Rigid syntomic regulators and the p-adic L-function of a modular form [Elektronische Ressource] / vorgelegt von Maximilian Niklas

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• Mathematics

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• Publié par: universitat_regensburg
• Publié le: 01 janvier 2010
• Nombre de lectures: 21
• Langue: English

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Rigid Syntomic Regulators and the p-adic
L-function of a modular form
DISSERTATION ZUR ERLANGUNG DES DOKTORGRADES
DER NATURWISSENSCHAFTEN (DR. RER. NAT.)
AN DER FAKULTAT FUR MATHEMATIK DER UNIVERSITAT
REGENSBURG
vorgelegt von
Maximilian Niklas
aus
Regensburg
2010Promotionsgesuch eingereicht am: 3. November 2010
Die Arbeit wurde angeleitet von: Prof. Dr. Guido Kings
Prufungsaussc huss:
Prof. Dr. Helmut Abels (Vorsitzender)
Prof. Dr. Guido Kings (1. Gutachter)
Prof. Dr. Kenichi Bannai, Keio University, Japan (2. Gutachter)
Prof. Dr. Uwe Jannsen
Prof. Dr. Klaus Kunnemann (Ersatzprufer)Contents
Introduction 5
Overview 10
Chapter I. Syntomic Eisenstein classes 13
Chapter II. The product of syntomic Eisenstein classes 23
II.1. Syntomic cup product with coe cients 23
II.2. Product structures on modular cohomology groups 28
II.3. The product of two Eisenstein classes 41
II.4. Rigid cohomology and overconvergent modular forms 45
II.5. A theorem of Coleman 47
II.6. Rigid and non-overconvergent forms 49
II.7. A formula for the product of two Eisenstein classes 57
Chapter III. The rigid realization of modular motives 59
III.1. Rigid cohomology and Hecke operators 60
III.2. Classical and p-adic modular forms 69
III.3. De nition of the linear form l 72f;rig
III.4. Panchishkin’s linear form l 73f
III.5. Comparison of the linear forms. 75
Chapter IV. Panchishkin’s measure 77
IV.1. Review of p-adic measures 77
IV.2. Convolution of Eisenstein measures 79
IV.3. Hida’s ordinary projection 86
IV.4. Relation to the p-adic L-function 88
Chapter V. The main theorem 99
V.1. Euler factors and the -projection 99
V.2. Proof of the main theorem 103
Bibliography 111
3Introduction
Fundamental objects studied in Arithmetic Geometry are schemes X of
nite type over Q. One way of obtaining interesting invariants of X is the
following: Assume X is smooth projective of pure dimension d: For each
0i 2d; one can de ne the formal product over all primes
Y
i s i I 1pL(s;hX) := det(1 q jH (X ;Q ) ) ;p let Q
p
where l is a prime = p; is a geometric Frobenius element at p and I isp p
the inertia subgroup at p: The polynomials
i Ipdet(1 TjH (X ;Q ) )p let Q
have coe cients in Q for all l =p such that p is a prime of good reduction
iand conjecturally this is true for all primes. Granted this, L(s;hX) de nes
a holomorphic function in s in some right half-space of the complex plane.
One expects that it can be continued meromorphically to a function on the
iwhole ofC and therefore it makes sense to consider the valuesL (n;hX) for
an arbitrary integer n: The superscript indicates that by value we mean
the rst nonvanishing coe cient in the Laurent series expansion at s =n:
Motivated by the class number formula
hR0
L(0;h SpecK) = ; K=Q a number eld
w
h = class number; w = number of roots of unity; R = regulator;
ione hopes that also for higher dimensions, the analytic invariantsL (n;hX)
are related to algebraic invariants ofX: Conjectures of Beilinson [Bei85],[DS91]
tell us more precisely what we should expect for these values, at least up to a
rational number: He considers higher Chern classes, so-called regulator maps
i+1 i+1
r :H (X;n)!H (X ;R(n))D Rmot D
from rational motivic cohomology into Deligne cohomology. For simplicity,
i
assume n> + 1: Beilinson conjectures that
2
5
666 INTRODUCTION
(1) The restriction ofr to a certainQ-subspace of "integral" elementsD
is an isomorphism after tensoring withR:
(2) The determinant of this isomorphism calculated relative to basis
i+1
elements inH (X;n) on the left hand side and a basis in a naturalmot
Q-structure of Deligne cohomology on the right hand side, is equal
ito L (i + 1 n;hX):
The full conjecture is only known for dimX = 0; where it is deduced from
results of Borel [Bor74] by a comparison of two regulators. The problem
for higher dimensions is that nite dimensionality of the motivic cohomol-
ogy groups involved is not known. It is however still interesting to consider
i+1
the weaker problem of nding a suitable subspace of elements of H (X;n)mot
whose determinant gives the desiredL-value. Let us generalize the situation
slightly and replace X by a (pure) motive M of weight i overQ which we
think of as given by a pair (X;); whereX=Q is smooth and projective and
is a projector in a suitable ring of correspondences. For suchM; we formally
set
1 i+1 1 i+1
