Annales de l’institut Fourier (AIF)

Rubin, Karl

A Stark conjecture “over ${\bf Z}$” for abelian $L$-functions with multiple zeros

Küchle, Oliver

ANNALES DE L’INSTITUT FOURIER
OLIVER KÜCHLE
Multiple point Seshadri constants and the
dimension of adjoint linear series
Annales de l’institut Fourier, tome 46, no 1 (1996), p. 63-71
<http://www.numdam.org/item?id=AIF_1996__46_1_63_0>
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Ann. Inst. Fourier, Grenoble
46, 1 (1996), 63-71
MULTIPLE POINT SESHADRI CONSTANTS AND
THE DIMENSION OF ADJOINT LINEAR SERIES
by Oliver KUCHLE
1. Introduction.
Starting with and motivated in part by the famous Fujita conjectures,
there has been a lot of activity recently concerning effectivity statements
for ample or adjoint line bundles on smooth complex projective varieties,
most prominently Siu's effective version of the big Matsusaka Theorem and
various effective numerical criteria for freeness or very ampleness of adjoint
linear systems due to Demailly, Ein-Lazarsfeld, Kollar, Siu and others (cf.
[De] and the references therein).
Here, less ambitiously, the concept of multiple point Seshadri cons-
tants is used to obtain "effective" Riemann-Roch type estimates on the
number of sections in adjoint linear series. Namely, for a nef and big
divisor L on an n-dimensional smooth projective variety Y with canonical
divisor Ky^ by Riemann-Roch and vanishing the number of sections
/z°(y,Oy(J^y + rL)) for r > 0 is a polynomial of degree n with leading
coefficient L71 jn\ in r. Our "effective" version here (cf. Corollary 3.4) is,
that for any r > n2 > 9, the estimate
^°(y,(M^y+rL))>J^-l
holds. This, together with some variants concerning spanned line bundles,
surfaces and minimal n-folds of general type, follows from universal lower
Supported by Max-Planck-Gesellschaft and Deutsche Forschungsgemeinschaft.
Key words: Seshadri constants — Adjoint linear series.
Math. classification: 14C20.
64 OLIVER KUCHLE
bounds for multiple point Seshadri constants at very general points which
are defined as follows.
Let L be a big and nef line bundle on an n-dimensional (irreducible)
complex projective variety X, and m an integer. For pairwise distinct
^i? • • • jXm € X define the multiple point Seshadri constant at a ; i , . . . , Xm
by
( T ^ • r ^Ce(L,x^,...,Xm) :=mf———7—-^,c ^ multa;, C
i
where the infimum is taken over all integral curves C with C D { r r i , . . . ,
Xm} ̂  0.
Another way of saying this is that e(L, a ; i , . . . , Xm) is the maximum
of all real numbers e such that
m
H=rL-e^E,
i=l
considered as an R-divisor is nef on the blow up /:B^^_^j(X) —^ X
of X along a;i,... ,a;yyi, where Ei denote the exceptional divisors. Since
nef divisors have non-negative self-intersection, this immediately gives the
upper bound
^/T^
£(L,a;i,...,a^) < —n='ym
It turns out that for m ^> 0 and very general points this bound is asymp-
totically sharp.
Here by very general points we mean that (a;i , . . . , Xm) is outside the
union of countably many proper subvarieties of X x - - ' x X, and by general
that (a;i , . . . , Xm) is outside a Zariski closed subset.
Write for short e(L^n^m) for the multiple point Seshadri constant of
L at m very general points. Note that, by the open nature of ampleness,
the multiple point Seshadri constants at general and very general points
are related in the following way: for any 6 > 0 one has
e{L, rri,..., Xm) > e{L, n,m) -6
for general points x\,... ,Xm (cf. [EKL], Lemma 1.4 for the precise argu-
ment).
It is convenient to state our result in terms of the 1-point constant
£(L, n, 1) at a very general point of X. The main result of [EKL] was to
establish the lower bound ^(L, n, 1) > 1/n for arbitrary X. It is conjectured
MULTIPLE POINT SESHADRI CONSTANTS 65
that even e{L, n, 1) > 1 might be true. Here we prove with an elementary
argument:
THEOREM 1.1. — Let L be a nef and big line bundle on an n-
dimensional complex-projective variety X and m > 2 an integer. Then
( T \^ • I ( T ^ ^Ln ^^(m-l)71-1}e(L,n,m) > mm< e{L,n, 1), ———, -v——v———/—— \.
