-1
votes
0answers
52 views
What is the exact mathematical formulation of a claim
The motivation to this question can be found in
http://mathoverflow.net/questions/103846/why-are-galois-representations-so-important-in-number-theory
My question is concerned wi …
1
vote
1answer
108 views
Convert a quartic to Weierstrass Form
How to convert this quartic to Weierstrass form?
$x^{2}y^{2}-2\left( 1+2\rho \right) xy^{2}+y^{2}-x^{2}-2\left( 1+2\rho
\right) x-1=0$
7
votes
1answer
246 views
Best bounds toward Serre’s uniformity conjecture
If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre
that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the
Galois r …
0
votes
0answers
95 views
How I can prove that Λ(C,s) have infinitely many simple zeros at non-positive integers?
Let $C$ be an elliptic curve. Then the full L-series of $C$ is given by
$$L(C,s)=\sum_{n=1}^{\infty}((a_{n})/(n^{s}))$$
where s=α+iβ and $a_{n}$ are the coefficients of Dirichlet …
8
votes
1answer
242 views
examples of “exotic” moduli problems for elliptic curves?
Let $\textbf{Ell}$ be the category of elliptic curves over various base schemes, and where a morphism between $E\rightarrow S$ and $E'\rightarrow S'$ is a cartesian diagram with th …
0
votes
0answers
92 views
r-torsion points on elliptic curve on finite field
Let $E(\mathbb{F}_q)$ - elliptic curve, $r$ is prime, $|E(\mathbb{F}_q)[r]| > 1$.
Let $r | q-1$. Is it true that $|E(\mathbb{F}_q)[r]| = r^2$?
3
votes
0answers
112 views
Is Hasse-witt map isomorphism?
Fix a level $N \geq 3$ and denote by $\Gamma(N)\subset SL(2,\mathbb{Z})$ the subgroup of matrices which are congruent to the identity modulo $N$. The open modular curve $Y(N)$ corr …
2
votes
0answers
161 views
Shafarevich’s theorem for elliptic curves defined over function field of algebraic curve over algebraically closed field
Let $K$ be a number field and $S$ a finite set of places of $K$. Then Shafarevich's theorem states that there are only finitely many isomorphism classes of elliptic curves $E$ over …
4
votes
2answers
141 views
elliptic curve with a degree 2 isogeny to itself?
I've come across the following question, which I think must be easy for experts: is there a complex elliptic curve $E$ with an isogeny of degree 2 to itself?
Of course one can as …
5
votes
1answer
211 views
Where do the product expansions of modular forms come from?
It is well-known that many modular forms can be expressed as infinite products. For instance, the most famous one is probably the expansion
$$\Delta(q) = q \prod_{n=1}^\infty (1-q …
13
votes
3answers
474 views
Elliptic curve over a scheme is a group scheme?
In Katz's article p-adic properties of modular schemes and modular forms in the Antwerp proceedings, the following definition of an elliptic curve over a base scheme $S$ is given:
…
0
votes
1answer
131 views
Can we find a set of elliptic curves over rationals associated with $f$?.
We know that for each elliptic curve over rationals, we can define the Dirichlet series of the Hasse–Weil $L$-function, i.e., the function associated with an elliptic curve over ra …
4
votes
1answer
189 views
equivalence between katz and classical modular forms
$\newcommand{\CC}{\mathbb{C}}$
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\PP}{\mathbb{P}}$
$\newcommand{\QQ}{\mathbb{Q}}$
$\newcommand{\hH}{\mathcal{H}}$
$\newcommand{\eE}{\mathc …
2
votes
0answers
76 views
A nice rigid analytic model for local systems over an elliptic curve?
For $E$ an elliptic curve, let $LS(E)$ be the group of line bundles on $E$ with a flat connection. This is an $\mathbb{A}^1$-torsor over $E^\vee$. By Riemann-Hilbert (since the gau …
3
votes
1answer
244 views
Is there an elliptic surface over $Y(1)$?
Actually I have a few related questions.
Here, by $Y(1)$ I mean the affine $j$-line $\text{SL}_2(\mathbb{Z})\backslash\mathcal{H}$.
I know $Y(1)$ is only a coarse moduli space, s …

