All Questions
Tagged with nt.number-theory ag.algebraic-geometry
1,746 questions
3
votes
1
answer
222
views
Large integral points on the quadratic twist $ D y^2=x^3+A x +B$
For integers $A,B,D$ and $D$ squarefree let $E : y^2=x^3+A x + B$
and $E_D$ be the quadratic twist of the elliptic curve $E$:
$$ E_D : D y^2=x^3+Ax +B$$
$E_D$ is isomorphic to $ E'_D : y^2=x^3+D^2 A ...
- 25.4k
0
votes
0
answers
54
views
Functional equations with coupled arguments and additive sructure
Let $G$ be a locally compact abelian group and let $f: G \to \mathbb{R}^+$ be a continuous function satisfying the functional equation
$$f(x + \phi(y)) + f(y + \phi(x)) = 1 + f(x+y)$$
for all $x, y \...
1
vote
0
answers
87
views
Equidistribution of Frobenius Classes
Let $G$ be a reductive group over $\mathbb{Q}$. Let $K$ be a maximal compact subgroup of $G(\mathbb{C})$. Let $S$ be a finite set of primes. For each prime $p$ not in $S$, let $Frob_p$ be a conjugacy ...
- 11
3
votes
1
answer
350
views
Elements of $\mathbb{F}_p$ represented by an irreducible polynomial $f(x) = x^3 +a_2 x^2 + a_1 x + a_0$, $f(x) \in \mathbb{F}_p[x]$
Let $p \equiv 1 \bmod 3$ be a prime, $\mathbb{F}_p$ be the finite field with $p$ elements, and $a_0$ be a generator of $\mathbb{F}_p^{\times}$ $(\mathbb{F}_p^{\times}$ the group of nonzero elements of ...
- 31
2
votes
1
answer
150
views
Closure of specialization of points of an affine group scheme with smooth generic fiber
Let $R$ be a henselian discrete valuation ring with residue field $k$, and let $G$ be an affine faithfully-flat finite type group scheme over $R$ with smooth generic fiber. Let $R'$ be the ring of ...
- 7,527
2
votes
1
answer
198
views
The action of $\operatorname{Gal}(\overline{k}/k)$ on $\pi_1^{\mathrm{ét}}(\mathbb{G}_m)$
$\newcommand{\et}{\mathrm{ét}}$Let $k$ be a number field. Let $\mathbb{G}_m=\operatorname{Spec}(k[t^{\pm1}])$. The homotopy exact sequence for the étale fundamental group is given by
$$1\...
- 71
4
votes
1
answer
338
views
Bounds on quadratic character sums
I asked this question on Mathematics stack exchange but didn't get a response, so I ask here too.
Let $\chi$ be the non-trivial quadratic character of $\mathbb{F}_q$, and let $f(x)$ be a square-free ...
- 153
2
votes
1
answer
161
views
Fixed $a_p=p+1-\#E(\mathbb{F}_p)$ and $a_p \ne 0$ on an elliptic curve infinitely often for fixed curve over the rationals?
In this and this question we show that if $p=27a^2+27a+7$ is prime, then the order of the elliptic curve
$y^2=x^3+2$ modulo $p$ is either $p$ or $p+2$.
Q1 Can we unconditionally show that the order ...
- 25.4k
5
votes
0
answers
234
views
Triviality of $\unicode{1064}(T_pE ⊗ T_pE)$ for elliptic curves and Bogomolov's lemma
Consider the case of an elliptic curve $E$ over Q, and let $S$ be a finite set of primes including all places of bad reduction and a place $p$ of good reduction.
Bogomolov's Lemma says that when $p$ ...
- 2,907
1
vote
0
answers
195
views
When does the formal group of an abelian variety possess integral coefficients?
I am looking for a sufficient condition that ensures the formal group associated with an abelian variety has integral coefficients.
In precise, let $A$ be an abelian variety over a number field $K$ ...
- 195
9
votes
0
answers
977
views
Geometrization of the global Langlands correspondence?
Fargues-Scholze famously describe arithmetic local Langlands via global geometric Langlands on the Fargues-Fontaine (FF) curve.
The FF curve acts like an algebraic curve over $\mathbb{C}_p$ (its ...
- 15.4k
18
votes
2
answers
2k
views
What is the taxicab number for rational fourth powers?
