482

DJ . Lorenzini

^ikal'H^if - i be a matrix row and column euivalent (over Z) to the matrix M. Raynaud has proven that the group Ф:= Z/cjZ x ••• x Z/e„_iZ is the group of components of the Néron model of the jacobian associated to the generic curve. We refer the reader to [8] for the geometric motivations in studying arithmetical graphs. We discuss in this paper the properties of such a matrix M and its associated group Ф and we hope that by presenting here our results without any references to geometry, some non algebraic geometers will take interest in this subject and bring new techniques to the study of these matrices.

n

With the above notations, 2jS — 2 = ^ {di — 2). For any arithmetical graph, we

i = l

n

define its linear rank g^ by Iqq — 2 = ^ г,.(^, — 2). Since it might happen that r^- > 1

1=1 when d, = 1, it is not clear from the definitions that ^o è ß- We prove this fact in 4.7. Both integers g^ and go — ß can be interpreted geometrically [8]. The reader will find tables for arithmetical graphs of linear rank one and two in [9] and [11]. Let l^'^^^''^ denote the exact power of the prime / dividing the integer a. Our guess relative to the structure of Ф (for simple graphs) is the following: if jS ^ w ~ 1, then ф'.^е^- •••'e^_^_ß satisfies ^ ordi {ф){1 — 1) ^ 2go — 2ß. This implies

I prime

in particular that e^ •-e^__^_p S 2^^°""^^. This can also be expressed by saying that Ф splits as a product Гх C^ x ••• x C^, where Ci,...,C^ are cyclic groups and Y^lje^Z x ••• X Z/e„-i_^Z is bounded by an explicit constant depending on éfo-Jîonly.

We prove this fact for a wide class of arithmetical graphs (6.2), including the cases where:

• G is a simple tree (3.5); I Ф| = П^''^ and ^ ord^(|Ф1)(/-l)g26fo-2)S. This

I prime

theorem complements a result of Oort and Lenstra [6], where a bound for Y, ordjd Ф|)(/ — 1) is discussed for the first time.

i prime

• Я = J (6,2); in this case | Ф\ equals к, the number of spanning trees of G.

• For any arithmetical graph, we show that i?:=f|rf'"^ is an integer (4.6) and satisfies the bound ^ oYai{v)(l-\)'£lgQ — 2ß (4.7). Moreover, we show that

I prime

\Ф\йvк ( Ъ . S ) ,

There exists only finitely many structures of arithmetical graph on any given graph G (1.6). For each such structure, we defined its volume v and its Hnear rank^fo- We do not know if these integers are related in any way to the standard numerical invariants associated to a graph.

1 . ТЫ Group Ф

Proposition !.!• Let {G,M,R) be an arithmetical graph. The matrix M satisfies the fallowing properties:

m It is symmetric and represents a positive semidefinite quadratic farm of rankn^ L Its kernel is gemmted over Q by R.