Remark on fundamental groups and effective Diophantine methods for hyperbolic curves

Abstract

Dedicated to the memory of Serge LangIn a few earlier papers ([8], [9], [10]) attention was called to the striking parallel between the ideas surrounding the well-known conjecture of Birch and Swinnerton-Dyer for elliptic curves, and the mysterious section conjecture of Grothendieck [6] that concerns hyperbolic curves. We wish to explain here some preliminary ideas for 'effective non-abelian descent' on hyperbolic curves equipped with at least one rational point. We again follow in an obvious manner the method of descent on elliptic curves and, therefore, rely on conjectures. In fact, the main point is to substitute the section conjecture for the finiteness of the Shafarevich group. That is to say, the input of the section conjecture is of the form section conjecture ⇒ termination of descent At a number of different lectures on the topic of fundamental groups and Diophantine geometry, the question was raised about the role of surjectivity in the section conjecture as far as Diophantine applications are concerned. This implication is intended as something of a reply.To start the descent, on the other hand, requires the use of p-adic Hodge theory and the unipotent Albanese map. In this process, in general, one another conjecture is unfortunately needed. It could be, for example, the Bloch-Kato conjecture on surjectivity of the p-adic Chern class map that has been referred to in [9]. In other words, via the construction of Selmer varieties and Albanese maps, one deduces an implication Bloch-Kato conjecture ⇒ beginning of descentThe main caveat here arises from lack of actual knowledge of computational issues on the part of the author. To avoid misleading anyone about what is being achieved here, we have in the following section separated out the questionable portions as hypotheses [H] and [H']. That is to say, the objects that mediate this process, namely Galois cohomology groups/varieties and maps between them, seem in principle to be computable. But even to the algorithmically illiterate perspective, it is obvious that actual computation would be daunting to the point of impossibility given the technology of the present day. Nevertheless, it is perhaps not entirely devoid of value to point out one direction of investigation in effective methods, in the hope that even incompetent strategies may eventually be improved through the focussing of sharper skills obviously available in the community. Hence, the present paper.One point of some theoretical interest concerns the comparison with 'effective Mordell conjectures' in the usual sense where upper bounds for heights are proposed. If we fix a point b on the curve and measure heights with respect to the corresponding divisor, the height of another point measures the distance from b at all places. So an upper bound for the height corresponds to a lower bound for the distance from b at all places. On the other hand, what the p-adic Hodge theory provides (in principle) is a lower bound for the p-adic distance between all pairs of points at one place. This lower bound is ...

Integral points on moduli schemes of elliptic curves

“…We mention that these methods, which in fact allow to deal with considerably more general Diophantine problems, are all a priori not effective. See also Bombieri [9] and Kim's discussions in [43]. The first effective finiteness result for solutions in Z × Z of Mordell's equation (1.2) was provided by Baker [6].…”

Arithmetic gauge theory: A brief introduction

“…On the one hand, the gauge-theoretical perspective has the potential to be applicable to a very broad class of phenomena encompassing many of the central problems of current day number theory [38]. On the other, unlike Faltings's proof, which is widely regarded as ineffective, the gauge-theory proof conjecturally leads to a computational method for actually finding rational solutions [37], a theme that is currently under active investigation [8,9,22,23]. It should be remarked that the map A that associates gauge fields to points has been well-known since the 1950s when the variety V is an elliptic curve, an abelian variety, or generally, a commutative algebraic group.…”

A non-abelian conjecture of Tate–Shafarevich type for hyperbolic curves

“…The relationship to the section conjecture has been explored before. For instance in [Kim6], the third author shows that if X (Z p ) n = X (Z p ) for some n, then the section conjecture would in principle allow one to obtain a computable bound on the height of rational points. Our present conjecture, however, is quite different in flavor from the section conjecture and its direct consequences.…”