Acknowledgements. — The idea of this paper grew out of the
authors joint work with Lawrence Ein and Robert Lazarsfeld on local
positivity of ample line bundles (cf. [EKL]). I would like to thank them and
Andreas Steffens for valuable conversation about the subject, and Geng Xu
for pointing out an inconsistency in an earlier version.
2. Proof of Theorem 1.1.
2.1. — The first part of the proof is along the lines of [EKL]. To
begin with we remark that there is no loss of generality in supposing that
X is in fact smooth. To see this choose a resolution
f : Y — — X
of singularities and consider the pullback L' = /*L instead of L. Finally
note that e(L, / (^ / i ) , . . . , f(y-m)) = e(V\ 2 /1 , . . . , ym) for any y\,...,ym such
that / is an isomorphism near the yi.
2.2. — Suppose the theorem is not true. Let X771 denote the m-fold
cartesian product ofX minus the diagonals. Then, as in [EKL], (3.3), (3.4),
the fact that, for any real number /3 > 0, the set of pairs
<1 (C, x) C C X an integral curve, x = (a - i , . . . , Xm) € X771,
m
/3-^mul t^G>(G-L)l
1=1
is parametrized by count ably many irreducible quasi-projective varieties
implies that there exist a 6 > 0 and a Zariski open U C X171 such that
for all x = (;TI, . . . ̂ Xm) € U there is an m-exceptional curve Cx based at
r c i , . . . , Xm, i.e. an integral curve satisfying
(1 - 6) Y^mult^ G, > max^ 1 . -2 m —— \(L' C^.
^ \£^n^)' ̂  ^(m-l^-ij' x )
66 OLIVER KUCHLE
Fix such a 6 > 0. It then follows that there is an irreducible variety S
and an irreducible family C C X x S of integral curves together with a
dominant quasi-finite morphism g = (^ i , . . . ,^m) : S —^ X^ such that
the fibre Cs C X of C over s € S is an m-exceptional curve based at
9i(s),...,gm(s).
Such a family C will be called m-exceptional.
2.3. — Next we claim that there exists an integer m7, 2 < m' < m,
and an m'-exceptional family C' C X x S" of curves whose members C'g
pass through each of the gi(s), 1 < i < m', for every s G S".
To prove this start with the m-exceptional family C from (2.2). First
we can assume that, for sufficiently general s G 5, the curve Cg passes
at least through two of the gi(s)^ since otherwise we would obtain a
contradiction to the definition of ^(L,n, 1).
Observe that C and the graphs I\ of the gi are closed in X x S and
therefore also pr^I^ H C), where pr^ denotes the projection X x S —^ S.
Renumbering we then can assume that pr^(r^ D C) == S for j = 1,..., m',
in other words gj{s) e Cs for all s e S and such j, where 2 <, m' < m
by the above. Choosing an appropriate dense open subset S" C S we can
arrange gk{s) ^ Cs for k = m' + 1,.. . , m and all s e S " .
Now pick a subvariety S/ C 5"' such that g ' = (^i , . . . ,5w) : 5" —)>
X771 is quasi-finite. It remains to show that all curves of the resulting
family C' = C\xxS' are indeed m'-exceptional. But this follows from the
elementary observation that
(m'-l)71-1 f 1 (m-l)71-1!
——-———— > min s —, -———-—— >
(m') ~ \271' m71 J
for all positive integers n and all 2 < m' < m. The latter can be shown
by minimizing the real function f(a) = (a - l)71"1^71 in the real interval
[2,m].
2.4. — We proceed by proving that m-exceptional families C with
the property of (2.3) do not exist. The idea is to find an m-exceptional curve
of our family which intersects a given divisor in X having high multiplicity
at 771 — 1 points of the curve properly, and use the fact that in this case
the products of the multiplicities at the points of intersection give a lower
bound for the local intersection numbers.
To this end consider for i = 1,..., m the functions
s ̂  mult^(s)Cs.
MULTIPLE POINT SESHADRI CONSTANTS 67
We claim that these have constant values TI on open dense subsets of S. This
can be proven e.g. by using the relative Samuel stratification (cf. [LeTe],
Theorem (4.15)) of the morphism C —^ S, which gives a finite partition of
C into locally closed subsets on which the relative Hilbert-Samuel functions
are constant. Restricting this stratification to the graphs 1̂  C C then gives
a partition of 1̂  into locally closed subsets on which the Hilbert-Samuel
functions of the local rings Oc^g^s) are constant, and this implies that the
multiplicities are also constant (cf. [Fu], §4.3). The arguments in [LeTe], §4
in fact show that s i—> mult^(s) Cs are Zariski upper-semicontinuous in 5'.