The taxicab number is the smallest integer that can be expressed as a sum of two positive integer cubes in two different ways, and it is equal to $1729=12^3+1^3=10^3+9^3$. There are generalizations to ...
- 7,182
1
vote
0
answers
127
views
Does the torsion points of abelian varieties transfer to their formal group laws (upon suitable choice of coordinates)?
Let $A$ and $B$ be two abelian varieties over any algebraically closed field. Let $A[p^n]$ and $B[p^n]$ denotes the set of $p$-power torsion points of $A$ and $B$. Assume that $A[p^n]$ and $B[p^n]$ ...
- 195
1
vote
0
answers
137
views
Syntomic f-cohomology for open varieties
Syntomic cohomology $H^{i+j}_{\mathrm{syn}}(X,n)$ of a proper variety $X$ with good reduction over a $p$-adic field $K$ is computed via a spectral sequence in terms of $H^i_{\mathrm{f}}(G_K;H^j_{\...
- 15.4k
7
votes
1
answer
361
views
Shouldn't we expect analytic (in the Berkovich sense) étale cohomology of a number field to be the cohomology of the Artin–Verdier site?
Let $K$ be a number field. Consider $X=\mathcal{M}(\mathcal O_K)$ the global Berkovich analytic space associated to $\mathcal O_K$ endowed with the norm $\|\cdot\|=\max\limits_{\sigma:K \...
- 804
17
votes
3
answers
2k
views
Are some congruence subgroups better than others?
When I first started studying modular forms, I was told that we can consider any congruence subgroup $\Gamma\subset\operatorname{SL}_2(\mathbb{Z})$ as a level, but very soon the book/lecturer begins ...
- 821
4
votes
0
answers
111
views
Group structure on $\mathbb{Z}$-points of an algebraic torus over $\mathbb{Z}[1/N]$
Consider the affine conic $C\subset\mathbb{A}^2_\mathbb{Z}$ cut out by $x^2 + axy + y^2 + b$, where $a,b\in\mathbb{Z}$.
Assume that $a\ne \pm 2$, and that $C$ admits an integral point $(x_0,y_0)$. The ...
- 7,527
3
votes
0
answers
270
views
Isomorphism between two K3 surfaces in characteristic $11$ and the action of $\operatorname{PSL}(2, \, \mathbb{F}_{11}) $
We work over a field $k$ with $\operatorname{char}(k)=11$.
In the paper [1], Lemma 3.5, it is shown that the K3 surface $X_0$ defined as the weighted projective hypersurface of degree $12$ $$X_0=V(t_0^...
- 66.3k
17
votes
2
answers
2k
views
How to think of algebraic geometry in characteristic p?
How does a working mathematician usually think about algebraic geometry in characteristic $p$? For the sake of concreteness, and to make things more "geometric" (whatever that means), let's ...
3
votes
0
answers
346
views
Modern integral $p$-adic Hodge theory and modularity lifting and Fontaine-Mazur
As a follow-up to a comment on this answer, I'm wondering if there are expected to be applications of the new point of view on integral $p$-adic Hodge theory, à la Bhatt-Morrow-Scholze and others, to ...
- 15.4k
3
votes
0
answers
174
views
On the sheaves-functions dictionary
Let $X$ be a variety over a finite field $k$. Let $\pi_{1}(X)$ be the arithmetic etale fundamental group of $X$, and $\rho:\pi_{1}(X)\to k^{\times}$ a continuous character. If $x: \text{Spec}(k)\to X$ ...
- 667
2
votes
1
answer
319
views
Bounding $H^4_{\text{ėt}}$ of a surface
Let $X\longrightarrow X'$ be a smooth proper map of smooth proper schemes defined over $\mathbb{Z}[1/S]$, where $S$ is a finite set of primes. Assume $X'$ is a curve of positive genus, and $X$ is a ...
- 2,907
5
votes
1
answer
228
views
Lifting mod $p$ representations of arithmetic fundamental groups of a non-affine scheme over a finite field of characteristic $p$
Let $X$ be a geometrically irreducible scheme (not necessarily affine) over $\mathbb{F}_{p}$ and let $ \pi_{1}(X) $ be the arithmetic etale fundamental group of $ X $. Let $ \overline{\mathbb{F}}_{p} $...
