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heights of ideal classes and reduction theory for Bhargava cubes

Suppose $K$ is a quadratic imaginary field with discriminant $D$; let $S$ denote the ring of integers in $K$. For a fractional $S$-ideal $J$, define the height of $J$, denoted $H(J)$, to be the ...
• 81
841 views

A cubic equation, and integers of the form $a^2+32b^2$

I am trying to determine whether there are any integers $x,y,z$ such that $$1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)$$ It is clear that $x$ is odd. We can consider this equation as quadratic ...
• 5,957
299 views

Are there natural isomorphisms $S^{(2,1)}(k^{m+1})\cong k^2\otimes W$?

In this popular 2019 MO question, user მამუკა ჯიბლაძე asked: The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism between ...
138 views

Let $f(x,y)= ax^2+bxy+cy^2$, or similarly denote it by $(a,b,c)$. This question is about the case $(1,0,p)$ where $p$ is prime. Suppose I have one solution $\bar{x}_1=(x_0,y_0)$ for $f(x,y)=m$ for ...
182 views

• 5,957
1 vote
65 views

Is there an available English translation for Artin's "Quadratische Körper im Gebiete der höheren Kongruenzen"?

Otherwise, is it reasonable to work through the German edition with only a basic knowledge of German?
• 11
154 views

Eisenstein series evaluated at $2i$

Consider the real analytic Eisenstein series defined by $$E(z,s) := \sum_{\gamma\in\Gamma_\infty\setminus\Gamma} Im(\gamma z)^s$$ where as usual, $\Gamma=SL(2,\mathbb{Z})$ and $\Gamma_\infty$ is the ...
• 537
502 views

Two conjectures for primes $p\equiv 1\pmod 8$

Motivated by my paper Quadratic residues and quartic residues modulo primes [Int. J. Number Theory 16 (2020), 1833-1858], here I pose two new conjectures for primes $p\equiv1\pmod8$ based on my ...
• 14.5k
361 views

Integers representable as binary quadratic forms

It is known that odd prime $p$ can be represented as $p=x^2+y^2$ if and only if $p \equiv 1$ mod $4$, represented as $p=x^2+2y^2$ if and only if $p \equiv 1$ or $3$ mod $8$, represented as $p=x^2+3y^2$...
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