All Questions
2,494 questions
3
votes
0
answers
65
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p-torsion in the Tate-Shafarevich group of supersingular elliptic curves
Let $E$ be a supersingular elliptic curve over $\mathbb{F}_p(t)$. Is something known on the $p$-torsion of the Tate–Shafarevich group in this case? In particular, I would like to know if (or if known ...
- 103
1
vote
0
answers
89
views
Generic reducedness of geometric generic fibre
Let $f:X\to Y$ be a surjective morphism between two projective schemes over a field of characteristic $p>0$. Also assume that $X$ is smooth,$Y$ smooth & irreducible and $f_*\mathcal{O}_X=\...
- 5,986
2
votes
0
answers
139
views
Effective Bombieri-Lang conjecture
The Bombieri-Lang conjecture is the following well-known conjecture:
Let $X$ be a projective variety defined over a number field $K$. Suppose that $X$ is general type. Then $X(K)$, the set of $K$-...
- 27k
1
vote
1
answer
121
views
Rational functions on elliptic curves over global fields with given support
Let $E$ be an elliptic curve over a global field $k$. Let $x_1, \dots, x_r$ be a set of generators of $E(k) / E(k)_{tor}$ (or more generally, a $\mathbb Q$-basis of $E(k)_{\mathbb Q}$), and let $x_0$ ...
- 77
1
vote
0
answers
84
views
Descent of isogenies between p-divisible groups
Let $\mathcal{G}$ be a $p$-divisible group over $K$, which is a finite extension of $\mathbb{Q}_p$. Let $\rho: \text{Gal}(\bar{K}/K)\rightarrow \text{GL}(T_p\mathcal{G})$ be the associated Galois ...
- 11
2
votes
1
answer
125
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Questions about elliptic curves with level-$n$ structure
Let $n$ be a positive integer, which is $4$ or a prime number $l$.
Let $E$ be an elliptic curve defined over a number field $K$.
Assume that all the $n$-torsion points of $E$ are defined over $K$, i.e....
- 77
3
votes
0
answers
77
views
primes that ramify in division fields for hyperelliptic jacobians
Let $C$ be a hyperelliptic curve $y^2=f(x)$ of genus $g\geq 2$ and $\Delta$ the discriminant of $f(x)$. Let $\ell>2$ be a prime that divides $\Delta$ to the order $e:=\operatorname{ord}_\ell(\Delta)...
- 685
3
votes
1
answer
367
views
Variants of Grothendieck section conjecture
Let $X$ be a smooth projective variety defined over a field $k$. We fix the following notations : $\overline{k}$ denotes the algebraic closure of the field $k$, $X_{\overline{k}}$ denotes the variety $...
- 443
2
votes
1
answer
127
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Changing the weight space for an eigenvariety
Let $G$ be an algebraic group (like $\operatorname{GL}_2$ or $\operatorname{GL}_2 \times\operatorname{GL}_2$ for example). Assume that there exists an eigenvariety $\mathcal{E}^G$ parameterizing ...
- 93
3
votes
1
answer
114
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Selmer complex and total complex
Thanks for your reading. I'm studying Selmer complex book by Jan Nekovar. For the definition of Selmer complex I meet a problem.
In the introduction(page 9, 0.8.0) the author gives us a definition of ...
- 319
4
votes
1
answer
183
views
About the reduction type of the Kodaira symbol of elliptic curves defined over p-adic local fields
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $E$ be an elliptic curve defined over $K$. Tate's algorithm can be used to compute the Kodaira symbol of the reduction type of $E$.
However, I ...
- 77
3
votes
1
answer
196
views
Surjectivity of specialization map
Let $S$ be a henselian DVR and $X/S$ be a flat and proper curve with $X$ being regular. Under what conditions the specialization map $Pic^0_{X/S}(S)\to Pic^0_{X/S}(Spec(k(s)))$ is surjective? Here $s\...
- 918
4
votes
0
answers
110
views
Euler factors from bad primes and the Beilinson-Bloch vanishing conjecture
The vanishing part of the Beilinson-Bloch conjecture asserts that for a smooth projective variety $X$ over a number field $K$, $\dim_{\mathbb{Q}} \operatorname{CH}^i(X) \otimes_{\mathbb{Z}} \mathbb{Q} ...