- 863
2
votes
0
answers
262
views
Abelian extensions of number fields generated by torsion points of elliptic curve (as analogy to Lubin-Tate theory)
According to a remark from wikipedia the motivation of Lubin-Tate theory arose from the analogy to the way in which elliptic curves $E/K$ over a number field $K$ with extra endomorphisms (ie those ...
- 6,028
1
vote
0
answers
88
views
Identification of different components of Hilbert modular surface?
I'm wondering whether the different components of the Hilbert modular surface can be (naturally?) identified with each other, or if they're at least abstractly isomorphic. (I'd also be interested in ...
- 2,044
2
votes
2
answers
270
views
Finding rational points on intersection of quadrics in affine 3-space
Consider the subvariety of Spec $\mathbb{Q}[x,y,z]$ cut out by the equations
\begin{eqnarray*} f_1: a_1x^2 - y^2 - b_1^2 & = & 0 \\
f_2 : a_2x^2 - z^2 - b_2^2 & = & 0
\end{eqnarray*}
...
- 7,527
296
votes
8
answers
143k
views
Philosophy behind Mochizuki's work on the ABC conjecture
Mochizuki has recently announced a proof of the ABC conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. Can someone briefly explain the philosophy ...
3
votes
1
answer
370
views
Bloch–Beilinson conjecture for varieties over function fields of positive characteristic
Is there a version of the Bloch–Beilinson conjecture for smooth projective varieties over global fields of positive characteristic? The conjecture I’m referring to is the “recurring fantasy” on page 1 ...
- 531
0
votes
0
answers
108
views
Isogeny classes for elliptic curves over quadratic field
Question. Is it possible for an elliptic curve $E$ over quadratic field $K$ to have two separate (yet connected) isogeny classes?
There are two $\mathbb{Z}/14\mathbb{Z}$ elliptic curves, $E_1$ and $...
- 913
1
vote
1
answer
469
views
Does $\sum_{n \geq 0} a_n x^n=\sum_{n \geq 0} b_nx^n$ imply $a_n=b_n$ for vector-tuple power series?
My reference is Infinite series in p-adic fields by Keith Conrad.
Corollary 5.6. If $f(x)=\sum_{n≥0} a_nx^n$ has a positive radius of convergence in the $p$-adic field $\mathbb Q_p$ then $f$ is ...
- 930
1
vote
1
answer
154
views
Divisors on product abelian fourfolds
Given a principally polarized abelian surface $A$ with CM of signature $(1,1)$ by an imaginary quadratic number field $K$, I am interested in studying the Néron-Severi group $\text{NS}(A\times A)$. ...
- 91
4
votes
0
answers
166
views
Semistability of Frey curves: why no additive reduction?
Let $(a,b,c)$ be a hypothetical nontrivial integer solution to the Fermat equation $x^p + y^p + z^p = 0$, where $p \geq 5$ is prime, and assume $a, b, c$ are (pairwise) coprime. From this solution, we ...
- 283
5
votes
3
answers
606
views
Can you describe all rational solutions to these simple-looking equations?
Can you describe, in parametric form or in any other explicit way, all rational solutions to any of the following equations:
$$
y^2 + z^2 = x^3+1,
$$
$$
y^2 + z^2 = x^3-1,
$$
$$
y^2+x^2y+z^2+1=0.
$$
...
- 7,182
0
votes
0
answers
126
views
Estimate of a Weil height
Found in a textbook. Since I am learning heights now, I thought it is a good exercise to pratice. But I am far away to solve it. Let $y\in{\overline{\mathbb Q}}^\times$. Prove that there exists $C>...
- 3,998
27
votes
4
answers
3k
views
Why do we care about the eigenvalues of the Frobenius map?
The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}_q$ for some $q=p^n$. Then the eigenvalues $\...
- 1,969
2
votes
0
answers
233
views
Representability of moduli problem of elliptic curves with complex multiplication
I'd like to know whether the moduli problem for elliptic curves with complex multiplication by a fixed imaginary quadratic number field $K$ (and with suitable level structure to be picked) is ...
- 91
2
votes
0
answers
150
views
Absolute Bloch-Kato Cohomology
The étale cohomology $R\Gamma_{\mathrm{ét}}(X;\mathbb{Z}_p(n))$ of a scheme $X/K$ can be computed by a Hochschild-Serre spectral sequence with terms of the form $H^i(K;H^j(X_{\overline{K}};\mathbb{Z}...
