... write M
s
(p
h
) for the number of solutions of the system (1.1) with
x ∈ (Z/p
h
Z)
s
.
THE HASSE PRINCIPLE FOR PAIRS OF DIAGONAL CUBIC FORMS
889
Lemma 12. Suppose that the linear forms L
1
(θ) ... derive an upper
THE HASSE PRINCIPLE FOR PAIRS OF DIAGONAL CUBIC FORMS
877
bound of the shape (4.5), subject to the constraints (4.4), wherein...
... case of Maass forms;
1
this estimate together with the Burgess bound (to
handle the contribution from the continuous spectrum) is sufficient to finish
the proof of Theorem 2.
Remark 1.5. We find rather ... handle the shifted convolutions sums (1.5). The proof
of Theorem 3 in a more general form is given in Section 6. In the appendix we
provide a proof of a subconvexity bou...
... +2)+O(1))}
so that the lower bound in the corollary follows with v
+
= w − ky for some
1 ≤ k ≤ [w/y]. The proof of the upper bound in the corollary is similar.
We now embark on the proof of Theorem 3.1. ... consider now the problem of providing a better localization
of the sign changes of E for small values of u. Our main result of this section
is the follo...
... (n,
n
2
).
Proof. The proof is by induction. The result for Q
0
is obvious, and the
result for Q
1
∼
=
P
1
is Property (D).
Except for the base cases in the previous paragraph, the argument for
the odd ... sum of the outer terms. The right term
provides a generator of degree (2n, n), and the left term provides the rest of
the generators.
The above proof...
... and on this subspace the formula for the dual
action of G is the same (this was used in the proof of Corollary 3.10).
Recall that βG (the Stone-
ˇ
Cech compactification of the discrete group G)
can ... xyx
−1
(see the proof of
THE DERIVATION PROBLEM FOR GROUP ALGEBRAS
225
Corollary 1.2), they write N for the closure of the elements of G belonging
to relativ...
... proof of Theorem 1.
(ii) Actually the above results show how one can read off the covering
properties of the family of intervals I
i,j
(µ) for i<jfrom the corresponding
overlappings of the ... important to keep track of exactly how the parts of the J
+
i
∩
J
−
j
’s are placed to cover G(µ) and J(µ). This has been more or less analysed in
the above proof except...
... eventually the initial path of length
N of C
k
, then eventually the initial path of length N of A
k
is the initial path
of length N of BC. Eventually, the terminal path of length N of A
k
is the
terminal ... sequence of paths
in G, then either {A
k
} is constant or, for all N, the initial and terminal paths
of A
k
of length N are eventually constant.
Proof...
... Z
p
)
×
, E
F
is the closure of the projec-
tion of the global units O
×
F
into U
F
, and C
F
is the closure of the projection of
the subgroup of elliptic units (as defined for example in §1 of [Ru3]) ... denote the formal group giving the kernel of reduction modulo p on E.
The theory of complex multiplication shows that
ˆ
E is a Lubin-Tate formal
group of...
... completes the proof of the claim,
and the proof of the lemma.
The next proposition gives the global estimate for the level set of the
constrained sharp fractional maximal function of u.
Proposition ... lower bounds will be specified
for k. These bounds are required for the proof to work, and depend only on
C and α. To realize the contradiction at the end of...
... influence on the types of
any of the other particles during [0,s] (and of course no influence on the paths
of these other particles), as long as we stay on I
1
∩I
2
(compare the argument
for (2.36) ... most of the estimates for the terms
in the right-hand side here for k ≥ 2. The basic inequalities remain valid for
k = 1 by trivial modifications which we again l...