Zero point clamping system SPEEDY hydratec 1

WM-020-290-00-en

Maximum productivity
nUsing the zero point clamping system SPEEDY hydratec you will increase your production times to a maximum. nThis clamping element stands for the highest requirements in automation. automated production Clamping

Maximum flexibility

nSPEEDY hydratec has a uniform interface that can be arranged variably. nSPEEDY hydratec features compact design, easy assembly, as well as high, fast changing cycles.

Maximum safety

nInsensitive to soiling due to central locking and effective clearing device. nHighest possible safety in automation due to integrated polling features (clamped, released, mount control).
SPEEDY hydratec an investment that pays for itself in a very short time.
Zero point clamping system
STARK clamping systems overview
Differentiation is by the method of actuation and varying size of the retractable nipples:
SPEEDY classic clamp mechanically / release hydraulically
Most complete and versatile zero point clamping system. n Four sizes and numerous variants
Catalogue order no. WM-020-217-00-en
Catalogue order no. WM-020-276-00-en
Catalogue order no. WM-020-278-00-en
Catalogue order no. WM-020-280-00-en
SPEEDY metec clamp / release mechanically
Robust, mechanical zero point clamping system for easy, low-cost solutions. n Three sizes
Catalogue order no. WM-020-282-00-en
Catalogue order no. WM-020-284-00-en
Catalogue order no. WM-020-286-00-en
SPEEDY airtec clamp mech. / release pneum.
Pneumatic zero point clamping system. n One size and numerous variants
Catalogue order no. WM-020-288-00-en
SPEEDY hydratec clamp / release hydraulically
Hydraulic, double action zero point clamping system for the highest clamping speeds. n One size and numerous variants
Catalogue order no. WM-020-290-00-en
system 3000 clamp / release hydraulically
Double action zero point clamping system with small size and high clamping force. n One size and numerous variants
Catalogue order no. WM-020-066-00-en
system 4000 custom systems
Flexible zero point clamping system for mechanical, pneumatic and hydraulic applications. n One size and numerous variants
Catalogue order no. WM-020-067-00-en

Information

For data sheets and 3D data see www.stark-inc.com
Stark Spannsysteme GmbH A-6840 Goetzis Tel.: +43 (0) / 39-0 Fax: +43 (0) / 39-7 E-mail: verkauf@stark-inc.com www.stark-inc.com Subject to modifications WM-020-290-00-en
Table of contents SPEEDY hydratec 1

Information

STARK Spannsysteme system overview Table of contents SPEEDY hydratec 1 Technical data Transparency from the start Technical data Continuously adjustable clamping force Technical data Tilting torque example calculation Function description SPEEDY hydratec 1 Positioning and clamping in one function
.. i.4.. i.5.. i.6.. i.7.. i.8.. i.9
SPEEDY hydratec 1 l Flush Mount, 115mm

l l l l

Order no. 6000 002.. 1.003.. 1.005.. 1.006.. 1.007.. 1.3.. 1.4
Flush Mount, 115mm, release check Flush Mount, 115mm, release check, with equaliser Flush Mount, 115mm, release check, without centring F lush Mount, 142mm, raised supports, release and clamping check
Hydraulic pump set Information

Retractable nipple

l l l l l l l l l l l
With zero point Without centring With equaliser With O-ring With zero point Without centring With equaliser Variant D Variant E Hardened, ground, 60mm Hardened, ground, 138mm
804 490. 2.485. 2.495. 2.289. 2.493. 2.207. 2.205. 2.125. 2.120. 2.298. 2.284. 2.6
Spacer washer Self-aligning nipple
Nipple fastening Bearing plates

Accessories

Retractable nipple key Nipple fastening key

AF 22 AF22, variant D

804 247. 3.254. 3.033. 3.2
Mechanical insertion force tester
Technical data Transparency from the start
SPEEDY hydratec order no. Catalogue page

Flush Mount module

Bearing surface
Raised supports Release check (2 bar)
Clamping check (2 bar) Element with zero point

Element with equaliser

Element without centring Maintenance interval Max. clamping force Retention force
Cycles [ N ] [ N ] [ bar ] [ bar ] [ N ] [ mm ] [ N ] [ Nm ] [ cm3 ]
750,000 20,000 38,140 20,000 4.5 7,14 80l/min 6bar +10 to +80 0.5 0.- 0.3 1.5 < 0.005 < 0.01 2.90
750,000 20,000 38,140 20,000 4.5 7,14 80l/min 6bar +10 to +80 0.5 0.- 0.3 1.5 < 0.005 < 0.01 3.20
750,000 20,000 38,140 20,000 4.5 7,(1150) 14 100l/min 6bar +10 to +80 0.5 0.- 0.3 1.5 < 0.005 < 0.01 4.10

at 140bar

Min. release pressure Max. operating pressure Extraction force at max. release pressure Extraction, retraction distance ** Lateral forces max. permitted Max. tilting torque (diagonally across supp.) Oil volume for clamping and releasing Air volume clearing device Operating temperature Min. permitted clamping time Min. permitted release time Radial pre-positioning
[ l/min. ] [ C ] [ s ] [ s ] [ mm ] [ mm ] [ ] [ mm ] [ mm ] [ kg ]
Axial pre-positioning Max. loading angle Repeatability System accuracy

