Let $K$ be an imaginary quadratic field of class number 1 and $E$ an elliptic curve over $\mathbf{Q}$ with CM by $\mathcal{O}_K$. Let $\psi$ be the Groessencharacter of $K$ attache …

In Noah Snyder's historical undergraduate thesis on Artin L-Functions, it mentions that Takagi proved Kronecker's Jugendtraum in the case of Q(i) in his doctoral thesis. Since I do …

Dear friends, I have some trouble finding a precise definition of what a modular form with complex multiplication. Could anyone provide such a definition and references for the st …

For an abelian variety $A$, it is said to be have $complex \ multiplication$ if $\mathrm{End}(A) \otimes_{\mathbb{Z}} \mathbb{Q}$ contains a number filed $F$ of degree $2 \cdot \m …

I have tried vainly to understand the significance of the following statement attributed to David Hilbert: The theory of complex multiplication is not only the most beautiful p …

The Kronecker-Weber Theorem, stating that any abelian extension of $\mathbb Q$ is contained in a cyclotomic extension, is a fairly easy consequence of Artin reciprocity in class fi …

This is related to my first MO question and Kevin Buzzard's conjecture at http://mathoverflow.net/questions/12486/integers-not-represented-by-2-x2-x-y-3-y2-z3-z In December 2010 m …

This would be a vague question, but I still want to ask here. Do you have any recommended book on complex multiplicaton. I know only 2 books: Shimura's book Abelian Varieties with …

In modular curves and modular forms, there is an adelic formulation, in which smaller open subgroups of some adelic group relate to higher level structure. As we know, higher level …

An elliptic curve $E$ with complex multiplication by an imaginary quadratic field $F$ has $\ell$-adic Galois representations that essentially encode the class field theory of $F$ - …

In Chapter II.10 of The Map of My Life, Goro Shimura mentions a certain problem: The second topic concerns a polynomial $F(x)$ with integer coefficients. Take $$ F(x) = x^3 …

Let $K$ be a quadratic imaginary field, and E an elliptic curve whose endomorphism ring is isomorphic to the full ring of integers of K. Let j be its j-invariant, and c an integral …

Dear MO Community, I am trying to understand Mazur's 1976 notes "Rational points on Modular Curves" (which can be found in Springer Lecture Notes in Mathematics 601). Let N be a …

I scoured Silverman's two books on arithmetic of elliptic curves to find an answer to the following question, and did not find an answer: Given an elliptic curve E defined over H, …

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polyloga …