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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

6 votes

Volumes of fundamental domains of maximal orders in definite quaternion algebras over Q

See chapters II and III of the classic book of Vigneras. Beware however that in the later parts of chapter III she assumes widely that C.E. (the "Eichler condition") is verified, and C.E. is explicitl …
stankewicz's user avatar
  • 3,625
3 votes

Quick proof of the fact that the ring of integers of $\mathbb Q(\zeta_n)$ is $\mathbb Z[\zet...

I don't know if this is what you're looking for, but I like the proof in Neukirch's Algebraic Number Theory Section I.10. Conceptually the idea (for $n$ a prime power) is that if the discriminant is a …
stankewicz's user avatar
  • 3,625
9 votes
Accepted

"Bad" reduction of Shimura curves via dual graphs

inkspot is indeed correct that the component graphs are indeed not generally trees. As you seem to have deduced for yourself, Cerednik–Drinfeld uniformization is a highly nontrivial concept, and it re …
stankewicz's user avatar
  • 3,625
5 votes

applications of Tate-Poitou duality

Take a look at the recent paper of Mazur and Rubin, "Ranks of twists of elliptic curves and Hilbert's 10th problem" Lemma 3.2, one of the indispensable lemmas of the paper, is a direct application of …