Weight

* With appropriate version and adjustments, accuracies in the -range are possible. ** Other extraction and retraction distances possible on request.
Technical data Terminology definitions
1 Clamping force Clamping force refers to the load up to which the zero point is guaranteed. The clamping force stated must not be exceeded. Due to the adjustable clamping force, there is a maximum, at the maximum clamping pressure (see table below). Retention force refers to the max. overload up to which the nipple will continue to be retained, but the zero point has already been left (designed for M10 screw). The loading device must, during manual and automated loading, yield without the application of force. Repeatability refers as a rule to the accuracy with which the same pallet in a specific orientation is changed on the same interface. System accuracy refers to the accuracy obtained on changing several pallets, e.g. on different machines.
2 Retention force 3 Radial pre-positioning 4 Repeatability 5 System accuracy
Variable clamping force with SPEEDY hydratec. The continuously adjustable clamping force makes it possible to optimally clamp different workpieces. By adjusting the clamping pressure, unnecessarily high forces are not applied to the workpiece. As a result the receptacles or pallets do not need to be as strong, depending on the machining - the result is advantages in handling and price. A key aspect, e.g. during milling, is ensuring an adequate, but not excessively high clamping force during the machining to be able to absorb the machining forces. Workpiece clamping based on shape and force is ideal. Depending on the clamping pressure, there is a clamping force of: Clamping force / clamping pressure diagram

[bar] 20 [kN]

Example calculation: Hydraulic clamping pressure = 70bar. According to the diagram this will give a clamping force of 10,000N.
Technical data Tilting torque example calculation Profit from our specialist competence
Example: Fast closing clamp plate 4x SPEEDY hydratec with 200 x 200 spacing and max. feed force of 7 kN with distance of 400 mm. Question: Due to the predominance of roughing work, the system is to be checked for double safety. Are the insertion force, number of fast closing clamps and the selected spacing right for this application? L2 L1

Spacing

D SPEEDY hydratec Jig with workpiece
Pivot point FE Solution: ME > 2 x MV ? MV = FV x LV = 7,000N x 0.4 m MV = 2,800Nm ME = 2 x (FE x L1) + 2 x (FE x L2) ME = 2 x FE x (L1 + L2) L1 = D / 2 L2 = D / 2 + Spacing

Machine table

MV : Moment from feed force ME : Moment from insertion force FV : Feed force (7,000N) FE : Insertion force (20,000N) Spacing = 200 mm = 0.20 m D (Bearing ring) : 60 mm = 0.060 m LV : 400 mm = 0.40 m
L1 + L2 = D + Spacing L1 + L2 = 0.060 m + 0.200 m = 0.260 m ME = 2 x FE x (L1 + L2) = 2 x 20,000N x 0.260 m ME = 10,400Nm ME / MV > 2 ? ME / MV = 10,400Nm / 2,800N ME / MV = 3.7 > 2
With this design, safety by around a factor of 3.7 is provided. Here the pressure could be reduced to 76bar to achieve a double safety margin. (All dimensions to be entered in SI units (metres, Newtons))
Function description SPEEDY hydratec Positioning and clamping in one function
s as roces com p entire ark-inc. e the w.st n se ou ca tion at ww Y ima an an

Principle of operation:

hydratec released: The highly effective clearing device and the central locking keep the bearing surface and the positioning hole clean.
Feed pallet: The central locking is pushed down out of the way and the pallet is positioned at a distance of exactly 4.5 mm * in the correct position for the clamping process. If couplings are used, these are then also in the required axial position for exact engagement. * Other distances possible as an option
hydratec clamped: The clamping pressure is now applied to the system. Protection is normally provided using a non-return valve that can be opened. The pipe for the clearing air is now used to poll whether the pallet is in contact. Clamping elements with further options such as release check and clamping check that poll the piston directly are available in the STARK range.

STANDARD

l l l l l
Module Bearing surface Clearing device Pneumatic mount control Central locking
Characteristics: Flush Mount fast closing clamp module made of high quality tool steel. Due to the compact dimensions, requires little space. Can be installed as a module, in plates or directly in the machine table. Double action fast closing clamp. Is clamped and released hydraulically. Due to its short cycle times, particularly suitable for automation. Positioning is via the precision bore (A) and via the bearing surface (B). Intelligent clearing system for cleaning the bearing surface and the precision bore. Mount control via differential pressure, blocking air possible. Application: For flush mounting in machines, machine pallets, plates, angles, cubes, mounting towers and swivelling yokes. Can be used for all common machining tasks such as milling, grinding, eroding as well as on test stands and mounting devices. Ideal for automated loading.

Z 6000 002

115 A 60 B
Order no. Retention force Clamping force Pressure max. * Weight Data sheet 38,000N 20,000N 140bar 2.90kg D070 * Pressure to release and to clamp the hydratec. See also page i.7.

Practical example:

Fast closing clamp plate with four SPEEDY hydratec. In the middle there is a double coupling for passing hydraulics to the jig. The hydratec is matched to the coupling travel and the axial feed of the Rmheld couplings.

SPEEDY hydratec

l l l l l l
Module Bearing ring Clearing device Pneumatic mount control Pneumatic release check Central locking
Characteristics: Flush Mount fast closing clamp module made of high quality tool steel. Due to the compact dimensions, requires little space. Can be installed as a module, in plates or directly in the machine table. Double action fast closing clamp. Is clamped and released hydraulically. Due to its short cycle times, particularly suitable for automation. Positioning is via the precision bore (A) and via the bearing surface (B). Intelligent clearing system for cleaning the bearing surface and the precision bore. Mount control via differential pressure, blocking air possible. The pneumatic release check provides additional safety in the overall system. Application: For flush mounting in machines, machine pallets, plates, angles, cubes, mounting towers and swivelling yokes. Can be used for all common machining tasks such as milling, grinding, eroding as well as on test stands and mounting devices. Ideal for automated loading.

