##
###
HIGHER ORDER CONGRUENCES AMONGST HASSE–WEIL -VALUES

DANIEL DELBOURGO, LLOYD PETERS

2014
*
Journal of the Australian Mathematical Society
*

For the (d + 1)-dimensional Lie group G = Z × p Z ⊕d p , we determine through the use of p-power congruences a necessary and sufficient set of conditions whereby a collection of abelian L-functions arises from an element in K 1 (Z p G ). If E is a semistable elliptic curve over Q, these abelian L-functions already exist; therefore, one can obtain many new families of higher order p-adic congruences. The first layer congruences are then verified computationally in a variety of cases. 2010
## more »

... tics subject classification: primary 11R23; secondary 11G40, 19B28. For a ring R, we denote by K 1 (R) its first algebraic K-group, in the sense of Milnor. There are three main objectives in this article: (I) to describe the structure of K 1 (Z p G (d) ∞ ) via p-power congruences; (II) to work out these congruences for a family of abelian p-adic L-functions; (III) to numerically verify the predicted congruences in some explicit examples. We should point out that (I) is already fully solved when d = 1 thanks to the results of Kato [11], so our theorems here generalise his method to the d > 1 situation. There already exists a large body of work due to Kakde, Hara, Ritter and Weiss [4, 9, 10, 16] devoted to the study of nonabelian Iwasawa Main Conjectures. The extensions we are considering differ from the 'admissible extensions' in [4] in two important ways: (a) the full Lie extension Q (d) ∞,∆ is not a union of totally real fields; ∞,∆ such that L/Q is pro-p of dimension d + 1. Part (a) obstructs the formulation of an Iwasawa Main Conjecture, as nobody has yet constructed abelian p-adic L-functions in this setting. Part (b) is not so serious. Another point of departure from [4] is that the congruences derived by Kakde, Hara, Ritter and Weiss are described in terms of ideals inside completed group algebras, whereas the congruences derived here (and by Kato in [11] ) are p-adic in flavour. While both approaches ultimately yield necessary and sufficient conditions, in terms of checking congruences via a computer program, the latter is the only one that can be easily implemented (and, even then, numerous computational headaches arise). Remarks. (i) As no Main Conjecture can be formulated over Q (d) ∞,∆ for Tate motives, the next obvious place to look for examples is from the theory of elliptic curves. If U (m) = Gal(Q(µ p ∞ )/Q(µ p m )), then sequences of p-adic L-functions belonging to the algebras Z p U (m) [p −1 ] have already been constructed in [1, [5] [6] [7] . (ii) Some weak congruences were established under technical hypotheses in [1, [5] [6] [7] , inspired by the numerical evidence of the Dokchitser brothers [8] . Following the seminal work of Kakde [4, 10] , there is now a precise recipe that, in principle, allows one to describe K 1 (−) of a noncommutative Iwasawa algebra. To construct theta-maps, one needs a 'dense enough' family of subgroups for G (d) ∞ . In Section 4 we build homomorphisms by applying the appropriate norm map and then quotienting out the commutator. Given any multiplicative character χ : H (d) ∞ → C × p of finite order p v with v ≤ m, one next forms the composition [3] Higher order congruences amongst Hasse-Weil L-values 3 where, for each J ∈ Z (v) ∞ , one defines Delbourgo and L. Peters [4] The following statement constitutes the main algebraic result derived in this article. Theorem 1.1. A collection of elements a J = a v,J ∈ Z p U (v) × lies in the image of K 1 (Z p G (d) ∞ ) under the theta-map if and only if, for all positive integers m: (i) the congruence (1.1) m,h holds at each nontrivial cyclic subgroup h ⊂ H (d) ∞ /p m ; (ii) the congruence (1.2) m holds. Furthermore, the kernel of the theta-map is trivial, that is, θ J is an injection. There is a localised version of this theorem, which works in the following manner. Let S denote a canonical Ore set in the sense of [13] . Then a necessary set of conditions for a system of a v,J ∈ Z p U (v) × S J to lie in the image of K 1 (Z p G (d) ∞ S ) under the S-localisation of the theta-map θ J is that the associated c J satisfy the congruences (1.1) m,h and (1.2) m for m ≥ 1. Conjecture 1.2. The family of congruences (1.1) m,h and (1.2) m is also sufficient to determine whether the elements a v,J As has already occurred with the d = 1 situation studied in [11], we have been unable to establish the sufficiency of these p-power congruences, and unfortunately the conjecture remains unresolved at this point (though almost certainly it is true). For a fixed value of d > 1, the number of cyclic subgroups of H (d) ∞ /p m is of type O(p m(d−1) ), so the system of congruences to be checked will grow rapidly with m. However, if d = 1, the system of congruences grows only linearly as a function of m. If d = 2, then we are dealing with the three-dimensional Lie group G (2) ∞ Z × p Z 2 p , and the result below has some surprising implications for Hasse-Weil L-functions. Corollary 1.3. If d = 2 and m = 1, then (1.1) m,h and (1.2) m are equivalent to: (i) (a 1, h ) p ≡ N 0,1 (a 0,H (d) ∞ ) p mod p 2 ; and (ii) J,[H (d) ∞ :J]=p (a 1,J ) p ≡ N 0,1 (a 0,H (d) ∞ ) p(p+1) mod p 3 , respectively. Suppose that E denotes an elliptic curve defined over Q, and let p 2 be a prime of good ordinary reduction. The Hecke polynomial of E at p factorises into

doi:10.1017/s1446788714000445
fatcat:u4d4nhl5qfa47ox3lcdv2cx7qu