# Tagged Questions

**4**

votes

**0**answers

74 views

### minimal conductors among elliptic curves with a fixed CM type

Let $K$ be a quadratic imaginary field. To simplify my life, let us assume
that $K$ has class number one.
Consider the following infinite set:
$S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an ...

**11**

votes

**0**answers

170 views

### What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?

I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that ...

**5**

votes

**1**answer

167 views

### Units of $\mathbf Z[X,Y]/(P(X,Y))$

Let $P(X,Y)\in \mathbf Z[X,Y]$ be an irreducible polynomial and let $A$ denote the quotient ring $\mathbf Z[X,Y]/(P)$.
What is known about the group of units of $A$?
It's not even clear to me that ...

**2**

votes

**1**answer

157 views

### How to obtain the Period matrix from the Igusa Invariants of a genus two curve?

I am looking for an algorithm to obtain the period matrix tau= (tau1 & tau12\ tau12 & tau2) from the Igusa invariants of a genus two curve. More precisely:
I consider a family of genus two ...

**1**

vote

**1**answer

181 views

### Weierstrass points on modular curves

What is knowns about Weierstrass points on modular curves? Are there any explicit formulas of them, or any information about Weierstrass gaps? I am interested in (compactifications of) the quotients ...

**1**

vote

**2**answers

380 views

### Equations of elliptic curves

First part of question I have asked on mathoverflow already: http://math.stackexchange.com/questions/467088/explict-form-of-the-equation-of-elliptic-curve
1) Let $E(\mathbb{F}_{q^2})$ is elliptic ...

**5**

votes

**1**answer

419 views

### Analogy between Jacobian of curve and Ideal class group

It is excerpt from "Algebraic Geometry Codes Basic ...

**4**

votes

**0**answers

247 views

### On Stickelberger's Theorem over function fields

Here is the setup to Stickelberger's theorem over number fields (following Washington's book Intro. to cyclotomic fields).
Let $M/\mathbb{Q}$ be a finite abelian extension with galois group $G$. ...

**7**

votes

**3**answers

809 views

### Quadratic reciprocity and Weil reciprocity theorem

I was told that Weil reciprocity theorem (one has two meromorphic function $f,g$ on a complex curve $C$, so $\prod\limits_{x\in C} g(x)^{ord_xf}=\prod\limits_{x\in C}f(x)^{ord_xg} \ $ where $ord_xf$ ...

**2**

votes

**1**answer

431 views

### Conductor of an elliptic curve

Given any elliptic curve over $\mathbb{Q}$ of conductor $N$, by modularity of elliptic curves,
there exists a surjective morphism from $X_0(N)$ $\rightarrow$ $E$.There may be several such 'N' and ...

**3**

votes

**1**answer

1k views

### Trace of Frobenius over $F_q$

Let $q_0$ be a prime and $q$ = $q_0^n$.
Let $a(F_q/F_{q_0})$ denote any integer which is trace of Frobenius over the field $F_q$ for some elliptic curve which can be defined over $F_{q_0}$.
It is ...

**4**

votes

**1**answer

927 views

### Bound for the number of rational points on the modular curve

By Mazur's theorems (Reference:- Modular curves and the Eisenstein ideal, 1977),
we know that the only rational points of X_0(N) for N any prime > 163 are
the two cusps (o) and (oo) (|X_0(N)(Q)| = 2 ...

**9**

votes

**0**answers

438 views

### Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation

Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...

**4**

votes

**0**answers

530 views

### Soft proof of multiplicity one for character groups of Shimura curves?

Is it not possible to prove mutiplicity one type statements for character groups of quaternionic Shimura curves by simply using Raynaud's description for character groups at primes dividing the ...

**15**

votes

**1**answer

464 views

### Does every hyperbolic curve over a finite field have an etale cover with a real Frobenius eigenvalue?

More precisely: let X/F_q be a smooth projective algebraic curve of genus at least 2. Does there always exist a curve Y/F_{q^d} with a finite etale projection Y -> X, such that one of the Frobenius ...

**21**

votes

**2**answers

1k views

### The work of E. Artin and F. K. Schmidt on (what are now called) the Weil conjectures.

I was reading Dieudonne's "On the history of the Weil conjectures" and found two things that surprised me. Dieudonne makes some assertions about the work of Artin and Schmidt which are no doubt ...

**2**

votes

**4**answers

501 views

### a question on function fields (extending my previous question)

Consider the extension Q(a,b) of the field of rationals, where a,b are algebraically independent transcendentals. To Q(a,b) adjoin the roots of the polynomials x^5+a^5=1 and y^5+b^5=1. The resulting ...

**10**

votes

**1**answer

563 views

### a question on function fields

Consider the transcendental extension Q(t) of the field of rationals.
To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting
field Q(t)[x] is a radical extension of Q(t). Is it true that ...