Lectures on Hilbert modular varieties and modular forms. With the assistance of Marc-Hubert Nicole (Q2770565)

Abelian varieties are compact, connected, projective algebraic groups that generalize elliptic curves, and they play an important role in algebraic geometry and number theory. If the ring of a totally real field \(L\) of degree \(g\) acts as endomorphisms of an abelian variety \(A\) of dimension \(g\), then \(A\) is said to have real multiplication by \(L\). The main topics discussed in this book are abelian varieties with real multiplication and their moduli spaces as well as modular forms on those moduli spaces. NEWLINENEWLINENEWLINEA Hilbert modular variety of dimension \(g\) is the quotient of the \(g\)-fold product \(\mathcal H^g\) of the Poincaré upper half plane \(\mathcal H\) by a discrete subgroup of \(\text{PGL}_2 (L)\). Abelian varieties with real multiplication can be parametrized by a scheme \(\mathfrak M\), whose complex points are in natural bijection with a union of Hilbert modular varieties. Hilbert modular forms can be regarded as sections of certain line bundles over such a scheme \(\mathfrak M\). NEWLINENEWLINENEWLINEAlthough this book is written primarily for graduate students, it assumes a fair amount of knowledge in number theory and algebraic geometry, including rudiments of the theory of elliptic curves and classical modular forms. NEWLINENEWLINENEWLINEChapter 1 reviews elements of the theory of algebraic groups, tori, and abelian varieties. Chapter 2 describes the construction of abelian varieties with real multiplication over \(\mathbb{C}\) and the moduli spaces as well as basic properties of Hilbert modular forms. Abelian varieties with real multiplication over an arbitrary field are treated in Chapter 3. Chapter 4 discusses \(p\)-adic elliptic modular forms and their connections with \(p\)-adic \(L\)-functions and Galois representations. These discussions are extended to the case of \(p\)-adic Hilbert modular forms in Chapter 5. Chapter 6 explains the local deformation theory of abelian varieties, concentrating on the case of abelian varieties with real multiplication. Finally, there are two appendices, one about group schemes and the other about calculations with cusps of modular curves.