Therefore we can pick a general s' € S such that TI = mult^(s/) Cs' <
mult^(s) Cs for all i and s e S. Assume that 7*1 < r^ <, . . . < r^i.
2.5. — Recall that it is p71jn\ + o(p77') conditions to impose multi-
plicity at least p at a given point, and that different points impose inde-
pendent conditions; hence the theorem of Riemann-Roch shows that, for
k ^> 0, there exists a divisor D' € \kL\ having multiplicity at least
.. ^ k^L-{1-6)-
at p2(5 /),. . . ,^m(5 /)
2.6. — Finally put T := g-^^X.g^s'),... ,gm(s1)). Since g is quasi-
finite and the gi are dominant, T is of dimension dim(^i(T)) = dim(X).
Therefore the subfamily {Ct)teT consists of m-exceptional curves having at
least multiplicity TI at gi(t) = gz{s') for i = 2 , . . . ,m, and their first base
points gi(t),t € T are dense in X. In particular, we can find a t € T such
that Ct ^ D ' . But this gives a contradiction because of
k^/T^ rn
k^C,)=D^C,>{l-6)-———T,r,
^m^^
. .(m-l)A:^^
>. (1 - 6 p — — ) — — V mult̂ ,) Ct
m^/m — 1 ~^ ' '
which proves the theorem.
3. Applications.
We start with a supplementary result concerning surfaces. In case X
is a smooth projective surface Ein and Lazarsfeld proved that ^(A, x} >_\
68 OLIVER KUCHLE
for an ample line bundle A off a countable subset of X (cf. [EL]). From this
one obtains easily:
LEMMA 3.1. — £(L,2,m) > 1 if and only if L2 > m.
Proof (Ein-Lazarsfeld). — Since we are only interested in very
general points the arguments given in [EL] in fact show that e(L, 2,1) ^ 1
holds for any nef and big line bundle L on a projective (possibly singular)
surface. Therefore, after blowing up X along a very general point via
the map fx : X' —> X with exceptional divisor E^ the line bundle
Lx = f^L - E^ is again nef with L2, = L2 - 1. Iterating this procedure
gives the desired result, n
One application of the concept of multiple point Seshadri constants
is to provide a fairly good estimate on the number of sections of adjoint
linear series.
PROPOSITION 3.2. — Let L be a nef and big line bundle on a
smooth projective variety X of dimension n > 2 and m > 2 an integer.
Then
h°{X^Ox(Kx+rL))^m
whenever
r > n • max< 1 2 m
e(L,n,lY ^/Zn'' ^/^(m-l)^
Proof. — Let / : Y —> X be the blowing up of X along m very
general points x^ , . . . , Xm with exceptional divisors E^ , . . . , Em- Since
m
H=f*L-s{L,n,m)^Ei
1=1
is nef and big if e{L, n, m)"- < L " ' / m , the same holds for
(r-—F"-——^}f^+—i-n——^(rL-e(L,n,m)YE-\\^ £(L,n,m)j" £(L,n,m)y v ' ' '^ ' j
if r >_ n/e(L, n, m). Then Kawamata-Viehweg vanishing gives
H\X,Ox(Kx + rL) ®I^ ® • • • ®I^)
m
=H\Y,OY(KY+rrL-n^Ei))=0
i=l
MULTIPLE POINT SESHADRI CONSTANTS 69
if r > n/e(L, n, m), or r = n/e(L, n, m) and e{L, n, m)"' < 2/Vm. In other
words the linear series \Kx + rL| separates the points r r i , . . . ,Xm under
these conditions which are implied by Theorem 1.1 and our assumption. In
particular one obtains h°(X, Ox{Kx + rL)) > m. D
Remark 3.3. — Suppose L is a big line bundle on X which is
spanned. Since L is spanned, the complete linear series |L| induces a
morphism (p : X —^ Z c ^N which is generically finite because L is
big. Therefore any curve through a sufficiently general point is mapped
by (p onto a curve in Z. Fixing such a pair x € (7, we can choose a
hyperplane section H C Z through y(x) meeting (p(C) properly, and
this gives rise to x € D = ^H e \L\ meeting C properly. Therefore
L ' C > multr D- multr C > multa; G, and this shows e(L,n,l) > 1 for
spanned and big L.
COROLLARY 3.4. — Let X be a smooth complex protective variety
of dimension n > 2.
(1) Suppose L is nef, n > 3 and r > n2. Then
h^X^O^Kx+rL))^7^-^-!.
