All Questions
2,494 questions
3
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0
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29
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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
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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
9
votes
1
answer
2k
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The definition of ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ in Inter-universal Teichmüller theory
$\newcommand{\Vect}{\mathit{Vect}}$I am reading Mochizuki's INTER-UNIVERSAL TEICHMÜLLER THEORY I to III and I hardly understand this theory, but there is a thing particularly I bother.
Given an ...
2
votes
0
answers
139
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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
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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 $x_0$ ...
- 105k
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....
- 8,945
1
vote
2
answers
197
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What are the finite étale coverings of a quasi-hyperelliptic surface?
Let $X$ be a quasi-hyperelliptic surface in characteristic 3 where the canonical bundle $K_X$ is trivial.
Question: Is there a finite étale covering $Y \rightarrow X$ such that
$Y$ is an abelian ...
- 103
3
votes
0
answers
77
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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
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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 $...
- 156
4
votes
1
answer
183
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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 ...
- 37k
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 ...
- 37k
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 ...
- 10.9k
2
votes
1
answer
308
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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 ...
- 149k
3
votes
1
answer
196
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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\...
- 16.2k
4
votes
0
answers
110
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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
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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 ...
- 149k
6
votes
2
answers
656
views
Finite flat group schemes over $\mathbb{Z}$ that are extensions of $\mu_p$ by $\mathbb{Z}/p\mathbb{Z}$
Suppose $X$ is a finite flat group scheme over $\mathbb Z$, killed by a prime number $p$ and such that there exists an extension as finite flat group schemes defined over $\mathbb Z$:
$$0\to \mathbb{Z}...
- 12.9k
0
votes
0
answers
190
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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 ...
- 4,713
7
votes
0
answers
141
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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
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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
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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 ...
- 35.5k
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
4
votes
2
answers
229
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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 $\...
CommunityBot
- 1
6
votes
1
answer
303
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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 ...
- 2,616
7
votes
0
answers
152
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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{...
- 10.7k
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 ...
- 12.9k
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 ...
- 12.9k
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
1
vote
1
answer
140
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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 ...
- 4,111
2
votes
0
answers
125
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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
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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 ...
- 8,945
3
votes
1
answer
153
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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 ...
- 11.3k
9
votes
1
answer
1k
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Coderivations of $S(V)$ correspond to linear maps $S(V) \to V.$ Only over characteristic $0$?
Definition. Let $k$ be a commutative ring. Let $V$ be a $k$-module. We turn the symmetric algebra $\mathrm{S}\left(V\right)$ of $V$ into a graded Hopf algebra by defining the comultiplication
\begin{...
- 33.8k
-1
votes
1
answer
342
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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
16
votes
5
answers
8k
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Is the ABC conjecture known to imply the Riemann hypothesis?
I once heard from a graduate student that the ABC conjecture implies the Riemann hypothesis. I can't find a reference for this, but given the department the student is from I tend to believe he might ...
- 13
5
votes
1
answer
365
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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 ...
- 56
7
votes
1
answer
549
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Finiteness of the Brauer group for a one-dimensional scheme that is proper over $\mathrm{Spec}(\mathbb{Z})$
Let $X$ be a scheme with $\dim(X)=1$ that is also proper over $\mathrm{Spec}(\mathbb{Z})$. In Milne's Etale Cohomology, he states that the finiteness of the Brauer group $\mathrm{Br}(X)$ follows from ...
CommunityBot
- 1
5
votes
1
answer
150
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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 ...
- 537
6
votes
3
answers
2k
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Direct proof of special case of Hasse's theorem for elliptic curves
Consider the elliptic curve $y^2 = x^3 + x$ over $\mathbb{F}_p$, where $p \equiv 1 \pmod 4$.
If memory serves correctly, the number of points (excluding the point at infinity) is $p - a$ where $a$ is ...
- 12.3k
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
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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 ...
- 63.9k
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 $1$-...
- 12.9k
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
1
vote
0
answers
82
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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
59
votes
1
answer
9k
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IMO 2017/6 via arithmetic geometry
The (very nice) final problem of IMO 2017 asked contestants to show:
If $S$ is a finite set of lattice points $(x,y)$ with $\gcd(x,y)=1$, then there is a nonconstant homogeneous polyonmial $f \in \...
- 1,207
0
votes
0
answers
81
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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 ...
- 111