{O})$ its field of fractions. Let $E / K$ be an elliptic curve and $\mathcal{E} / \mathcal{O}$ its Neron model and $\mathcal{E}^\...
- 3,730
16
votes
2
answers
2k
views
Applications of integral p-adic Hodge theory
What are some geometric applications of integral p-adic Hodge theory (as opposed to rational p-adic Hodge theory)? Answers which understand Hodge theory as the study of Galois stable $\mathbb{Z}_p$-...
- 375
4
votes
1
answer
647
views
A comparison theorem between crystalline cohomology and étale cohomology
Suppose $X/\mathbb F_q$ is a smooth projective variety. Katz-Messing (eudml) shows that the characteristic polynomial of the Frobenius on $H^i_{et}(\overline{X},\mathbb Q_\ell)$ and $H^i_{crys}(X)$ ...
- 7,746
1
vote
1
answer
229
views
Purity for proper varieties
Let $X$ be a proper, geometrically connected, geometrically integral variety over $\mathbf{F}_q$. There exists a finite field extension $k/\mathbf{F}_q$ of degree $d$ and an alteration $X'\to X_k$ ...
user480741
1
vote
0
answers
46
views
Weight of adjoint action on a lower central series extension
Let $\mathcal{U}$ be a unipotent Lie $\mathbb{Q}_p$-group scheme, whose associated gradeds from the lower central series filtration are $\mathcal{U}_0 = \mathcal{U}^{\text{ab}}$, $\mathcal{U}_1 = [\...
- 2,907
3
votes
1
answer
719
views
Number of points of a quadric hypersurface over a finite field
Let $k = \mathbb{F}_q$ be a finite field with $q$ elements and $Q\subset\mathbb{P}^n_k$ a quadric hypersurface defined over $k$.
By the Chevalley-Warning theorem if $n\geq 2$ then $Q$ has a point. Is ...
- 8,998
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
241
views
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