- 15.4k
11
votes
1
answer
1k
views
Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions
Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions:
Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...
- 533
6
votes
1
answer
536
views
How to see that Eisenstein series are eigenfunctions of the laplacian?
Let $\Gamma$ be a discrete subgroup of $PSL_2(\mathbb{R})$ of finite type. Let $c_1,\ldots,c_h\in\mathbb{R}\cup\{\infty\}$ be a set of representatives of the $\Gamma$-equivalence classes of cusps. For ...
- 7,527
40
votes
1
answer
10k
views
What actually is the idea behind the condensed mathematics?
Condensed mathematics is the (potential) unification of various mathematical subfields, including topology, geometry, and number theory. It asserts that analogs in the individual fields are instead ...
- 1,525
7
votes
1
answer
488
views
Relationship between Serre-Tate coordinates of ordinary elliptic curves and Tate curves
Let $K$ be a complete extension of $\mathbb{Q}_{p}$ with valuation $v$ over $p$, valuation ring $R$, maximal ideal $\mathfrak{m}$ and residue field $k$. It is well known that if $E/K$ is an elliptic ...
- 305
65
votes
2
answers
9k
views
Who is the "young student" André Weil is referring to in his letter from the prison?
I am reading a nice booklet (in Italian) containing the exchange of letters that André and Simone Weil had in 1940, when André was in Rouen prison for having refused to accomplish his military duties.
...
- 66.3k
70
votes
7
answers
28k
views
Have there been any updates on Mochizuki's proposed proof of the abc conjecture?
In August 2012, a proof of the abc conjecture was proposed by Shinichi Mochizuki. However, the proof was based on a "Inter-universal Teichmüller theory" which Mochizuki himself pioneered. It was known ...
3
votes
1
answer
180
views
Approximating $p$-adic power series by polynomials
Let $p$ be a prime, and let $f \in \mathbb{Z}_p[[x_1,\dots,x_d]]$ be a power series convergent on all of $\mathbb{Z}_p^d$. We make the following definition concerning the approximation of $f$ by ...
0
votes
0
answers
76
views
Largest set of monomials whose span is "co-prime" to a given polynomial
Let $K$ be a number field, and let $F \in K[x_1, \cdots, x_n]$ be a polynomial. For a positive integer $d \geq 3$, define $M(F;d)$ to be the largest positive integer such that there exists a set $S$ ...
- 26.9k
4
votes
1
answer
478
views
Is there an elliptic curve analogue to the 4-term exact sequence defining the unit and class group of a number field?
Let $K$ be a number field. One has the following exact sequence relating the unit group and ideal class group $\text{cl}(K)$:
$$1\to \mathcal{O}_K^\times\to K^\times \to J_K\to \text{cl}(K)\to 1$$
...
- 221
0
votes
0
answers
175
views
Why $k((x,t))$ can not be a local field?
If $k$ is a finite field, then $k((x))$ is a local field, and we can define a discrete valuation on $k((x))$ with respect to which it is complete. It is sometimes called a 1-dimensional local field.
I ...
- 930
1
vote
0
answers
156
views
Does this subset of elliptic curves over $\mathbb{Q}$ have positive proportion?
Let $E: y^2 = x^3 + Ax + B$ be a quasi-minimal elliptic curve over $\mathbb{Q}$, i.e. $\gcd(a^3, b^2)$ is $12$th power free. Furthermore, let $\operatorname{rank}(E) = 1$ and $j(E)=\frac{1728 \times ...
- 61
1
vote
2
answers
643
views
Describe all integer/rational solutions to $x^3+y^3+z^3+t^3+s^3=0$
The question is in the title.
Equation $\sum_{i=1}^n x_i^3 = 0$ has no non-trivial integer solutions for $n=3$. For $n=4$, there are known descriptions of all integer/rational solutions, see
Choudhry, ...
- 7,182
9
votes
0
answers
394
views
Tate's thesis and Riemann-Roch - $\mathrm{GL}_n$ or twisted version?
I recently learned why the Tate's thesis, especially Poisson summation formula, over a function field $F = \mathbb{F}_q(X)$ of a smooth projective curve $X_{/ \mathbb{F}_q}$ implies Riemann-Roch ...
- 2,215