# Questions tagged [reference-request]

This tag is used if a reference is needed in a paper or textbook on a specific result.

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### What is the modular curve for level 1, 2?

An elliptic curve over a scheme $S$ is the data of a proper smooth morphism
of schemes $\pi: E\rightarrow S$ whose geometric fibres are connected curves of genus $1$ and a section $e: S \rightarrow E$....

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### Known and named DAG (Graph Theory)

I found in a book a type $L_n$ and $A_n$ of quivers, so I ask if there is any references of known and named classes of directed acyclic graphs?
Thanks.

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170 views

### Density of twin square-free numbers

It is well-known how to compute the density of square-free numbers, to get
$$ \lim_{N\to\infty} \frac{\#\{ n \leq N : n \text{ square-free}\}}{N} = \frac{6}{\pi^2}.$$
What is the density of numbers ...

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237 views

### Texts on moduli of elliptic curves

I want to study FLT (Fermat's Last Theorem), and now I'm studying moduli of elliptic curves.
I've heard that Deligne-Rapoport, Katz-Mazur, Mazur's "Modular curves...", and Katz's "p-adic..." are very ...

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26 views

### Modulus estimate with intersecting annuli (quasi-additivity)

In general for annulus $A\subset \mathbb{C}$ if $A_{1},A_{2}....\subset A$ are disjoint annuli inside it, then we have
$$mod(A)=\frac{1}{2\pi}\int_{A}\int_{A} \frac{1}{|z|^{2}}dz>\frac{1}{2\pi}\...

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126 views

### Probability that a Random Monic Polynomial Has Few Real Zeros

In the paper https://arxiv.org/pdf/math/0006113.pdf, it is shown that the probability that a random polynomial $a_0 + a_1x + \cdots + a_n x^n$ has $o(\log n / \log\log n)$ real zeros is $n^{-b + o(1)}$...

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112 views

### Existence of pointwise Kan extensions in $\infty$-categories

This answer by Emily Riehl mentions a to-be-published proof of the fact that $\infty$-categorical (pointwise) Kan extensions exist when the target category admits (co)limits indexed by the relevant ...

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53 views

### Semi-discrete Wasserstein distance to uniform

Does the $p$-Wasserstein distance have a simpler expression when applied to these two distributions :
A uniform distribution on $[0,1]^d$
A discrete distribution with $N$ equally-weighted point mass ...

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120 views

### “Fundamental theorem for Hopf modules”

I am studying Hopf algebras in categories, and I hope, somebody could help me with the following.
Joost Vercruysse in his paper Hopf algebras---Variant notions and reconstruction theorems writes (...

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81 views

### The properties of almost all directed graphs

A mathematician on the forum previously requested a reference on human brains modelled as directed graphs. This makes sense as neurons are mostly unidirectional and I have been thinking about similar ...

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140 views

### Comparing cardinalities of sets in different universes [on hold]

Suppose I interpret the independence of Continuum Hypothesis (CH) as:
- There is an universe $V_1$ with no cardinalities between $\omega$ and c (Cardinality of continuum) in that universe.
By an abuse ...

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113 views

### Stacks project for Galois representations and automorphic forms

Is there anything like Stacks project for Galois representations and automorphic forms? I am not asking people to start something like Stacks project, just asking if something like Stacks project ...

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138 views

### The function $\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{(2n+1)^s}$: reference request or particular values at integers and abscissa of convergence

We denote for integers $m\geq 1$ the M?bius function as $\mu(m)$. With the help of a CAS, Wolfram Alpha online calculator, I was calculating certain values of $$\sum_{n=0}^\infty\frac{(-1)^n\mu(2n+1)}{...

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87 views

### Browder's Fixed Point Theorem in uniformly convex Banach spaces with non-identical image

This is in fact an exercise from Dirk Werner's book "Funktionalanalysis", but I do think that the result is quite interesting and up to now, I can only partly solve this problem. From the point of my ...

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180 views

### Computing the $2$-adic volume of a special orthogonal group

Let $n \geq 0$ be an integer, let $A = (a_{ij})$ be the $(2n+1) \times (2n+1)$ matrix defined by $a_{ij} = 0$ unless $i + j = 2n+2$, in which case $a_{ij} = 1$. Let $G$ be the group scheme over $\...