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Say $K=\mathbf{Q}(\sqrt{p^{*}})$ is the unique quadratic field in which $p$ is the only ramified finite prime. That is, $p^*=(-1)^{(p-1)/2}p$. There are two $L$-functions one can concoct out of $K$, …

This question is about automorphic forms for the group $\mathrm{GL}_2$, over a rational function field. Let's say $\mathbf{F}_q$ is a finite field, and $X=\mathbf{P}^1_{\mathbf{F}_q}$ is the projecti …

The statement that the Hecke algebra acts irreducibly on $S_k(\Gamma(1))$ is known as Maeda's conjecture, and it is still open. So an affirmative answer to your question about eigenforms with integer …

I was going to leave this as a comment, but I have a firm conviction about this, so here's my answer. The fields in your (1) are called (a) locally compact non-archimedean fields, or (b) non-archimed …

In number theory, base change can also refer to an operation on automorphic representations. If L/K is an extension of number fields, and pi is an automorphic representation of a reductive group G ov …

Let $F$ be a nonarchimedean local field, and let $D/F$ be the central division algebra of invariant $1/d$. Let $k$ be the algebraic closure of the residue field of $F$ and let $\pi$ be a uniformizer. …

Let $\mathbf{F}_q$ be a finite field of odd characteristic. Let $X_t$ be the hyperelliptic curve over $\mathbf{F}_{q^2}(t)$ with affine equation $$y^2 = \left((x^{(q+1)/2}-(x-1)^{(q+1)/2})^2 - t\righ …

If $E$ is an algebraic closure of $F$, then $D\otimes_F E\simeq M_2(E)$. (In fact this is also true if $E$ is taken to be, say, the unramified quadratic extension field of $F$.) We get an algebraic …

Let $\mathbf{F}_q$ be a finite field, and let $A=\mathbf{F}_q [[ x^{1/q^\infty} ]]$ be the completion of $\mathbf{F}_q[x^{1/q^\infty}]$ with respect to the $x$-adic topology. Then the $q$th power Fro …

I have typeset Deligne's letter, and placed the result here: http://www.math.ias.edu/~jaredw/DeligneLetterToPiatetskiShapiro.pdf I have made some minor edits so that the text reads more naturally to …

The proof involving Gauss sums always seemed the best to me. I'm going to run my own undergrad number theory students through that proof, right after we develop some experience with roots of unity. …

Let $K$ be a complete nonarchimedean field. The classical Tate algebra $K\langle T \rangle$ has lots of automorphisms, e.g., any substitution $T\mapsto a_1T+a_2T^2+\cdots$, where $a_1\in \mathcal{O}_ …

This question is about Tate's 1963 paper "Algebraic Cycles and Poles of Zeta Functions". Here he announces a conjecture (now known as "the Tate conjecture") which states that certain classes in the c …

Yoshida considers the Lubin-Tate tower in his geometric realization of the depth zero supercuspidals for $GL(n)$. For unitary groups, I'm sure that the answer to your question will be found in a simi …

The short answer to the question is that all currently known proofs of the local Langlands correspondence (and I'm just referring to GL(n) here) are "global" in the sense that they involve embedding t …