All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
2
votes
0
answers
114
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Rational points on a weighted projective surface
The equation
$$\displaystyle y^2 = f(x_1, x_2, x_3)$$
with $f$ a non-singular quartic form in three variables defines a del Pezzo surface of degree 2. I am interested in the similar construction
$$\...
- 26.9k
2
votes
0
answers
258
views
Is a reductive group scheme always parahoric?
Let $R$ be complete (or, more generally, Henselian) discrete valuation ring with fraction field $K$. Let $G$ be a reductive $R$-group scheme. Is $G$ a parahoric in the sense of Bruhat-Tits? If so, ...
- 407
7
votes
0
answers
118
views
Explicit algebraic constructions of Parshin covers
Let $X$ be an algebraic curve defined over a number field $K$ and has genus $g \geq 2$. Let $P$ be a $K$-point of $X$. We say that $X_P$ is a Parshin cover of $X$ if $X_P$ is defined over a finite ...
- 26.9k
2
votes
0
answers
183
views
Solving solutions to systems of polynomial equations over $\mathbb Z$
Macaulay Resultants help identify common root in $\mathbb P^{n-1}(\mathbb K)$ of $n$ homogeneous polynomials in $n$ variables when $\mathbb K$ is algebraically closed. Is it possible that some type of ...
- 13.9k
12
votes
2
answers
1k
views
How to visualize finiteness of class number?
As the question title asks for, how do others "visualize" the finiteness of class number with algebro-geometric insight? I just think of it as a result in algebraic number theory and not one in ...
11
votes
2
answers
1k
views
mod p etale cohomology of the special fiber and the generic fiber
Let $O$ be a valuation ring with fraction field $K$ of characteristic zero and residue field $O/m=k$ of characteristic $p>0$, and $X$ be a proper smooth scheme over $O$. Then can we control the mod ...
- 6,622
3
votes
0
answers
132
views
Is there a way to reduce this problem to two variables through functions coming from arithmetic?
Consider following diophantine equation in $\mathbb Z[x,y,z]$ in three integer variables $x,y,z$
$$x^2+L(y,z)x+L_1(y)L_2(z)=0$$ where $L(y,z)$ is a non-homogeneous linear polynomial in $y,z$ and $L_1(...
- 13.9k
10
votes
0
answers
286
views
Published reference on the automorphism group of modular curves $X_1(N)$?
I wish to cite that the automorphism groups of $X_1(N)$ have already been completely calculated, and what they are, but I am having difficulty finding this calculation in the literature.
I have ...
- 3,539
2
votes
0
answers
95
views
Polynomials passing through points with tangential conditions
In corollary here http://math.mit.edu/~lguth/Exposition/erdossurvey.pdf on polynomial methods it is said "(Parameter counting) If $S\subset\mathbb F^n$ is a finite set, then there is a non-zero ...
- 13.9k
9
votes
0
answers
684
views
Why are the open and closed adic discs defined the way that they are?
The closed adic disc is defined as $Spa(\mathbb{Q}_p\langle T\rangle,\mathbb{Z}_p\langle T\rangle)$, and the open adic disc is defined to be the fiber $Spa(\mathbb{Z}_p[[T]],\mathbb{Z}_p[[T]])_{\eta}$ ...
- 2,071
9
votes
0
answers
327
views
What role, if any, do Archimedean valuations play in adic spaces?
I've been reading about adic spaces, and I couldn't help but wonder what would happen to the theory if one included in the definition of $Spa$ Archimedean valuations as well...?
Is there a weird ...
- 2,071
20
votes
1
answer
902
views
Double Counting: Motivic Edition
One of the most important proof techniques in combinatorics is double counting: proving that both sides of an identity count elements of some set in two different ways. This question is an attempt at ...
- 85.6k
5
votes
1
answer
300
views
How to show that a hypersurface is a diagonal intersected with hyperplanes?
Suppose I have a hypersurface $V(F) = \{ \mathbf{x} \in k^n: F(\mathbf{x}) = 0 \}$, where $F$ is a homogeneous form of degree $d > 1$. I would like to show that there exists some diagonal form $D(...
- 3,625
11
votes
1
answer
3k
views
Books with exercises to learn Langlands program, Galois representations, modular forms
I want to develop a basic background in number theory with a goal towards contributing to some part of the Langlands program (which I know is vast, and has many different aspects; I want to do ...
- 785
36
votes
2
answers
3k
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 ...
13
votes
2
answers
2k
views
What is the best reference for motives?
I want to learn about homotopy theory on number fields, and I heard that the theory of motives made it possible, so I want to know what is a good textbook for motive theory.