(2) Suppose L is spanned and L71 > (2n/(n + 1))". Then
h°(X^Ox(Kx + (n+ 1)L)) > L71 + 1.
(3) Suppose L is nef and X is a surface. Then
/i°(X,(9x(^x+2L))^L2-!.
Proof.
(1) This follows directly from Proposition 3.2 and the bound
^-^^
plus some elementary estimates. One has n" ^ (8n)"(8n— I)1"" for n ̂  3,
and therefore a first estimate h°(X, Ox^Kx+n^L)) >_ 8n. Then one proves
__m^__ / 1
(m-1)"-1 - V c-L^^^m
if m > en with a c > 8 by expanding (m/(m — I))71"1. Now given r > n2
we can determine the wanted m satisfying ^L71jn^ > ̂ /(m - I)72"1 as
70 OLIVER KUCHLE
follows. We already know that m > 8n, so we can choose m to be the
largest integer satisfying
Tr71!/1
-——— > m.
8 n71 ~
(2) This follows from Proposition 3.2 and Remark 3.3.
(3) This follows in the same spirit from Lemma 3.1. D
Remark 3.5. — Let L be big and nef and X be smooth. According
to Corollary 3.4 the multiple point approach improves the estimate
h\X, Ox(Kx + n{n + s)L)) > (n + s}
\ n )
for the number of sections in adjoint linear series which was obtained in
[EKL] using the generation of s-jets at one very general point. However,
for L71 and r in a certain range it is possible to obtain better lower bounds
for h°{X,Ox(Kx + rL)) by interpolating between the two methods. For
example, if
1 2 mr > (n+ k) • max^
e(L,n,lY ^/Ln' ^/^(m-l)71-1 J 5
then as above one proves that \Kx +rL\ generates A-jets at m very general
points, in particular
^(X.Ox^x+rL^m.f71"^)
\ n /
for such r.
Finally we state a variant of Corollary 3.4 which gives estimates of
the dimension of some linear series on varieties with mild singularities.
PROPOSITION 3.6. — Let X be a n-dimensional normal projective
variety with at most log-terminal singularities, and r a positive integer such
that rKx is a nef and big C'artier-divisor (which is the case e.g. if X is a
minimal n-fold of general type and index r). Then, for all q> n2,
h°{X^Ox{qrKx)) ̂  ̂ ^^-[rKxT - 1.
o Ti
Sketch of proof (cf. also [EKL], (4.6)). — Let
f ' ' Y — > X
MULTIPLE POINT SESHADRI CONSTANTS 71
be a resolution of singularities of X such that
^y+A=E/*JCx+P
with a fractional divisor A supported on a divisor with normal crossings
and P an integral effective /-exceptional divisor.
Now KY + A + {qr - l)f*Kx is numerically equivalent to an integral
divisor, and since
e((qr-l)rKx^^m)=qr-^e{r{rKx)^m)^
a variant of Proposition 3.2 using vanishing for Q-divisors implies that
/i°(y, OY(KY + A + (qr - l)/*^x)) = h\Y, Oy^rKx + P))
=h\X,Ox{qrKx))>m
whenever
. . f 1 2 m \qr — l > r n ' max< , __———-, —, — — — >.
[e{rKx,n,l) ^/(rKxV ^/(rKx^^m - I)71-1 J
Then the claim follows as in the proof of Corollary 3.4. D
BIBLIOGRAPHY
[De] J.-P. DEMAILLY, I^-vanishing theorems for positive line bundles and adjunction
theory, Prepublication de 1'Institut Fourier, Grenoble (1994).
[EKL] L. EIN, 0. KUCHLE and R. LAZARSFELD, Local positivity of ample line bundles,
J. Diff. Geom. (to appear).
[EL] L. EIN and R. LAZARSFELD, Seshadri constants on smooth surfaces, Journees de
Geometric Algebrique d'Orsay, Asterisque, vol. 218 (1993), 177-186.
[Fu] W. FULTON, Intersection Theory, Springer, 1984.
[LeTe] M. LEJEUNE-JALABERT and B. TEISSIER, Normal cones and sheaves of relative
jets, Comp. Math., vol. 28 (1974), 305-331.
Manuscrit recu Ie 17 novembre 1995,
accepte Ie 5 decembre 1995.
