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Algebraic number fields, Algebraic integers, Arithmetic Geometry, Elliptic Curves, Function fields, Local fields, Arithmetic groups, Automorphic forms, zeta functions, $L$-functions, Quadratic forms, Quaternion algebras, Homogenous forms, Class groups, Units, Galois theory, Group cohomology, Étale cohomology, Motives, Class field theory, Iwasawa theory, Modular curves, Shimura varieties, Jacobian varieties, Moduli spaces
2
votes
density of primes such that both $f(x)$ and $g(x)$ have roots
There are two cases: when $f, g $ are coprime polynomials and the other case is when they are not. In the first case, there are integer polynomials $f_1,g_1$ and a non-zero integer $c$ such that $f(x) …
5
votes
0
answers
258
views
Sums of two integer squares in arithmetic progressions
Is there an explicit formula in the literature for the number of representations of a positive integer $n$ as a sum of two integer squares, the second of which is divisible by $5$? So this means to co …
10
votes
2
answers
1k
views
What are the current trends in class field theory?
Being far from an expert in the subject I was wondering if people can hint towards a modern exposition of the developments in the last 10 years ? Or if not then suggest some sub-subjects in CFT that a …
3
votes
0
answers
165
views
Averages of $L(s,\chi)$
Let $(\frac{m}{n})$ denote the usual quadratic Jacobi symbol.
What is the
abscissa of convergence
of the double Dirichlet series ?
$$
\sum_{\substack{m,n \in \mathbb{N} \\ \gcd(m,n)=1 \\m,n\equiv 1 \m …