Z 6000 003

Element with zero point (standard) Order no. Retention force Clamping force Pressure max. * Weight Data sheet 38,000N Element with equaliser Order no. Retention force Clamping force Pressure max. * Weight Data sheet 38,000N Element without centring Order no. Retention force Clamping force Pressure max. * Weight Data sheet 38,000N to 20,000N 140bar 2.70kg D071 to 20,000N 140bar 2.70kg D071 to 20,000N 140bar 2.70kg D071
* Pressure to release and to clamp the hydratec. See also page i.7.

Information:

Depending on the application, there can be significant advantage if all retractable nipples are of the same type. This configuration will make automatic fitting considerably easier and, in many cases, actually make it possible in the first place. For this purpose STARK has a range of hydratec in which the required equalising functions are already incorporated. An increasing number of machining concepts include direct workpiece clamping in the automation. STARK is the right partner for you.

Module Raised supports with clearing nozzles Clearing device Pneumatic mount control Pneumatic clamping check Pneumatic release check Central locking
Characteristics: Flush Mount fast closing clamp module made of high quality tool steel. Due to the compact dimensions, requires little space. Can be installed as a module, in plates or directly in the machine table. Double action fast closing clamp. Is clamped and released hydraulically. Due to its short cycle times, particularly suitable for automation. Positioning is via the precision bore (A) and via the bearing surface (B). Intelligent clearing system for cleaning the raised supports and the precision bore. Mount control via differential pressure, blocking air possible. The pneumatic clamping and release check provides additional safety in the overall system. Application: For flush mounting in machine pallets, plates, angles, cubes, towers and swivelling yokes. For all common machining tasks such as milling, turning, grinding, eroding as well as on test stands for mounting devices. Ideal for automated loading.

A 21,5

142 B 16

Z 6000 007

Order no. Retention force Clamping force Pressure max. * Weight Data sheet 38,000N 20,000N 140bar 4.10kg D070 * Pressure to release and to clamp the hydratec. See also page i.7.

Hydraulic pump set

Double action version With control panel Ready to connect
Characteristics: Every application has different requirements on function and safety. The hydraulic and pneumatic operation and monitoring can be combined in one unit. An experienced engineer from STARK will be available to assist you during design. Options: With integrated hydraulic and electrical control via remote control with 7m cable (with Harting connector). Function triggered using control panel with illuminated pushbuttons. Including hydraulic oil HLP32 and electrical connection with 10m long cable and CEE 5/16 connector. PLC controller for release check, clamping check, system control. Outputs and inputs for the CNC control of the machine. And much more! Talk to us. STARK offers individual solutions for your application

Symbol photo

Practical example: Fully integrated control system on a STARK set-up station with slewing functions in two axes and zero point clamping system.

Retractable nipple

With zero point
Characteristics: Retractable nipple with zero point. Application: For positioning and clamping on machine pallets, machine vices, chucks, jigs, direct workpiece clamping.

M10 26

Z 804 490
Order no. Screw quality min. 10.9
Tightening torque Tightening torque Weight Data sheet at the nipple at the screw 73Nm 48Nm 0.13kg D027

Characteristics: Retractable nipple without centring. Application: For positioning and clamping on machine pallets, machine vices, chucks, jigs, direct workpiece clamping.

27,M10 26

Z 804 485

With equaliser

Characteristics: Retractable nipple with equaliser in one axis (blade shape). Application: For positioning and clamping on machine pallets, machine vices, chucks, jigs, direct workpiece clamping.

22 M10

Z 804 495

Spacer washer

With O-ring
Characteristics: Spacer washer with seal Application: Is used for equalising the raised supports (e.g. 6000 007). The O-ring incorporated seals the centre bore with the pallet clamped so that the system can then be checked pneumatically via the raised supports. Fitting: The spacer ring is fitted and fastened with the retractable nipple. The O-ring is fitted on the hydratec side.

O-ring on hydratec side

Z 804 289

Order no. 804 289

Weight Data sheet 0.02kg Retractable nipple

Self-aligning nipple

Characteristics: Self-aligning retractable nipple with zero point. Application: For positioning and clamping on machine pallets, machine vices, chucks, jigs, direct workpiece clamping. Has advantages on assembly, e.g. in unfavourable angular situations. For this purpose the front part of the nipple is equipped with a self-aligning function for gentle loading.

Z 804 203

Order no. 804 493
Self-aligning func. 1.5mm
Tightening torque Screw 5Nm

Weight 0.30kg

Data sheet D085
Characteristics: Self-aligning retractable nipple without centring. Application: The self-aligning nipple compensates for manufacturing tolerances and thermal expansion. For positioning and clamping on machine pallets, machine vices, chucks, jigs, direct workpiece clamping. Has advantages on assembly, e.g. in unfavourable angular situations. For this purpose the front part of the nipple is equipped with a self-aligning function for gentle loading.

27,30 9

Z 804 207

Order no. 804 207

Characteristics: Self-aligning retractable nipple with equaliser on one axis. Application: The equaliser function compensates for manufacturing tolerances and thermal expansion. For positioning and clamping on machine pallets, machine vices, chucks, jigs, direct workpiece clamping. Has advantages on assembly, e.g. in unfavourable angular situations. For this purpose the front part of the nipple is equipped with a self-aligning function for gentle loading.

Z 804 205

Order no. 804 205

Nipple fastening D

Characteristics: Makes it possible to fasten the nipple with one clamping action. As a result the highest accuracy is achieved. The integrated O-ring prevents the entry of coolant during machining. Application: Machine pallets, machine vices, chucks, jigs, direct workpiece clamping. Spanner for installation (p. 3.1) is used to lock. Tightening is to be undertaken via the nipple.