H (M(n)) :=p H (X;n); H (M(n)) :=p H (X ;R(n)): Rmot mot D D
i
Here, we always assumen> +1. The weak Beilinson conjecture as formu-
2
lated above can now be extended to the case of motives in an obvious way
and has been proven in a number of cases, for example for motives attached
to Dirichlet characters [Bei85], Hecke characters of imaginary quadratic
elds [ Den89], and Hecke cusp eigenforms of weightk 2 [Bei86], [SS88],
[DS91,x5], [Gea06]. By the modularity theorem, the latter class of exam-
ples includes all elliptic curves overQ:
One can ask if this philosophy relating the complexL-function to regulators
can also be found in the p-adic world, where p is a xed nite prime. For
this, letM be a motive overQ and for simplicity let it have good reduction
at p: One can attach to M p-adic invariants which are of algebraic nature
like its p-adic etale realization or the crystalline realization of its reduction
mod p: Conjecturally, there should also exist a p-adic analytic invariant of
M; thep-adicL-function attached toM: Thep-adicL-function should be a
p-adic analytic function
L (;M) : Hom (Z ;C )!Ccont p(p) p p
on the space of p-adic characters ofZ which is characterized by a certainp
interpolation property with respect to the complexL-function. L (;M) is(p)
an important object in arithmetic and conjecturally is closely related to the
Iwasawa theory ofM: The interpolation property implies that for an integer
nn which is critical in the sense of Deligne, the number L (y ;M) (where(p)INTRODUCTION 7
y :Z ,!C is the obvious inclusion) is algebraic and essentially equal topp
L(n;M) divided by a period coming from the comparison of Betti and de
nRham cohomology. For a noncritical integern; the valueL (y ;M) is much(p)
more mysterious and is a priori just a possibly transcendental p-adic num-
ber. One can ask if it has an interpretation in terms of regulator maps as in
the case of the complexL-function. For this one needs to nd a good target
space for ap-adic regulator map which is analogous to Deligne cohomology.
Deligne cohomology can be thought of as "absolute Betti cohomology". This
means roughly that a complex computing Deligne cohomology is obtained
0from a complex computing Betti cohomology by rst taking the F -part
of the Hodge ltration and then invariants under complex conjugation, the
in nite Frobenius. (Here, taking invariants under a map is used in the
sophisticated sense of taking the shifted mapping cone of 1 :) Therefore,
in order to get a p-adic analogue of Deligne cohomology, we should rst
look for a p-adic Betti cohomology, i.e. a "geometric" p-adic cohomology
theory. Betti cohomology can be considered as the cohomology which is
computed using real-analytic di erential forms on X(C): A natural candi-
date for p-adic Betti cohomology is therefore Berthelot’s rigid cohomology
which is computed using p-adic analytic (overconvergent) di erential forms
0on the rigid analytic space associated toX : If one takes theF -part of theQp
Hodge ltration and then the Frobenius invariants of suitable rigid cohmol-
ogy complexes (this is much more complicated than we make it seem here)
one obtains rigid syntomic cohomology, which has been developed by Besser
in [Bes00]. For a nite extension K of Q with ring of integersO andp K
any smooth scheme overO ; he de nes rigid syntomic (or simply syntomic)K
icohomology groups H (X;n) with Tate twist coe cients which are inde-syn
pendent of auxiliary data. He also de nes higher Chern classes with values
in syntomic cohomology which give a syntomic regulator map
i i
r :H (X;n)!H (X;n):syn mot syn
As in Deligne cohomology one can generalize this to a motiveM and obtain
a regulator map
1 1r :H (M)!H (M):syn mot syn
The purpose of this thesis is to relate this regulator map to the p-adic
L-function of M in case M =M(f)(k +l); where M(f) is the motive con-
structed by Scholl [Sch90] associated to a cusp newform of weightk 2 and
l is a natural number. We assume thatf has good reduction modp and that
p 5: Let us furthermore only for this introduction thatf has ratio-
nal Fourier coe cients. Our strategy for relating the p-adic L-function and8 INTRODUCTION
the syntomic regulator is to imitate the proof of the complex weak Beilin-
son conjecture forM =M(f)(k+l); which consists essentially of three steps:
(1) Describe the image of speci c K-theory classes Eis (’); (themot
Eisenstein symbols) under the regulator map.
(2) Compute explicitly the cup product of these images in order to get
elements in the correct degree.
(3) Relate this product to theL-function using duality and the Rankin-
Selberg method.
In the p-adic case, step 1) has been solved by Bannai-Kings [BK]. We
build on their work and obtain step 2) as our rst main result: Proposition
II.7.1 g