- 531
4
votes
1
answer
286
views
Known cases of Tate conjecture for varieties which are smooth over a curve
What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth morphism to a smooth projective curve? I am ...
- 658
0
votes
0
answers
190
views
About Chern classes via Atiyah class
I am trying to understand a construction of the Chern classes of a vector bundle $\mathcal{E}$ via the Atiyah class, like is done in this text and here in section 1.4. I am interested in the case ...
- 348
2
votes
1
answer
308
views
Can the number of elements of order 4 in the Tate–Shafarevich group grow arbitrarily large?
Let $E/K$ be a number field. For quadratic field extensions $L/K$, it is known that $\operatorname{Ш}(E/L)[2]$ can be arbitrarily large (cf. P. L. Clark and S. Sharif, "Period, index and ...
- 1,541
7
votes
0
answers
141
views
Average number of $\mathbb{F}_p$-points over twists of a variety
Let $p \gg 1$ be a sufficiently large prime. I recently stumbled across a fascinating fact about the number of $\mathbb{F}_p$-points on elliptic curves over finite fields. Specifically, we have:
Fact ...
2
votes
0
answers
126
views
Full level structure Deligne-Rapoport v.s. Katz-Mazur
For modular curves over schemes there are two main references that I use, namely Deligne Rapoport [DR], and Katz-Mazur [KM]. However I recently noticed that there is a difference in conventions in ...
- 2,224
1
vote
0
answers
157
views
Degeneracy of the Cassels-Tate pairing $\operatorname{Ш}(E/K)[n]\times \operatorname{Ш}(E/K)[n]\to \Bbb{Q}/\Bbb{Z}$
$\DeclareMathOperator{\Sha}{Ш}$ Let $E/K$ be an elliptic curve over a number field $K$. Let $\Sha (E/K)$ be the Tate-Shafarevich group, and let $n\ge 2$ be an integer. According to Theorem 15 in the ...
- 1,541
6
votes
1
answer
303
views
A Simplification of the computation of local heights in Gross-Zagier
At the end of the following document, https://public.websites.umich.edu/~asnowden/seminar/2014/gz/L07.pdf , it was stated that to prove the formula of Gross and Zagier, it is not necessary to compute ...
- 213
7
votes
0
answers
152
views
Discriminants and lattices in Algebraic geometry vs Geometry of numbers
(Post-writing, this question ended up being way more rambly than I intended. Sorry for that. There's a lot of closely related ideas I'm trying to unravel and it's hard to extract an individual ...
- 473
6
votes
1
answer
407
views
Good reduction for the universal elliptic curve
Let $X$ be a modular curve, i.e. a quotient of the upper half plane $\mathbb{H}$ by a congruence subgroup $\Gamma$. When $\Gamma=\Gamma_0(N)$, it is known that $X$ has a smooth model denoted $\mathcal{...
- 2,907
3
votes
0
answers
111
views
Weil pairing for an abelian variety uniformized as torus modulo lattice over a non-archimedean field
Suppose I have an abelian variety $A$ with split toric reduction over a non-archimedean field $K$, so that $A$ can be realized as $T / \Lambda$ where $\Lambda$ is a lattice in the torus $T$. Letting $...
- 1,298
3
votes
0
answers
150
views
$p$-adic points of open subschemes of complete intersections
I'm currently studying the $3\times 3$ magic square of squares problem for a research project. The variety is initially given by the intersection of $8$ quadrics in $\mathbb{P}^8$, but via Gröbner ...
- 31
2
votes
1
answer
240
views
Cohomology of torsion points on elliptic curves
$\DeclareMathOperator\Gal{Gal}$Let $E$ be an elliptic curve defined over a number field $K$. Put $G:=\Gal(\bar{K}/K)$, and for each valuation $v$ of $K$, put $G_v:=\Gal(\bar{K_v}/K_v)$. Consider the ...