27,8 O-ring 9,M10 22

Z 809 125

Order no. 804 125

Weight Data sheet 0.09kg D027

Nipple fastening E

Characteristics: Makes it possible to fasten the nipple in one clamping action. As a result the highest accuracy is achieved. Application: Machine pallets, machine vices, chucks, jigs, direct workpiece clamping.

29,8 17

Z 809 120

Order no. 804 120

Weight Data sheet 0.07kg D027

Bearing plate 60

Hardened Ground
Characteristics: Hardened and ground bearing plate. Application: Bearing plate for SPEEDY hydratec. For usage with unhardened pallet surfaces. The high durability guarantees the highest accuracy over a long period. Ideal in combination with the standard option of system control.

Z 804 298

Order no. 804 298
Weight Data sheet 0.12kg D085

Bearing plate 138

Fastening screws (countersunk head screw Torx M4x12) included with the items supplied

137,8 7

Z 804 284

Order no. 804 284

Weight Data sheet 0.80kg D085

Retractable nipple key

Characteristics: Retractable nipple key for fitting and removing SPEEDY retractable nipples. Application: Screwing in place the retractable nipples. (not shown)

Z 804 247

Order no. Across flats Weight AF22 0.15kg AF22 (for nipple fastening 804 125) 0.15kg
Pay attention to torque during fitting, as per technical data.
Fast closing clamp plate with four SPEEDY hydratec. In the middle a quadruple coupling for passing hydraulics to the jig. The hydratec is matched to the coupling travel and the axial feed of the Rmheld couplings. A particularly effective clearing nozzle keeps the couplings clean.

Accessory

Direct force indication in kN
Characteristics: With the aid of the SPEEDY insertion force tester the insertion force on SPEEDY hydratec fast closing clamps is reliably checked. Using the insertion force tester you can measure directly how much force is applied and can be absorbed by the SPEEDY hydratec zero point clamping system. This method is significantly more reliable than a pressure measurement and can save a lot of time during checking. STARK recommends an annual preventive check on the fast closing clamps (note operating manual). Application: Installers, fitters and operators who assemble, fit, service, maintain and operate the SPEEDY hydratec zero point clamping systems. They should be familiar with the use of hydraulic equipment. Attention:all service and maintenance work is only allowed to be undertaken by personnel trained by STARK.

Order no. Description Mechanical insertion force tester SPEEDY hydratec Bearing ring SPEEDY hydratec retractable nipple adapter
Weight 2.70kg 0.50kg 0.80kg 4.00kg

Set weight:

The system accuracy depends not only on the geometry of the joint, the insertion force of the zero point clamping systems also has a significant impact on a positive, stable joint. For safety reasons regular insertion force checks are recommended on the SPEEDY hydratec zero point clamping system. For this purpose, STARK offers a fully mechanical insertion force tester. Its principle of operation is based on a linear change in the length of components that is proportional to force. The length change is converted into a force and indicated on an analogue dial gauge. The mechanical insertion force tester is characterised by its accuracy (measurement precision 3%) and robustness. Low cost and reliable, without any electronics. We supply the mechanical insertion force tester calibrated, with inspection stamp and measuring instructions in a high quality plastic box. STARK provides training courses for training your operators and service personnel. Training courses are held either on site or at Stark Spannsysteme GmbH. Please ask for information, we would be pleased to advise you. Supplied in plastic case: (L390 x W280 x H110)
Insertion force tester (1) with calibration certificate and operating manual in plastic box, with bearing ring (2), retractable nipple adapter (3) and optional spacer washer (4)
Calibration: STARK recommends annual calibration of the insertion force tester. For calibration the insertion force tester can be sent to STARK in the original plastic box.

Only the original.

.fits together!
For this reason our customers receive:
manufacturer's guarantee n function guarantee A n Warranty protection n range of fits A
STARK zero point clamping systems
Consultation, planning, design, production, installation, service all from a single source!
Cost savings in manufacturing are these days increasingly only possible during machine set-up and by shortening the process times. Your production will be significantly faster by using zero point clamping systems. Key aspects such as focusing on bottlenecks (TOC), shortening of cycle times, batch sizes and inventory reduction, to name but a few advantages, are implemented quickly in manufacturing by using STARK zero point clamping systems. Utilise the extensive experience and flexibility of specialists in zero point clamping technology to optimise your production. The double action clamping system SPEEDY hydratec is robust in use and designed for the fastest possible changing processes. Due to its compact design, the SPEEDY hydratec only requires little space for installation, as a result very small spacings can be realised easily.

Due to the special retractable nipple contour and the matched radii, the bore is not damaged on insertion in the SPEEDY. No swarf can be jammed in the cylindrical bore and due to the optimal application of force the retractable nipples are fixed positively and highly accurately by the clamping mechanism bending or lifting is not possible and as a result high positioning accuracy is guaranteed. Positioning, clamping, releasing SPEEDY hydratec integrates everything in an intelligent hydraulic zero point clamping system.
Stark Spannsysteme GmbH Kommingerstrasse 48 A6840 Goetzis Tel. +43 (0) 5523 64739-0 Fax +43 (0) 5523 64739-7 verkauf@stark-inc.com www.stark-inc.com

Introduction to the Abelian Stark Conjectures
Outline Notes, Version of 9/12/04 D. Solomon May 16, 2005

