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I'm not an expert, but this is what I learned from Lenstra's Algorithms in Algebraic Number Theory (Bull AMS, 1992) and Kirschmer and Voight's paper Algorithmic enumeration of ideal classes for quater …

This seems unlikely to me. Philosophically unique factorization means something like the following: if $R$ is a (say completely reducible) representation of a group $G$ then the trace of $R$ determin …

To address the conceptual question, the $L$-function essentially characterizes the automorphic representation and can be studied locally (associating local $L$-functions to local components of the glo …

Here's something within algebraic number theory, but is sort of a "special topic" in the sense that it's not treated in most algebraic number theory courses, but could easily be. It also has some com …

Leopoldt's conjecture seems to have been proved now by Mihailescu: http://arxiv.org/abs/0905.1274 In fact before that, I think Fujiwara had done significant work on it (maybe the case of totally rea …

Here's an approach for a really bad bound. Updated below based on comments and further reflection, but still giving a very bad bound. First, $K_f$ is contained in the field generated by the eigenvalu …

For simplicity, let's consider the case of holomorphic modular forms over $\mathbb Q$ of squarefree level and trivial nebentypus. Then one knows from Iwaniec, Henryk; Luo, Wenzhi; Sarnak, Peter. …

Revised. The global Waldspurger packet of $\pi$ is indeed finite, as you say in your comment. It's elements are metaplectic representations which are in bijection with the Vogan packet of $\pi$, i.e. …

Hecke conjectured that $\theta_I$ form a basis for the space $S_2(p)$, but this was found to fail for $p=37$ by Eichler. In fact, Gross realized that whenever you get vanishing central $L$-values you …

Consider the space of modular forms $M_k(N)$. Any modular form $f \in M_k(N)$ is determined by a finite number of Fourier coefficients (e.g., Sturm's bound), thus there is a finite set of Hecke opera …

Let $f(x)$ be a rational function which is a ratio of two integral polynomials, and $n \in \mathbb Z$. Then the sequence of iterates $n, f(n), f(f(n)), f(f(f(n)), ...$ will be an infinite sequence of …

At least it should be true under suitable assumptions. Jacquet-Shalika proved nonvanishing of automorphic $L$-functions of unitary cuspidal representations of GL($m$) at $s=1$. An automorphic $L$-f …

Let me give at least a partial answer. If you want to view the 1 and 2 dimensional theories uniformly, you should look at everything adelically. In dimension 1, Dirichlet characters can be viewed as …

I know I'm late to the party, but it seems to me the other answers don't directly answer the last part of the question: Is it true that an integer in a field of greater class number will have more …

For $n=4$, in addition to what Jeremy mentioned, there is some addition information in the answers to sum of squares in ring of integers. John Goes has also done some work but it seems his preprint i …