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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
0 answers
293 views

Elliptic curves: about a passage in J. Silverman's "Advanced topics of elliptic curves"

Reading the proof of the main theorem of complex multiplication for elliptic curves over number fields in J. Silverman's book "Advanced topics of elliptic curves" I got stuck at a passage which looks …
Hair80's user avatar
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2 votes
0 answers
101 views

Does same group of units imply surjective contraction map on spectra

Let $B\subset A$ be an inclusion of Krull dimension 1 rings, and assume $A$ a Dedekind domain. I think that $\mathfrak{p}\in Spec B$ is in the image of the contraction map $\mathfrak{P} \mapsto \mathf …
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  • 675
2 votes
0 answers
103 views

On equidistribution of primes in positive characteristic

In S. Lang's book "Algebraic Number Theory" (1986), page 317, Theorem 6 states essentially that given $P$ a set of primes, let $\tau:P\longrightarrow J$ be the typical idèle map taking values in the i …
Hair80's user avatar
  • 675
1 vote
1 answer
162 views

Completion reducing to localization on Noetherian rings

It is quite easy to show that if $A$ is a Dedekind domain and $\mathfrak{p}\in \operatorname{Spec} A$, then if $A_{\mathfrak{p}}$ is the completion of $A$ at $\mathfrak{p}$ and $A_{(\mathfrak{p})}=(A\ …
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  • 675