Oliver KUCHLE,
Mathematisches Institut
Universitat Bayreuth
D-95440 Bayreuth (Germany).
kuechle@btm8xl.mat.uni-bayreuth.de


Multiple point Seshadri constants and the dimension of adjoint linear series

Vigué, Jean-Pierre

Sur les domaines hyperboliques pour la distance intégrée de Carathéodory

Balser, Werner

Beck, Andreas

ANNALES DE L’INSTITUT FOURIER
WERNER BALSER
ANDREAS BECK
Necessary and sufficient conditions for
matrix summability methods to be stronger
than multisummability
Annales de l’institut Fourier, tome 46, no 5 (1996), p. 1349-1357
<http://www.numdam.org/item?id=AIF_1996__46_5_1349_0>
© Annales de l’institut Fourier, 1996, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
46, 5 (1996), 1349-1357
NECESSARY AND SUFFICIENT CONDITIONS FOR
MATRIX SUMMABILITY METHODS
TO BE STRONGER THAN MULTISUMMABILITY
by W. BALSER and A. BECK
0. Introduction.
Frequently one finds a "solution" of a (non-linear) system of differ-
ential equations in the form of a power series with radius of convergence
equal to zero. This raises the following natural problem which recently has
attracted much attention: Find summability methods summing such formal
solutions (on certain sectorial regions) to an analytic function which also
solves the underlying differential equation.
Recently, J. Ecalle has defined such a summation method which he
named multisummability. Roughly speaking, his method is an iteration of
the classical Borel summability method, and it depends upon finitely many
positive parameters. Given a system of differential equations possessing a
formal solution, one can explicitly compute the "correct values55 for these
parameters and then apply the corresponding method to sum the formal
series. In addition to solving the differential equation, the sum so obtained
has other natural properties; e.g. it is asymptotically equal to the formal
series which we started with (when the variable tends to the origin in the
corresponding region). For a self-contained presentation of the definitions
and results from the theory of multisummability, we refer to [I], or the
survey article of B. Malgrange [6]. Here we will use only some notation and
Key words: Multisummability — Power series — Matrix methods.
Math. classification: 34A25 - 40D25.
1350 W. BALSER & A. BECK
basic results from this theory, which are contained in [I], 6.3, Theorem 1
and 4.3, Theorem 2.
The content of this article is as follows: Given an infinite matrix A, we
give necessary and sufficient conditions under which the summation method
defined by A (in series-to-sequence form) sums every formal power series
which is (&i , . . . fep)-summable (in Ecalle's sense) to the same sum, provided
that kp > 1/2. Up to now, the only example of a matrix A satisfying these
conditions has been given by Jurkat [5]. In a second part we then show
existence of formal power series which are summable by Jurkat ̂ s method
but not ( fc i , . . . fcp)-summable for any choice of the parameters A; i , . . . kp.
This paper contains results of the second author's doctoral disserta-
tion [2], written under the first author's direction.
1. Matrix summability methods for formal power series.
For arbitrary d € R, a € R4" and p € R4' U {00} , let
S{d,a,p) := {z=rei(/> : 0 < r < p, \d - (j)\ < a/2}
denote a sector (on the Riemann surface of the logarithm). A closed sector
S is a set of the form
5(d, a, p) := [z = re^ : 0 < r < p, \d - (f)\ < a/2}
with d and a as above, but p e R4" (so a closed sector is automatically
bounded, but does not contain the origin).
Given an infinite matrix A = (am,n) with am,n € C for m,n e No,
we call a formal power series f(z) = ̂  fnZ" (with complex coefficients fn)
A-summable in a direction d € R, if there exist a > 0 and p C R"^, such
that the following two conditions hold:
00
1) The series fm(^) := S ̂ nfn^ have radius of convergence greater
n=o
than or equal to p, for every m € No.
2) The limit f(z) := lim fm(^) exists uniformly on every closed
m—>oo
subsector of 5(d, a, p).
The so-defined function /, which is analytic on S'(d,a,p), will be
called the A-sum of / on S(d, a, p). We sometimes say that / is A-summable
MATRIX METHODS AND MULTISUMMABILITY 1351
on the sector S{d, a, p) if S(d, a, p) can be chosen as the sector that appears
in 2). We define the kernel functions corresponding to A by
00
km(z) :=^a^,^71,
n=0
provided that the series converge. We say that A = (am,n) is power series
regular, or for short: p-regular, if every power series f(z) = ̂ /n^71, with
radius of convergence R > 0, is A-summable on the disc of radius R about
the origin to its correct value. We call A strongly regular, if it satisfies the
condition
(The kernel-functions km{z)^ m € No are all entire functions, andwe have(S) J"^"1 !̂̂
locally uniformly on C \ {.r G M : x >, 1}.