Prerequisites

A rst course in Algebraic Number Theory (for number elds): integers, ideals, absolute values, class groups, units, Dirichlets Theorem, behaviour of primes in Galois extensions, basic theory of Cyclotomic elds. See e.g. [F-T], [La], [Wa] Acquaintance with main theorems of (abelian) class-eld theory in terms of ideals: Mainly ray-class groups/elds and the Artin map. See e.g. [La] or (especially) appendix to [Wa] Basic understanding of representation theory of nite groups over a eld F of characteristic 0, almost exclusively abelian groups, mostly F = C. For the latter, deeper understanding of characters, orthogonality relations, idempotents, eigenspaces (isotypic parts) of modules, connection with the ring/module theory of the group-ring etc. See e.g. [F-T], [Se] Basic familiarity with Riemann zeta-function and Dirichlet L-functions (denitions, Euler product and acquaintance with the functional equation will probably suce. Some knowledge of equivalents for Dedekind zeta-function and Hecke L-functions helpful). See e.g. [F-T], [La], [Wa], article by Martinet in [Fr] General Algebra: Basic theory of rings and modules. Tensor product and exterior powers over commutative rings. Group-rings. Basic notions of complex analysis (analytic continuation, Dirichlet series, Gamma function) Knowledge of p-adic numbers and very basic p-adic analysis. See e.g. [Wa, Ch. 5], [Ko]
Motivation: L-functions of Cyclotomic Fields

Some Denitions

Let f := exp(2i/f ) for f Z1 Set Kf := Q(f ) and Gf := Gal(Kf /Q). Gf (Z/f Z) = a a

a where a (f ) = f.

Identify character group Gf with Dirichlet characters modulo f i.e. ( : Gf C ) ( : (Z/f Z) C ) For s C, (s) > 1, set Lf (s, ) =

n1 (n,f )=1

( ) n ns

() p 1 s p

Primitivity
Note that Lf (s, ) may be imprimitive i.e. there may exist f properly dividing f and Dirichlet character dened modulo f such that ( mod f ) = ( mod f ) a a whenever (a, f ) = 1. In any case, there exists a unique minimal f |f (the conductor of denoted f ) and character dened modulo f satisfying (1) (called the primitive character associated to , denoted ). The usual primitive Dirichlet L-function L(s, ) is just 1

() p ps

= Lf (s, )
and so, since f divides f , we have Lf (s, ) =

p f p|f

L(s, )

For example Lf (s, 0 ) =

L(s, 0 ) =
Analytic Facts about L(s, ) and Lf (s, )
For the following analytic facts about primitive Dirichlet L-functions, see e.g. [Wa, Ch. 4]: (PDL1) L(s, ) has a continuation to C that is analytic at all s C (except for L(s, 0 ) = (s) at s = 1) (PDL2) ords=1 L(s, ) = 0 (except that ords=1 L(s, 0 ) = ords=1 (s) = 1) (PDL3) L(s, ) is related to L(1 s, 1 ) by a functional equation (also involving (s), Gauss sums. ) For any Dirichlet character modulo f we dene r () := 1 if (1) = 1 (say is even) and = if ( = 1 (say is odd ) or if = 0 1)
Then the precise form of the functional equation plus (PDL2) gives : (PDL4) ords=0 L(s, ) = r () Returning to Lf (s, ) for Gf : Eqs. (2)+(PDL1)+(PDL4) give: Theorem 1.1 Suppose that is a Dirichlet character modulo f then Lf (s, ) has a meromorphic continuation to C which is actually holomorphic except that ords=1 Lf (s, 0 ) = 1. Moreover ords=0 Lf (s, ) = r () + #{p : p|f, p f , () = 1} p (3) Note: If = 0 then R.H.S. of (3) becomes simply #{p : p|f }.

Leading Terms at s = 0

Theorem 1.2 For any Lf (0, ) =

a=1 (a,f )=1

a 1 f 2 a f

() a 1 2

(a ) P
Proof: see [Wa, Thm. 4.2] (assumption that primitive is not used).

One can check directly R.H.S.=0 whenever is even and = 0 (or = 0 but f > 1), 3
agreeing with (3). (Harder: Check directly R.H.S.=0 if p s.t. p|f, p f , () = 1.) p Theorem 1.3 If is even and f > 1 (so that Lf (0, ) = 0) then Lf (0, ) = = 1 2

f a a log |1 f |()

log |a (1 f )|(a )
Proof: If is primitive: use [Wa, Thm. 4.9] for Lf (1, 1 ), then the functional equation (and ibid., Lemmas 4.7, 4.8). General case can be deduced from this using norm relations for cyclotomic units, or proved directly, see [Ha, Lec. 3] or refs. on [Tate, p. 79]. P Basic Aim of the Stark Conjectures: Formulate (and prove?) qualitative analogues of Theorems 1.2 and 1.3 when: Kf /Q Y Galois extension of number elds K/k with group G Gf Y group character of G i.e. character of cx. rep. of G leading term of Lf (s, ) at s = 0 Y leading term of Artin L-function LK/k,S (s, ) at s = 0 Extensions and Variations: Replace s = 0 by s = 1; Replace number elds by function elds; Replace L-functions by p-adic L-functions;. Connections with: Explicit Class Field Theory and Hilberts 12th Problem; Stickelbergers Theorem and Generalisations; Additive and Multiplicative Galois-Module Structure; Euler Systems; K-Theory of K;. We shall consider almost exclusively the abelian case, i.e. Gal(K/k) is abelian.

