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This article analyzes AlphaZero-type algorithms quantitatively from the viewpoint of local and global optimal sequences of play on a 7×7 board. Through targeted evaluation of the AI agent, the authors reveal the strategic, that is, winrate-dominated, nature of such algorithms, and expose thereby certain inherent obstacles against optimal play. Possible remedies are then explored, leading to techniques that may help further quantitative analysis of those algorithms and for the search for optimal solutions, on 7×7 as well as larger boards.

The subject of this paper is a quaternion algebra over a quadratic extension of a totally real algebraic number field. We generalize an important technical lemma of Shimura, on which a certain period of automorphic forms depends.

Let E/F be a totally real quadratic extension of a totally real algebraic number field. The author has in an earlier paper considered automorphic forms defined with respect to a quaternion algebra BE over E and a theta lift from such quaternionic forms to Hilbert modular forms over F. In this paper we construct adelic forms in the same setting, and derive explicit formulas concerning the action of Hecke operators. These formulas give an algebraic foundation for further investigations, in explicit form, of the arithmetic properties of the adelic forms and of the associated zeta and L-functions.

Introduction. Let E/F be a totally real quadratic extension of a totally real algebraic number field. To a suitably defined automorphic form h defined with respect to a quaternion algebra BE over E, we can associate a Hilbert modular form I(z, h) defined with respect to the field F . Such a lifting has been explicitly considered in the author’s recent paper [D99], via a convolution with a theta function, and the Fourier coefficients of the theta lift have been computed in terms of certain periods of the original form. The purpose of the current paper is to establish an explicit formula relating the actions of the Hecke operators on the original automorphic form and its theta lift. This result is analogous to theorems of Shimura in, for example, [Sh82] and [Sh88], where he has shown commutativity properties with respect to these Hecke operators. In addition to their inherent interest, these commutativity results are necessary in obtaining algebraicity results concerning Shimura’s period invariants as well as certain special L-values which naturally arise. The reader can consult Shimura’s [Sh88] for further motivation in this regard. See also [Y]. It is the author’s hope to discuss applications in this direction in the near future. Since it is essential for our purposes to consider everything explicitly, the quantity of technical details is considerable. In the interest of space, and also in order not to obscure essential ideas, a large amount of computation has been suppressed. Instead we have endeavored to point out key ingredients in the computations and to give precise references. It is then hoped that

Let F be a totally real algebraic number field, and let E be a totally real quadratic extension of F . In this article we establish a theta correspondence between certain automorphic forms defined with respect to a quaternion algebra over E and Hilbert modular forms defined with respect to F . Given such a quaternionic form, say h , the main theorem expresses the Fourier coefficients of its theta lift in terms of periods of h . The results in this paper generalize some theorems of Shimura.

In the last chapter, we studied the analytic properties of L-functions, in particular their zero distributions. If we simply regard an L-function as a complex analytic function, then its values at different points seem to be equally important or unimportant. However, since our L-functions all come from arithmetic geometric or automorphic origins, it turns out that the function values at certain points are more transparent in revealing the associated arithmetic or automorphic information, and thus deserve special attention. It is for this reason that such L-function values are called special values.

In this chapter we introduce various notions connected to periods of automorphic forms; all of these have received extensive treatment in Shimura’s work. The period conjectures of Shimura occupy a central position in number theory today, and will be discussed in more detail in Chapter 3 and Chapter 6. In this chapter we concentrate on the concepts themselves and give examples.

In this chapter we summarize a part of the first author’s recent work on the period conjectures of Shimura. Compared with the other chapters, the writing here is more terse. This chapter should more properly be viewed as a guide to the relevant literature, where elaborations on technical aspects can be found.

In the first three sections of this chapter we introduce the concept of lifting automorphic forms, the base change lift in particular. A few applications are discussed in relation to the period invariants. Shimura’s period conjectures and some of Yoshida’s work are summarized in the last two sections of this chapter.

The concept of modular form are based on very natural considerations. In this chapter we recount some rudiments of the theory of modular forms without assuming any previous knowledge of the subject on the reader’s part. The number theoretic interest of the subject becomes apparent when we describe the Hecke operators on the spaces of modular forms and the L-functions attached to eigenforms. The connection between elliptic curves and modular forms of weight 2 is briefly described towards the end in order to state the celebrated Shimura-Taniyama Conjecture, which is now a theorem of A. Wiles, et al. See [Wi95] and related articles.

In Chapter 1, we introduced the L-functions of modular forms and indicated briefly how useful they are in our study of arithmetic problems. In fact, the converse theorem of Weil asserts that an L-function, together with its twists, uniquely determines the modular form itself. Hence the study of modular forms can be largely reduced to that of their L-functions. Since L-functions are simply certain complex analytic functions, we can employ our familiar theory of complex analysis to study them, thereby facilitating our study of arithmetic problems for modular forms themselves.

The main purpose of this paper is to establish the existence of fundamental periods of primitive cusp forms of Hilbert modular type of several variables, as well as the relationship between those fundamental periods and the special values of the associated L L -functions. These results, together with some recent results of Shimura, give us the means of translating with ease results concerning periods of automorphic forms derived from various points of view. We also verify several conjectures of Shimura on the properties of such fundamental periods.

In this paper we discuss several related topics in the theory of automorphic forms, on which the author’s current research is focused. Roughly speaking, we shall deal with the questions of arithmeticity regarding special values of certain zeta and L-functions, as well as algebraicity results concerning periods and Fourier coefficients of certain automorphic forms.