To be honest, I don’t ...
- 153
15
votes
0
answers
665
views
Étale cohomology of varieties in positive characteristic via singular cohomology
Suppose $X$ is a smooth scheme, not necessarily projective, over $\mathbb{Z}[1/N]$ for some integer $N\neq 0$. I would like to understand the cohomology groups $H^i_{ét}(X_{\overline{\mathbb{F}}_p}, \...
- 666
14
votes
1
answer
2k
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 ...
5
votes
1
answer
184
views
Is the function field of every congruence modular curve generated by $j,j\circ g$ for some $g\in\text{GL}_2(\mathbb{Q})^+$?
So I've heard in passing that for any congruence modular curve $X$ (over $\mathbb{C}$), there is a $g\in\text{GL}_2(\mathbb{Q})^+$ such that $X$ is birational to a plane curve in $\mathbb{C}^2$ given ...
- 10.7k
6
votes
1
answer
705
views
Faltings theorem and number of singularities
The Faltings theorem states that the number of rationals over an algebraic curve is finite if the genus is greater than 1. The genus decreases by increasing the number of singularities. My question is ...
- 1,207
2
votes
1
answer
568
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}_{\...
- 445
19
votes
3
answers
1k
views
Points of elliptic curves over cyclotomic extensions
Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It ...
- 2,305
6
votes
1
answer
486
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 ...
- 3,337
5
votes
1
answer
284
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 ...
- 71
5
votes
1
answer
1k
views
Torsion subgroup of the group of points of an elliptic curve over local field
Let $K$ be a local field with residue field $k$ and $E/K$ an elliptic curve. I'm interested for which $K$ and $E$ the group of torsion points on the curve is finite. I can prove that this group is ...
- 2,305
1
vote
1
answer
106
views
Which positive integers can occur as the genus of a numerical semigroup minimally generated by 3 (or 2) elements?
Let $S$ be a numerical semigroup. Let $g(S)=|\mathbb N \setminus S |$, where $\mathbb N$ here denotes the set of non-negative integers. Let $e(S)$ be the embedding dimension of $S$, i.e. the ...
user111492
2
votes
1
answer
520
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 ...
- 371
3
votes
1
answer
226
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$ ...
- 31
5
votes
0
answers
201
views
The Geometry of Jacobi Forms and their Asymptotic Expansions
A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying
$$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}...
- 1,701
5
votes
0
answers
275
views
Reference request: Tate's conjecture for L functions of motives
What's a good reference for the most general form of Tate's conjecture for the order of poles of the L function of a motive? Thanks!
- 1,362
0
votes
1
answer
125
views
Polyhedral conditions for quadratic inequalities in fixed dimension
Denote $\mathcal T$ be set of $(T_1,T_2,T_3,T_4)\in\mathbb Z^4$ that satisfy
$$0<T_1,T_2,T_3,T_4$$
conditions?
Define the level set $$M_{\gamma}(Q,\mathcal T)=\{(T_1,T_2,T_3,T_4)\in\mathcal T:Q(...
- 13.9k
42
votes
3
answers
5k
views
The Origin(s) of Modular and Moduli
In mathematics and in physics, people use the terms "modular..." and "moduli space" very often. I was puzzled by the etymology, the origins and the similarity/equivalence/differences for these usages/...
- 10.5k
7
votes
1
answer
366
views
index of smooth varieties
What are the simplest examples of smooth, projective varieties defined over the fraction field of an Henselian DVR of characteristic $0$ which have index $>1$?
EDIT: Also assume that the residue ...
- 1,981
4
votes
2
answers
1k
views
Frobenius actions on de Rham cohomology, clarify questions on a paper of Kedlaya
I am reading the paper "$p$-adic cohomology: from theory to practice" by K. S. Kedlaya. I have several naive questions about section 2: Frobenius action on de Rham cohomology. As a physicist, I lack ...
- 2,971
3
votes
1
answer
424
views
Moduli problem of stable nodal curves over the integers
Over an algebraically closed field of characteristic zero, e.g. $\overline{\mathbb{Q}}$, the Deligne-Mumford stack $\overline{\mathcal{M}}_{g,n}$ represents the functor $$\overline{\mathcal{M}}_{g,n}(...
- 7,952
1
vote
0
answers
110
views
Complex multiplication of Jacobians of quadratic twists of a fixed hyperelliptic curve
Let $C: z^2 = f(x,y)$ be a hyperelliptic curve defined over $\mathbb{Q}$, with $f$ a binary form with integer coefficients and non-zero discriminant. Let $A_C$ denote the Jacobian variety of $C$, and ...