In other words, this means that A is strongly regular if and only if
the geometric series is A-summable to its natural sum on 5(7r, 27T, oo) and
on 5(0, £,1), for some e > 0. It is a well-known fact (see [4], Th. 135,
that a strongly regular method of summation sums all convergent power
series (i.e. series with positive radius of convergence) in their whole Mittag-
Leffler-star to their analytic continuation. It will turn out later that this
is the right kind of regularity if we want to compare our methods with
multisummability.
For k > 0 we say that A = (am,n) satisfies the order condition {Ok), if
the corresponding kernel functions are all entire functions of (exponential)
order < k. Furthermore, we say that A = (am,n) satisfies the growth
condition (Gk) (for k > 1/2), if the following holds:
(The corresponding kernel functions are all entire functions, and
for every K, > k and arbitrary e € (0,27T — T T / K ) there exist
ci,C2 € R"1", such that
(Gk) <
\km(z)\ <C^\
[for every z € 5'(7r, 27T — e — T T / K , oo) and for every m € No.
It is important to notice that the order condition (Ok) does not imply
the growth condition (GA;)) since the constants ci and 02 in the growth
condition have to be independent of m. Finally, we say (for k > 0) that
1352 W. BALSER & A. BECK
A satisfies the comparison condition (Cfc), if for arbitrary p 6 N, d € M
and ^i > /^2 > ... > i^p > k, whenever f(z) is (^i, . . . , ̂ p)--summable
in direction d to sum /(^), then we also have A-summability of f(z) in
direction d to the same sum.
2. The main result.
In what follows, we give necessary and sufficient conditions for a
summation method A to satisfy the comparison condition (Gk)i for k > 1/2.
LEMMA 1. — Let k > 1/2 and A be an infinite matrix, satisfying
(5), (Ok) and (Gk). Then A satisfies (C^).
Proof. — Let d € R, p € N and M > ^2 > • • • > i^p > k be given
00
and consider a series f(z) = ^/n^, which is ( /^ i , . . . ,/tp)-summable in
o
direction d. Using basic theorems from the theory of multisummability (cf.
[I], Theorem 1 and 4.3, Theorem 2) we can decompose the series / into
a convergent series plus finitely many so-called moment series; so without
loss of generality we can restrict / to be one of these moment series, i.e.
'--L^-
where a € C \ {0}, 7(0) is the path from 0 to a along argw = arga, and '0
is a function, L-integrable on 7(0), satisfying
(2) |^(w)[ < de-52!"!"'
for some ci, 02 > 0 and some k > k. It remains to show that this series is
A-summable to the sum
[ ^(w)
(z) = / —-—- dw,W  I
J^(/'y(a) ^ - Z
for z € «S'(arg a + TT, 27r — Tr/fc, oo). Using (1) and (2) one can easily see that
for m € No and e > 0, condition (0^) implies existence of ci, 03 > 0 with
00 yl"! p/^T^^g"^ °° \y\n
EK,"/nN"</ y( , ) Sl̂ l'ir^
^-n ^0 'L ^-n xn=0 </u ' x 'n=0
fHcie-W ,,m^/^I ie^l^ cJ .< / ——————Cle2!35! drc.
Vo ^
MATRIX METHODS AND MULTISUMMABILITY 1353
The right hand side is finite if we take e small enough, so we can conclude
that
E^/n^Y ^U^d^
^0 A(a) ^ ^/
for every m € No and every z € C.
Now let S'i be a closed subsector of 5'(arga + TT, 27T — TT/^, oo). Then
we can choose r > 0, £• > 0 and K, € (fc, fc) so that 5 C 5(arga + TT, 27r —
£ — T r / K ^ r ) . Doing so, we conclude from (Gk) existence of ci, 02 > 0
(independent of m) so that
\km (z/w)\ < ci e^l^l^6 V^ € 5i, Vm e No.
