Algebraic Number Theory

Abstract

Algebraic number theory f Serge Lang. -2nd ed. p. cm.-(Graduate texts in mathematics; 110) Includes bibliographical references and index.

Trace formulae and applications to class numbers

“…Replacing the integrand by its absolute value we obtain the finiteness of the integral by modifying the argumentation above. The main modifications that have to be made concern the analogues of (8) and (11). In the case that the Eisenstein series appearing in the counterpart of (8) belong the same cusp we can apply the Maaß-Selberg relations directly.…”

Eigenvalues of Frobenius and Hodge Numbers

“…∈ S is a place of F, we denote by Frob v ∈ G F,S the corresponding geometric Frobenius automorphism, which is well defined up to conjugation in G F,S [La,§1.5]. Set q v = |κ(v)|, where κ(v) denotes the residue field of v.…”

Riemann’s zeta function and beyond

“…This will necessitate some algebraic background; accordingly this section requires some familiarity with the concepts involved. However, it is not essential to the rest of the paper, and readers may wish to skip directly to Section 2.3, or instead to consult [93], [142] for definitions and examples.…”

“…The above construction can be generalized to an arbitrary number field -or even "global field" -F to obtain its adele ring A F (see [93], [142]). Most constructions involving A Q generalize to A F , though we will mainly focus on F = Q for expositional ease.…”