- 622
1
vote
0
answers
82
views
Behavior of translation functors in characteristic $p$
Let $G$ be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic $p>0$, and let $\mathfrak g$ be the Lie algebra of $G$. Let $U_\chi(\mathfrak g)$ ...
- 2,974
2
votes
0
answers
125
views
Topological Hochschild Homology and $p$-adic étale cohomology of $\mathbb{Q}$-schemes
Recent progress in $p$-adic geometry has produced an interesting comparison isomorphism between the crystalline cohomology of a smooth algebra $A$ over a perfect field $k$ in characteristic $p$, and ...
- 2,907
1
vote
1
answer
120
views
Reference request for the isomorphism $H^1(G_{K_v},E)[n]\cong (E(K_v)/nE(K_v))^*$ in the context of Tate-duality
Let $E/K$ be an elliptic curve over a number field $K$. Let $M_K$ be the set of all places of $K$. Let $K_v$ be a completion of $K$ at $v$.
I'm searching for a reference for the statement of the ...
- 1,541
1
vote
1
answer
140
views
Specialization of points on the generic fiber in a prestable model of $\mathbb{P}^1$
Let $R$ be a complete DVR with uniformizer $\pi$, fraction field $K$ and residue field $k$. We assume $k$ is algebraically closed.
Let $X$ be a prestable model of $\mathbb{P}^1_K$ over $R$, so $X$ is ...
- 7,547
5
votes
1
answer
365
views
Unramified fppf cohomology
Let $F$ be a number field, $\mathcal{O}_F$ its ring of integers and $G$ an fppf $\mathcal{O}_F$-group scheme.
See the question Unramified Galois cohomology of number fields for unramified cohomology ...
- 265
5
votes
1
answer
150
views
Does an abelian variety $A$ have a model over a finite field if its $p$-divisible group $A[p^{\infty}]$ does?
Let $A$ be an abelian variety over an algebraically closed field $k$ of characteristic $p>0$. Let $X := A[p^{\infty}]$ be the associated $p$-divisible group. Assume that $X$ admits a model over a ...
- 769
2
votes
0
answers
110
views
Galois action on the cohomology of a curve over a local field with bad reduction
Let $C/\mathbb Q_p$ (or a p-adic local field more generally) be a smooth projective curve with split semistable reduction over $\mathbb Z_p$. What can we say about the action of the Galois group $\...
- 7,746
2
votes
0
answers
89
views
Conjecture on ordinary reductions of smooth complex projective varieties and Its context
I am interested in the conjecture suggesting that many reductions of a smooth complex projective variety are ordinary, as mentioned in Remark 5.1 of the paper by Mustaţă and Srinivas:
Ordinary ...
2
votes
0
answers
168
views
When do we have $\bigoplus_{i + j = n} R^i f_* \mathbb{Q}_\ell \otimes_{\mathbb{Q}_\ell} R^j g_* \mathbb{Q}_\ell \cong R^{i + j} h_* \mathbb{Q}_\ell$?
Milne, Étale Cohomology, theorem 8.5 states the following version of the Künneth formula (in slightly greater generality). Let $\Lambda$ be a finite commutative ring. Let $X, Y, S$ be schemes with $S$ ...
- 531
3
votes
2
answers
284
views
Definition of $M_{1,0}$
Is there an explicit construction of the moduli space $M_{1,0}/\mathbb{Q}$ of genus $1$ curves whose set of $R$-points, for a $\mathbb{Q}$-algebra $R$, is the set of isomorphism classes of genus $1$-...
- 2,907
1
vote
0
answers
82
views
Bounding the Bloch-Kato Selmer group of a twisted symmetric power of a Tate module
Let $E$ be an elliptic curve over $\mathbb{Q}$, with good reduction at $p$, and let $V = H^1_{et}(\overline{E}, \mathbb{Q}_p)$ be (the dual of) its (rationalized) Tate module.
Let $S^nV$ denote its $n$...
- 2,907
-5
votes
1
answer
153
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On Mordell equation $y^2=x^3+k$ [closed]
Have the Mordell equation $y^2=x^3+k$ solved for all integer $k$ or not?
Please Could you tell me about a good review papers about such equation.