From this estimate we find
^ f Wk^ (i) dw = /(.),
m•^OOJ^a) w vw7
uniformly on 5i. D
In the following converse to Lemma 1, it is interesting to note that
we only assume our matrix A to sum a very special one-parameter family
of formal power series. From this we conclude that for multisummability
these series play the same role as the geometric series for the question of
p-regularity:
LEMMA 2. — For k > 1/2 and a p-regular matrix A, assume that
the series
00
/^):=^r(i+n//^"
n=0
are A-summable in direction d, for every K > k and d € (0,27r). Then A
satisfies (5), (Ok) and (Gk)-
Proof. — Under the above assumptions, the series
00
fm,.(z) := E a^F(l + n/^
71=0
have positive radius of convergence (and using that K, > k is arbitrary, this
radius of convergence must even be infinite), thus represent entire functions,
for every m G No and K as above. Termwise integration of the series can be
justified to show
(3) km(z) = —— ( U^fm^u)^^ d(u-^
2m A.,e(r)
1354 W. BALSER & A. BECK
with a path of integration 7^,e(T) from 0 along aigz = r + (e + 27r)/(4^)
to a finite point z\ in 5(71-, 27T - T T / K — £, l/3r(27r - T T / K - £•)), then along
the circle \z\ = |2;i| to the ray argz == r - (e: + 27r)/(4/t) and from there
along this ray back to the origin (with arbitrary real r and sufficiently small
e>0).
By assumption, for every d as above there exist e(d), r(d) > 0 so that
the functions fm^(z), as m -> oo, converge uniformly on 5(d, e(d), r(d)) to
a function f(z) which is independent of d. For every a (small), we conclude
(using the compactness of the interval [a, 27r-a]) existence ofr(a) such that
the convergence is uniform (hence the sequence is uniformly bounded) on
5(7r, 2(7r-Q;), r(a)). Choosing r such that 7/c,e(T) lies in 5(7r, 2(7r-a), r(a)),
we can estimate (3) in a straightforward manner, and varying r, we so
obtain (Gk). Moreover, letting m —^ oo in (3), and observing that uniform
convergence allows interchanging limit and integration, we find that the
kernel functions tend to 1/(1 - z) in S'(r,£/(2^),oo). Varying r and using
that K can be taken arbitrarily large, one can easily prove (S) (note that
p-regularity implies convergence of km(z) in the unit disc). D
Lemma 1 and Lemma 2 together now imply our main result:
THEOREM 1. — Let k > 1/2. A p-regular matrix A satisfies the
comparison-condition (Ck) if and only if it satisfies the conditions (S),
(Ok) and (Gfc).
The following result shows that from matrices A satisfying (Ck) we
can construct other matrices satisfying (C^), for some 0 < k < fc, by
deleting certain columns of A.
THEOREM 2. — Let k > 0 be given and A = (a^n), m,n e No
satisfy (Ck). If we define B = (bm,n) by bm,n '-= am,2n, m,n € No, then B
satisfies (Ck.).
Proof. — Let d € R, (^i, . . . , Kp) € R^ with ̂  > ... > K,p > fc/2
00
be given. If f(z) = ^ /n^ is (/^i, . . . ,^p)-multisummable in direction
n=o
d to sum f(z), then it is well-known (see [1]) that the series f(z2) is
(2/ti, . . . , 2/<p)-multisummable in direction d / 2 to sum f(z2). Since 2^p > k
and A satisfies (Ck), there exist a,r > 0, such that
00
S ̂ mftnfnZ^ -^ f(z2) (m -^ Oo)
n=0
MATRIX METHODS AND MULTISUMMABILITY 1355
uniformly on closed subsectors of S(d/2,a,r). Hence we get that
00 00
^ bm^nfn^ = ̂  O'm^nfnZ71
n=0 n=0
converges to f(z) uniformly on closed subsectors of S(d, 2a; r2). D
As an application of Theorem 2, consider a matrix Aa depending
upon a parameter a, and assume that (Cfc) holds for every a. If for every
a there exists an a such that for every m, n
0"m,n(oi) = am,2n(o),
then Theorem 2 implies that each Aa satisfies (C^/s). Repeating this
argument we eventually obtain (Co). For Jurkat's method (see the following
section) one can use this technique to prove (Co).
3. Jurkat's method.
In 1993, W.B. Jurkat [5] studied the matrices Ja = (jm,n(^)) with
Om,n(<^)) = exp(-6m A(an)),
where 6m may be any positive sequence tending to zero as m —> oo, a is a
positive real parameter, and
\(u) = u \og(u + 3) loglog(n + 3).
This method had already been introduced by G.H. Hardy in [3] in connec-
tion with summing special power series with rapidly growing coefficients.
Jurkat showed that his method satisfies (Co), but it was open whether his
method was strictly stronger than, or even equivalent to multisummability.
The following Theorem 3 settles this question.
Let S be a fixed but arbitrary sector (of small opening). By C(Jo, S)
we denote the set of all formal power series that are Ja-summable on
S'(d, a, r). Given a formal power series f(z) = ̂  fn^ € C(Ja, S) we define
/m(^):=^Jm,n(a)/^71.