The Function K/k,S (s)

Motivation and Denition
K/k Galois extension of number elds, G := Gal(K/k) abelian CG = group-ring G extends C-linearly to ring hom. : CG C Note: in case K/k = Kf /Q we can rewrite: R.H.S. in Theorem 1.2 =

a f a

R.H.S. in Theorem 1.3 = 1 2

log |a (1 f )|a

and parentheses contain elements of CG which are independent of Idea: In Theorem 1.2 use partial -functions to express L.H.S. as () for some CGf independent of. Then Theorem is a determination of. (Similarly for Theorem 1.3, but only even characters involved. ) To do this in the general case K/k: Sram = Sram (K/k) := {(non-zero) prime ideals p Ok : ep (K/k) > 1} = {p : p|f(K/k)} Fix S nite set of places of k containing Sram as well as S = S (k), the set of infnite (archimedean) places of k. Dene IS := {fractional ideals a of k supported away from S} Thus IS If(K/k) and Artin map restricts to a surjection K/k : IS G, a a = a,K/k For (s) > 1 get an absolutely convergent sum in CG K/k,S (s) :=

a integral

1 N as a

So K/k,S (s) : {s :

(s) > 1} CG is analytic. (s) > 1):
The denition gives the following Basic Facts (currently only for (1) In terms of partial zeta-functions: K/k,S (s) =

K/k,S (s, g)g 1

where K/k,S (, g) : {s :
(s) > 1} C is the partial zeta-function dened by K/k,S (s, g) :=

aIS ,a =g

(2) Euler product in CG: Unique factorisation of fractional ideals gives

K/k,S (s) =

pS n=0

1 (N pn )s pn

1 N ps p
(3) In terms of L-functions: Apply 1 for any G (also considered as a hom. : IS G C ) to get 1 K/k,S (s) =

(p) N ps

=: LK/k,S (s, )
(thus LK/k,S (s, ) is a not-necessarily-primitive Hecke L-function). In other words, setting e =
(g)g 1 (idempotent of CG) we have, for 1 (K/k,S (s))e1

(s) > 1:

K/k,S (s) = =

LK/k,S (s, )e1

Example: K/k = Kf /Q, G = Gf
Sf := {pZ : p|f } {} Sram S
(inclusion is an equality unless f 2 (mod 4)). If n 1 then the integral ideal nZ lies in ISf i (n, f ) = 1 in which case nZ,Kf /Q = n in Gf
(explicit Class Field Theory over Q). So, in terms of zeta-functions: Kf /Q,Sf (s) =

a Gf 1 Kf /Q,Sf (s, a )a

where Kf /Q,Sf (s, a ) =

n1 (n,f )=1,n =a

1 = ns

n1 (mod f )

(f s (s, a/f ), Hurwitz -function) Also, for any Gf , fact (3) gives (Kf /Q,Sf (s)) = LKf /Q,Sf (s, ) =

pf 1 1

= Lf (s, )
(s) > 1 thus in terms of L-functions: Kf /Q,Sf (s) =

Lf (s, )e1

Thus Theorem 1.1 gives a meromorphic continuation Kf /Q,Sf : C CGf and
Theorem 2.1 Kf /Q,Sf (0) =
Proof Suces to show 1 (L.H.S.) = 1 (R.H.S.) for all Gf. But 1 (Kf /Q,Sf (0)) = LKf /Q,Sf (0, ) = Lf (0, ) = 1

a f 1 2

by Theorem 1.2.
Analytic Facts about LK/k,S (s, ) and K/k,S

We need meromorphic continuation of these functions and behaviour at s = 0 in the general case K/k, G, S. Let m = fz = m(K/k) = f(K/k)z(K/k) be a cycle and use the Artin map to identify G with a character of the ray-class group of k modulo m i.e. : Clm (k) C. (See Basic Facts and Notations) Just as for Lf (s, ), we reduce the treatment of LK/k,S (s, ) to the primitive case: unique minimal cycle m() = f()z() dividing m (the conductor of ) and character : Clm() (k) C (the primitive character associated with ) s.t. factors through via the hom. Clm (k) Clm() (k). Note: Let K := K ker so factors through Gal(K /k). Then m() = m(K /k). Thus denes hom. : If() C and the usual primitive Hecke L-function is L(s, ) =
On the other hand If() If IS and |IS = so, by defn. in (3): LK/k,S (s, ) =

pS p f()

The following well-known facts about primitive Hecke L-functions generalise (PDL1) (PDL4): (PHL1) L(s, ) has a continuation to C that is analytic at all s C (except for L(s, 0 ) = k (s) at s = 1) 7
(PHL2) ords=1 L(s, ) = 0 (except that ords=1 L(s, 0 ) = ords=1 k (s) = 1) (PHL3) L(s, ) is related to L(1 s, 1 ) by a functional equation. (This involves a -type factor for each v S (k), Gauss sums. ) For more details see [Tate, 1.3] or [Ha, Lecture 2]). (PHL4) The precise form of the functional equation plus (PHL2) gives : ords=0 L(s, ) = r () where r () := #{v S (k) : v z()} if = 0 #{v S (k) : v z(0 ) = } 1 = #S (k) 1 if = 0
Returning to G: For any nite set T of places of k, let rT () := #{v T : (Dv (K/k)) = {1}} if = 0 #{v T : (Dv (K/k)) = {1}} 1 = #T 1 if = 0
Theorem 2.2 If S contains Sram S and G then LK/k,S (s, ) has a meromorphic continuation to C (holomorphic except for = 0 at s = 1). Moreover ords=0 LK/k,S (s, ) = rS () Proof Eqs. (5)+(PHL1)+(PHL4) give the meromorphic continuation of LK/k,S (s, ) and the formula ords=0 LK/k,S (s, ) = r () + #{p S : p f() and (p) = 1} But an innite place v doesnt divide z() i it is complex or it is real and remains real in K , i.e. i (Dv (K/k)) = {1}. Similarly, p f() satises (p) = 1 i (Dv (K /k)) = {1} i (Dv (K/k)) = {1}. The result follows. P From equation (4) we get Corollary 2.1 K/k,S (s) can be continued to a function C CG that is analytic except at s = 1. P Consider the following hypothesis on a set S Sram S and an integer r 0: H(S, r) : (i) S contains at least r places that split (completely) in K, and (ii) #S r + 1

