Yuichiro Taguchi

2-ADIC PROPERTIES OF CERTAIN MODULAR FORMS AND THEIR APPLICATIONS TO ARITHMETIC FUNCTIONS

It is a classical observation of Serre that the Hecke algebra acts locally nilpotently on the graded ring of modular forms modulo 2 for the full modular group. Here we consider the problem of classifying spaces of modular forms for which this phenomenon continues to hold. We give a number of consequences of this investigation as they relate to quadratic forms, partition functions, and central values of twisted modular L-functions.

Ramifications Arising from Drinfeld Modules

In this paper, we study various ramifications arising from division points of Drinfeld modules, abelian T -modules, formal modules, etc.. A motivation for this is to know how many isogeny classes and isomorphism classes of Drinfeld A-modules exist over a finite extension of the fraction field of A . We will see ( cf. Remark (3.4) ) that, modulo the isogeny conjecture, an isogeny class can contain infinitely many isomorphism classes and, without any restriction on ramification at the infinite places, there can be infinitely many isogeny classes. To explane some of the results, let F be a function field in one variable over a finite field, ∞ a fixed place of F , A the ring of elements of F which are regular outside ∞ , and K a finite extension of F . Given a Drinfeld A -module φ over K and a prime v of A , we denote by K(φ; v) the field of v -division points of φ . Then it turns out ( Corollary 1.6 ) that the ramification at various primes in the tower (K(φ; v)/K)n≥1 is bounded at the places over ∞ by a divisor depending only on φ , and at the finite places, it is controlled in a fairly precise way in terms of the “discriminant” ∆(φ). Roughly speaking, ∆(φ) is the coefficient of the leading term of the defining equation of φ . For finite places, this result is analogous to the case of abelian varieties over number fields. ( At least one has the HermiteMinkovski theorem for number fields, which assures the existence of an estimate of discriminants. ) But at infinite places, there occur new phenomena, which we describe by example in §2. We construct explicitely an infinite family of Drinfeld modules with everywhere good reduction and with ramification at infinity becoming arbitrarily large ( Example 2.1 ), as well as an infinite family of mutually nonisomorphic Drinfeld modules with everywhere good reduction and with bounded ramification at infinity ( Example 2.2 ). In §3, we give a proposition on v -adic Galois representations ( a positive characteristic version of a theorem of Faltings ), and discuss how many isomorphism and isogeny classes can exist. §4 and §5 are generalizations of §1 to the cases of finite submodules of higher dimensional formal modules. Theorem (4.6) is an A -module version of Théorème 1 of [5].

Induction formula for the Artin conductors of mod l Galois representations

A formula is given to describe how the Artin conductor of a mod l Galois representation behaves with respect to induction of the representation.

On the finiteness and non‐existence of certain mod 2 Galois representations of quadratic fields

Some finiteness and non‐existence results are proved of 2‐dimensional mod 2 Galois representations of quadratic fields unramified outside 2.

A SIMPLE PROOF OF THE MODULAR IDENTITY FOR THETA SERIES

We characterize the function spanned by theta series. As an application we derive a simple proof of the modular identity of the theta series.

Computing zeta functions for ordinary formal groups over finite fields

We describe an algorithm to compute the one-dimensional part of the zeta function ZG of an ordinary formal group law G of finite dimension d over a finite field of pN elements and evaluate its time computational complexity. Assume G is given as d formal power series in 2d variables. Our algorithm computes ZG mod pt with O(d2p(t-1)(2d+3)N2(log p)2) bit operations.

REGULAR SINGULARITY OF DRINFELD MODULES

In analogy with the classical theory of ordinary differential equations (see e.g. [6], [1], [5]), we define in this paper the notion regular singularity of a Drinfeld module at infinity. It turns out (Theorem (2.2)) that a Drinfeld module with regular singularity can not have a complex multiplication if the infinite place is of degree one, and has a tamely ramified period lattice which is of diamond shape. In §3, we study regular sigularity of φ-modules, which have more similar formalism to D -modules. We express the regularity of the singularity of φ-modules over a local field in four ways (Theorems (3.8) and (3.9)); (1) naively in terms of the valuations of the coefficients of certain polynomials, (2) by the existence of φ-stable lattices, (3) by the tameness of Galois actions, and (4) in terms of the norm of connections. For a field K , we denote by K a fixed separable closure of K , and let GK := Gal(K/K), the absolute Galois group of K .

Moduli of Galois Representations

We develop a theory of moduli of Galois representations. More generally, for an object in a rather general class A of non-commutative topological rings, we construct a moduli space of its absolutely irreducible representations of a fixed degree as a (so we call) “f-A scheme”. Various problems on Galois representations can be reformulated in terms of such moduli schemes. As an application, we show that the “difference” between the strong and week versions of the finiteness conjecture of Fontaine-Mazur is filled in by the finiteness conjecture of Khare-Moon.

On Congruences of Galois Representations of Number Fields

We give a criterion for two l-adic Galois representations of an algebraic number field to be isomorphic when restricted to a decomposition group, in terms of the global representations mod l. This is applied to prove a generalization of a conjecture of Rasmussen-Tamagawa under a semistablity condition, extending some results of one of the authors. It is also applied to prove a congruence result on the Fourier coefficients of modular forms.