# Tagged Questions

721 views

### Upper bound on answer for Pell equation

A user on MSE, @martin , asked http://math.stackexchange.com/questions/1611411/pell-equations-upper-bound about an upper bound for $x$ in $x^2 - p y^2 = 1,$ when $p$ is prime. I checked, it appears ...
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### Binary Quadratic Forms with coefficients in $F_q[T]$

I aim to study the binary forms $ax^2 + bxy + cy^2 = (a,b,c)$ where $a,b,c \in {F_q}[T]$ (charasteristic of $F_q$ not 2) in particular those such that the discriminant $D = b^2 - 4ac \in F_q[T]$ has ...
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### Lower bound on class number of binary quadratic forms of discriminant of the form $n^2+4$

While searching for a use for the "sum invariant" of indefinite binary quadratic forms of discriminant $D = n^2 + 4$ (see https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html), I believe I ...
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### Recursions for some binary theta series in characteristic 3

Define $A(0), A(1), A(2) \dots$ in ${\bf Z}/3[[x]]$ as follows. For $n$ in $\bf N$ let $s=3^{2n+1}$. Then $A(n) = \sum a_kx^k$ where $a_k$ is the mod 3 reduction of the number of representations of $k$...
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### Does the Diophantine equation $(x^2+ay^2)(u^2+bv^2) = p^2+cq^2$ admit a complete solution?

In this MSE question/thread, I have been discussing the equation $$(x^2+ay^2)(u^2+bv^2) = p^2+cq^2, \tag{\star}$$ where $x,a,y,u,b,v,p,c,q$ are integers. I posed a conjecture which turned out to ...
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### References for Gauss Composition using Galois Cohomology

Note: I have already posted this on stackexchange, but have not yet gotten a response. What are some good references for the Gauss composition law on binary quadratic forms in terms of Galois ...
228 views

### Representing primes explicitly with binary quadratic forms

This is probably quite naïve, maybe even stackexchange-worthy. Consider a quadratic form such as $Q(x,y) = 3x^2+y^2$. We know that, for primes $p \equiv 1 \pmod{3}$, there exist integer solutions to ...
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### Question about Gauss composition law over PID.

Let $m$ be a square free integer, $\mathbb{Q}(\sqrt{m})$ a quadratic field extension of $\mathbb{Q}$, $\Delta$ is its discriminant and $O_{\mathbb{Q}(\sqrt{m})/\mathbb{Q}}$ its ring of integers. We ...
544 views

### Heegner Points and Binary Quadratic Forms

I've been trying to read Gross' paper on Heegner points on $X_0(N)$ and I am stuck on a few details. The definition he is working with is that a heegner points is a pair $y=(E,E')$, where $E$ and $E'$ ...
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### Number of solutions of a binary quadratic form.

Given a binary quadratic form with negative discriminant, such as $3x^{2} + y^{2} = k$, is there an efficient algorithm to compute a value $k$ (or all the values) for which the form has exactly $n$ ...
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### Representations by positive definite binary quadratic forms

It's known that the number of representations of an integer $k$ by sum of two squares is $$4\;\sum_{d|k}\left(\frac{-4}{d}\right)$$ or  4\sum_{d|k,\; d \textrm{ odd}} (-1)^{\frac{d-1}{2}}= 4(d_1(k)...
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### What is the identity class of the set of equivalence classes of binary cubic forms of discriminant $D$ ?

Let $\Delta = \sigma + 4 m$ be the fundamental discriminant of a quadratic field, where $\sigma \in \{ 0, 1 \}$. The binary quadratic form $Q(x, y) = A x^2 + B x y + C y^2$ of discriminant $\Delta$ ...
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### If the discriminant of a binary quadratic form has high valuation, is the form “almost a square”.

For a binary quadratic form $ax^2+bxy+cy^2$ over a field (characteristic not 2), the discriminant $b^2-4ac$ is 0 if and only if that form is the square of a linear form. I am curious about an ...
914 views

### Canonical form for a pair of quadratic forms

Could anyone recommend a reference to a canonical form for a pair of quadratic forms over R (not necessarily positively definite)? This is probably related to the Weierstrass elementary divisors (but ...
343 views

### Symmetric Power and Gauss quadratic form

Gauss in his book "Disquisitiones arithmeticae" considered only forms $ax^2+bxy+cy^2$ where $b$ is even, apparently because he had some notion of integral matrix in his mind even though he did not ...
I have two related questions on the representability of integers by quadratic forms in two variables : (1) Let $f: {\mathbb Z} \times {\mathbb Z} \to {\mathbb Z}$ be such a quadratic form, i.e. we ...