To say that v splits completely is equivalent to Dv (K/k) = {1} so H(S, r) rS () r G, so
Corollary 2.2 Suppose H(S, r) holds, then there exists K/k,S (0) RG (unique) s.t. K/k,S (s) = K/k,S (0)sr + o(sr )

in CG as s 0

( i.e. H(S, r) K/k,S (s) has at least an rth order zero at s = 0). Proof Since rS () r G, Equation (4) and the Theorem imply the existence of (r) K/k,S (0) in CG. Now the denition shows that K/k,S is real valued on R>1 hence on the whole of R (since meromorphic on C) from which it follows that K/k,S (0) actually lies in RG. P Notes: (i) K/k,S (0) =
(r) 1 dr | . r! dsr s=0 K/k,S (r)
(Warning: Note the implied factor of

in this notation.)

(ii) K/k,S (s) can have an rth order zero at s = 0 without H(S, r).
Stark Conjectures at s = 0
K/k nite abelian extension of number elds G = Gal(K/k), S a nite set of places of k, S S Sram Idea for an rth order Stark Conjecture at s = 0 (r 0): Assume H(S, r) and make conjecture about the form of K/k,S (0)
Basic Zeroth Order Conjecture
#S 1 so H(S, 0) is always satised. So what is K/k,S (0) = K/k,S (0)? Theorem 2.1 answers this for K/k = Kf /Q, S = Sf. One generalisation is Theorem 3.1 (Siegel-Klingen, Shintani) K/k,S (0) QG P Thus the Basic Zeroth Order Stark Conjecture at s = 0 is a theorem! Note: Theorem 3.1 is non-trivial K/k,S (0) = 0 H(S, 1) fails (i) S contains no place splitting in K, or (ii) S = {v} for some v splitting in K. Case (i) : then S contains no place splitting in K so k is totally real, K is totally complex. In this 9
case the element K/k,S (0) QG still contains a lot of interesting information. Case (ii) (less interesting, because (K/k,S (0)) = 0 for = 0 ): then S = {v}, Sram = k = Q = K or k = Q( D), K Hk
Basic First Order Conjecture
v S splitting in K and |S| 2
Assume that H(S, 1) holds, i.e.
+ Motivating Example: K/k = Kf /Q for f Z, f > 1 where + 1 Kf := Q(f + f ) = Q(cos(2/f ))
Take S = Sf = {pZ : p|f } {} as before (minimal unless f 2 (mod 4) or f = 4)
+ Kf is the maximal real subeld of Kf so that the real place v = : Q R C splits in + Kf.
Since also f > 1, hypothesis H(S, 1) is satised.
+ The restriction map Gf G+ := Gal(Kf /Q) yields an isomorphism f
G+ (Z/f Z) /{ 1} = f a |K + [] a
Hence G+ {even Dirichlet characters mod f } f and for any such character 1 (K + /Q,Sf (s)) = LK + /Q,Sf (s, ) =
Thus 1 (K + /Q,S (0)) = Lf (0, ) and we get the following analogue of Theorem 2.1:
Theorem 3.2 K + /Q,S (0) =

log |a (1 f )|[a ]1

1 log |g((1 f )(1 f ))|g 1 gG+ f
(Last equality for f > 2 only.) 10
Proof For rst eq. suces to show 1 (L.H.S.) = 1 (R.H.S.) for all G+. This f follows as for Thm. 2.1 (using Thm. 1.3 in place of Thm. 1.2). If f > 2, second eq. follows by pairing a with a. P

+, 1 Important Remark: (1 f )(1 f ) is element of Kf. Moreover its a unit away from primes of K above those in Sf.
More precisely: + if f = pl , p prime, then its a unit away from the (unique) prime P of Kf above p (which it generates).
+ If f = pl its a genuine cyclotomic unit in E(Kf ). (See e.g. [Wa, Prop. 2.8].)
For any number eld L and nite set T of places we dene the (group of) T -units of L to be UT (L) := {x L : |x|w = 1 w T } = {elements of L which are local units away from T } where | |w denotes the (normalised) absolute value associated to a place w. (The second statement only makes sense if T S (L).)
Thus U (L) = (L) (roots of unity), US (L) (L) = OL = E(L) (unit group) and UT (L) is always f.g. (see below). Often, but not always, one assumes T S (L): If T = S (L) T then UT (L) = OL,T.
Return to K/k, S, G as above and set SK := {places w of K above those in S} S (K) Then SK is stable under action of G on places. Since |gx|w = |x|g1 w , follows that USK (K) is also G-stable so a f.g. module over ZG. Henceforth we write US (K) to mean USK (K). Conjecture C(K/k, S, 1): The Basic First Order Stark Conjecture at s = 0 Suppose K/k is abelian extension of number elds with Galois group G and S is a nite set of places of k containing S Sram. Suppose H(S, 1) is satised. Choose v in S splitting completely in K and a place w dividing v in K. Then there exists US (K) and Q such that K/k,S (0) =
gG + Remark. Our Cyclotomic Example: Take K/k = Kf /Q, S = Sf so H(S, 1) is satised + as above. Take v = : Q R C, w : Kf R C. If f > 2 then Theorem3.2 says 1 that (6) is satised with = 1 , = (1 f )(1 f ) 2 (1)