- 26.9k
5
votes
0
answers
440
views
Why does Faltings' Siegel lemma imply Siegel lemma?
Recall the Siegel lemma:
Let $A = (a_{ij})$ be an $N \times M$ matrix with rational integer coefficients. Put $a = \max_{i,j} |a_{ij}|$. Then, if $N < M$, the equation $Ax = 0$ has a solution $x\in\...
- 3,998
8
votes
1
answer
796
views
Tate modules of commutative group schemes over finite field
Let $G$ be a finite type commutative group scheme over a finite field $k=\Bbb F_q$ with $\Gamma=Gal(\bar k/k)$ and $l$ be a prime such that $(l,q)=1$, we can also define the Tate module $T_lG =\...
- 6,622
4
votes
1
answer
382
views
Reference to the conjecture about injectivity of Abel-Jacobi map
Suppose $k$ is a number field, and $\sigma:k \rightarrow \mathbb{C}$ is an embedding. Then there is the (generalised) Abel-Jacobi map
\begin{equation}
\text{CH}^j(X)_0 \rightarrow \frac{H^{2j-1}((X \...
- 2,971
3
votes
1
answer
671
views
Endomorphisms of abelian varieties with real multiplication
Let us work over $\mathbb{C}$ to make life easier.
I've came across to the following definition. Let $F$ be a totally real number field of degree $g$, with ring of integers $\mathcal{O}_F$. An ...
user120787
6
votes
2
answers
781
views
Books building up to the Gross-Zagier formula
I am an undergrad extremely interested in some applications of the Gross-Zagier formula for elliptic curves. I have a strong foundation in group theory and abstract algebra, and an understanding of ...
- 169
7
votes
0
answers
432
views
Failure of integral comparison between crystalline and de Rham cohomology over a highly ramified base
Let $K$ be a finite extension of $\mathbb{Q}_p$ with the ring of integers $\mathcal{O}_K$ and the residue field $k$.
By a theorem of Berthelot and Ogus(https://link.springer.com/article/10.1007%...
- 7,377
4
votes
1
answer
370
views
Poincare duality for mixed motives
Suppose $k$ is a field of characteristic zero (and we assume it is a number field if necessary). If $U$ is a smooth quasi-projective variety over $k$, then there is Poincare duality,
\begin{equation}
...
- 2,971
4
votes
0
answers
192
views
A question on Nekovar's paper Belinson's Conjectures
In Section 2 of Nevovar's paper "Beilinson's Conjectures", for a pure motive $M$ of the form $h^i(X)(n)$ where $X$ is a projective smooth variety over $\mathbb{Q}$ and $n$ is an integer such that the ...
- 2,971
13
votes
1
answer
760
views
Infinitely many integer solutions to $X^4+Y^4-18Z^4= -16$
We found infinitely many integer solutions to
$$X^4+Y^4-18Z^4= -16 \qquad (1)$$.
The interesting part in this diophantine equation is the sum of
the reciprocals of the degrees is $3/4 < 1$, which ...
- 25.4k
0
votes
0
answers
328
views
Mixed motives and motivic cohomology
In Scholl's paper "Remarks on special values of $L$-functions", he defines that an object $M$ of $\textbf{MM}_{\mathbb{Q}}$ (the conjectured abelian category of mixed motives with coefficients $\...
- 2,971
4
votes
1
answer
415
views
How to check whether a mixed motive is defined over $\mathbb{Z}$
Suppose $M$ is an object of the (conjectured) abelian category of mixed motives over $\mathbb{Q}$, $\textbf{MM}_{\mathbb{Q}}$, Scholl defines that $M$ is defined over $\mathbb{Z}$ if the following ...
- 2,971
4
votes
1
answer
489
views
A question on Deligne's paper "Valeurs de fonctions L et périodes d'intégrales"
I have asked this question on Math StackExchange, but have not got any reply.
In section 1.7 of Deligne's paper "Valeurs de fonctions L et périodes d'intégrales", which has an English translation (...
- 2,971
8
votes
3
answers
1k
views
Sufficient conditions for a polynomial to be reducible over the integers
There are several well-known criteria for a polynomial with integer coefficients to be irreducible over $\mathbb{Z}$, e.g., Eisenstein's criterion. I'm looking for the opposite: other than ...
- 1,703
3
votes
0
answers
162
views
On what varieties are the conjectures on $L$-functions true
In Peter Schneider's paper "Introduction to the Beilinson conjectures", he lists some hypothesis (conjectures) on the $L$-function associated to the pure motive $H^i(X)$ where $X$ is a smooth ...
- 2,971