71=0
Hence / e C(Ja, S) implies that
f(z) := lim fm(z)
1356 W. BALSER & A. BECK
exists uniformly on closed subsectors of S(d,a,r). Now let ~Sk, k € No
denote a sequence of closed subsectors of 5'(d,a,r), that satisfies ~SQ :=
S'(d,_a/2, r/2), ̂  C Sk+i VA; e No and V^ e ^(d, a, r) 3ko e No such that
z € 5'fco. Furthermore, let pk be a sequence in (0, r) with p^ —)• r. It is easy
to see that for arbitrary fc e No and m e No
||̂  := sup sup \fm(z)
m€No ^Sk
and
ll/IL,m := SUp \fm{z)\
\^<Pk
are norms on C(J^S). The linear space C{Ja,S) can be seen to be a
Frechet-space with respect to the above norms. To show the existence of
/ € C(Ja,S) that is not multisummable in any direction, we proceed as
follows: For lm := ^/\og\og(m-}-3), define
' g-2m log(m+3)Zm ^rg-2mlog(m+3)  ^
lin ^ —————.———, -——I
^ m! ^m! J
c^ := mm< —. , ——  ^ m C No,
I ! ii ^m \
where
hm := sup _sup ^jp^(a)r(l 4- Zm^)^
p€No ^eS'(d,a,r) n=0
(note that ̂  is finite since ]^r(l + ̂ n)^ € (7(^,5')). Furthermore,
the series
00
fn ••= ̂  c^r(l + l^n)
m=0
converges for all n e No and therefore
fW ••= E (£ c'"^(l + ̂ 7l)) 's"
71=0 \m==0 /
is a well-defined formal power series. By checking that there exist no
positive constants C, K, k such that \fn\ < CK^l + n / k ) for all n e No,
we conclude that / cannot be multisummable in any direction. Consider
now the sequence fj of formal power series
j / oo \
w ' ' = ' E E^1^-71)^1
m=0 \n=0 /
which is a linear combination of series being in C(JQ,S), hence itself is
in C(Ja,S), for every j e No. Because of our definition of the Cm the
MATRIX METHODS AND MULTISUMMABILITY 1357
series fj may be shown to be a Cauchy sequence in C(Ja'S) (with respect
to the above norms). Since C{Ja^S) is a Prechet-space, this shows that
/ C C(Ja, S). Altogether we get the following
THEOREM 3. — There exist formal power series, that are not
multisummable, but Ja-summable by Jurkat's method.
BIBLIOGRAPHY
[1] W. BALSER, From Divergent Power Series to Analytic Functions, Lecture Notes in
Mathematics 1582, Springer-Verlag, 1994.
[2] A. BECK, Matrix-Summationsverfahren und Multisummierbarkeit, Dissertation,
Ulm, 1995.
[3] G.H. HARDY, Note on a divergent series, Proc. Cambridge Phil. Soc., (1941), 1-8.
[4] G.H. HARDY, Divergent Series, Oxford, 1956.
[5] W.B. JURKAT, Summability of asymptotic power series, Asympt. Anal., 7 (1993),
239-250.
[6] B. MALGRANGE, Sommation des series divergentes, Expo. Math., 13 (1995), 163-
222.
Manuscrit recu Ie 5 fevrier 1996,
accepte Ie 29 mars 1996.
W. BALSER & A. BECK,
Abt. Mathematik V
Universitat Ulm
89069 Ulm (Germany).


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ANNALES DE L’INSTITUT FOURIER
DOUGLAS L. ULMER
Slopes of modular forms and congruences
Annales de l’institut Fourier, tome 46, no 5 (1996), p. 1519
<http://www.numdam.org/item?id=AIF_1996__46_5_1519_0>
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Ann. Inst. Fourier, Grenoble
46, 5 (1996), 1519
CORRIGENDUM
SLOPES OF MODULAR FORMS AND CONGRUENCES
by Douglas L. ULMER
(Article paru dans Ie tome 46 (1996), fascicule 1, pp. 1-32)
The second full paragraph on page 6 should begin "On the other
hand, every normalized eigenform of weight 2 and level pN is congruent
after a twist to a form of level N and weight w with 2 < w < p + 1". (The
words "after a twist" are missing in the published version.)
Also, in Remark 2) of page 9 "character ( - » should be replaced
with "character ^".
It is a pleasure to thank Chandrashekhar Khare and Barry Mazur for
bringing these problems to my attention.
Tucson, Sept. 24'96.
Douglas L. ULMER,
University of Arizona
Department of Mathematics
Tucson AZ 85721 (USA).


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