- 17
0
votes
0
answers
81
views
What is the action of the Galois group due to Lubin-Tate formal group on the respective Tate module?
It is a well-known fact that a Tate module $T_p(A)$ of an abelian group (abelian variety or commutative group scheme) $A$ over a field $K$, equipped with a continuous action of the respective absolute ...
- 195
1
vote
1
answer
93
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A correspondence between pairs of isogenies and representation numbers
This question is re-posted from MSE because it didn't seem to get any traction/responses there.
This is a question from this paper about a correspondence between representation numbers of quadratic ...
- 185
2
votes
1
answer
154
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Extending line bundle to regular model
Let $S$ be the spectrum of an excellent discrete valuation ring with field of fractions $K$ and $C$ be a proper integral regular curve over $K$. Assume, $C$ admits a proper regular flat model $\...
- 5,986
4
votes
2
answers
229
views
Arithmetic application: Complete group ring and group ring for infinite group
Let $G$ be a profinite(infinite) group, $\Lambda$ be the complete group ring(Iwasawa algerbra) of $G$ over a unity ring $R$. My first question is that do we know something about the relation with $\...
- 319
4
votes
0
answers
178
views
Splitting of counit-trace map for $\ell$-adic sheaf $\Bbb Q_{\ell}$
I have a question about following argument on page 3 in paper arxiv.org/abs/1702.04404 by Will Sawin (Proposition 3):
The claim is that for a finite, dominant map $f:X \to Y$ between varieties $X,Y$ ...
- 5,986
4
votes
0
answers
100
views
Structure of points of elliptic curves in field with restricted ramification
Let $k$ be a finite field of characteristic $p$ and let $C$ be a curve over $k$. Let $E$ be a non-constant elliptic curve over $k(C)$. Taking the Néron model of $E$ and removing the singular fibers ...
- 177
2
votes
0
answers
280
views
Why is the weight monodromy hard in mixed characteristics?
I know very little about the conjecture, beyond Grothendieck's monodromy theorem perhaps (a dense open subgroup of inertia acting unipotently on pure motives). But I heard that it was completely ...
- 2,907
3
votes
1
answer
187
views
Reference Request: Preservation of étale maps under rigid analytic GAGA
Let $K$ be a finite extension of $\mathbb{Q}_p$. As the title says, I am looking for a reference in which it is shown that given an étale map $f:X\rightarrow Y$ between smooth algebraic $K$-varieties, ...
- 541
3
votes
1
answer
241
views
Could I get an interpretation for application of Euler characteristics in number theory？
As a beginner who just get in touch with Euler characteristics in this field, could I get some intuition for the arithmetic meaning of Euler characteristics of bounded complexes, for example Selmer ...
- 319
7
votes
1
answer
233
views
Proven results for the refined Birch Swinnerton-Dyer conjecture over rationals when rank at most $1$
For an elliptic curve, $E$, defined over $\mathbb{Q}$ and with the Mordell-Weil group, $E(\mathbb{Q})$, having rank, $r$, the Birch Swinnerton-Dyer (BSD) conjecture states that
$$
c_{E} = \lim_{s \...
- 73
1
vote
0
answers
85
views
Action of Atkin--Lehner involution on CM points
In their first paper on Heegner points and derivatives of $L$-series, Gross and Zagier describe the action of Atkin--Lehner involutions on certain CM $\mathbf{C}$-points of the modular curve $X_0(N)$. ...
- 265
1
vote
0
answers
137
views
Crystalline at $\ell=p$ implies unramified at $\ell\neq p$
Let $X$ be a smooth, projective variety defined over some $p$-adic field $K$. Is it true that if the etale cohomology $H^i_{et}(X_{\overline{K}},\mathbb{Q}_\ell)$ is crystalline at $\ell=p$, then $H^...
- 141
2
votes
0
answers
96
views
On the root numbers of quadruples of quadratic twists of elliptic curves
We got strong numerical evidence for the root numbers and analytic ranks
of quadruples of elliptic curves over the rationals.
Related to this question.
Let $k,k_1,k_2$ be squarefree pairwise coprime ...
- 25.4k