log |g|w g 1

Remark. Alternative Formulation of (6): In terms of partial zeta-functions: K/k,S (0, g) = log |g|w 11 for all g G
Remark. Variation of w: Fix v and suppose , satisfy Conj. for some w|v. Then w |v w = hw for some h G |g|w = |gh|w , h satisfy Conj. for w. Remark. Variation of v: If S contains another splitting place v = v then we shall see that Conjecture is trivial unless S = {v, v } in which case it follows from Analytic Class Number Formula for k (using either v or v ).
Basic rth Order Conjecture
How can we generalise the basic rst order conjecture when H(S, r) holds? We need to know more about US (K). More generally, for L a number eld and any T as above containing S (L), we dene a logarithmic embedding LT of UT (L): LT : UT (L) RT :=

Rw log ||w w aw w to aw

Let (RT )0 be the kernel of the map RT R sending
Thus (RT )0 is a hyperplane in RT and the Product formula implies im(LT ) (RT )0. Theorem 3.3 (Dirichlets Theorem for T -units) Suppose that T contains S (L). Then (i) ker(LT ) = (K) (ii) im(LT ) is a lattice of full rank in (RT )0. Proof (Sketch) It is easy to show (i) and that im(LT ) is discrete, hence a lattice. So suces to show rkZ (UT (L)/(K)) = #T 1. For T = S (L) this is classical Dirichlet. Now use induction on #(T \ S (L)), the niteness of Cl(L) etc. P Corollary 3.1 UT (L) is isomorphic as an abelian group to Z#T 1 (L). Moreover LT extends by R-linearity to a map LT : R Z UT (L) RT sending i i to i LT (i )

and giving an R-isomorphism of R Z UT (L) onto (RT )0. Return to the case K/k, S, G: the G action on places makes RSK into a natural RG-module and RSK =

R[G/Dv (K/k)]

Notation: If M is a ZG-mod. and R a comm. ring, write RM for R Z M considered as R-module and G-module so RG-module. E.g. RUS (K) is an RG-module and LSK : RUS (K) RSK is RG-linear restricting to RG-iso. RUS (K) (RSK )0. Thus we have an exact sequence of RG-modules 0 RUS (K) RSK R 0 (with trivial G-action on R). Corollary 3.2 For all G dimC (e CUS (K)) = rS () = ords=0 LK/k,S (s, ) = ords=(K/k,S (s)) Proof Exercise: tensor (7) and (8) with C and apply a little character theory. P (8)
In vague terms: the bigger the rank of CUS (K), the higher the order of vanishing of K/k,S (s) Now suppose H(S, r) holds and choose an (ordered) set V = {v1 ,. , vr } S such that vi splits in K for i = 1,. , r. We write S = V V so that RSK = RVK RVK (as RG-modules) and let V denote the projection from RSK to RVK with kernel RVK. We shall write LS for the map LSK and LS,V for the composite V LS :
S V LS,V : RUS (K) RSK RVK
Choose W = {w1 ,. , wr } VK such that wi |vi for each i. Then RVK is RG-free of rank r with the basis {w1 ,. , wr } so for any x RUS (K) we can write uniquely

LS,V (x) =

wi (x)wi
where wi (x) RG One easily checks that wi : RUS (K) RG is the unique RG-linear map satisfying wi (1 u) =

log |gu|wi g 1

Taking rth exterior powers over the commutative ring RG gives an RG-linear map

r RG LS,V

r RG RUS (K)

r RG RVK

= RG(w1 . wr )
Since w1 . wr is a free generator we can dene a unique RG-linear regulator RK/k,W : r r RG RUS (K) RG by RG LS,V (x) = RK/k,W (x)(w1 . wr ). 13
Explicitly, every element of r RUS (K) is a nite sum of terms of form x1 . xr with RG xi RUS (K) and RK/k,W (x1 . xr ) = det(wi (xj ))r i,j=1 Conjecture C(K/k, S, r): The Basic rth Order Stark Conjecture at s = 0 Suppose K/k is abelian extension of number elds with Galois group G and S a is nite set of places of k containing S Sram. Suppose H(S, r) is satised. Choose a set V = {v1 ,. vr } S of places splitting completely in K and a set W = {w1 ,. wr } VK with wi |vi i. Then there exists

QUS (K)

RUS (K) such that (9)

K/k,S (0) = RK/k,W ()

Remark. Variation of W : Fix V and suppose satises the conjecture with the choice W = {w1 ,. , wr }. Let W = {w1 ,. , wr } be another choice. Then wi = hi wi for some hi G for all i so wi (hx) = wi (x) and we nd RK/k,W (x) = RK/k,W (h1. hr x). Thus h1. hr satises the conjecture with the choice W. Remark. Variation of V : Permuting the vi (and hence the wi ) multiplies RK/k () by 1 so still satises the conjecture. This is the only variation possible unless S contains r + 1 splitting places in which case the conjecture can be shown to hold for any choice of V. When r = 1, V = {v} and W = {W }, we have

1 QG QUS (K)

= QUS (K), so = |g|w g 1. So for r = 1,
for some Q and US (K). Hence RK/k,W () = w () =
C(K/k, S, r) indeed agrees with rst order conjecture denoted C(K/k, S, 1) enunciated above.

Uniqueness of

In order to rene the conjecture (for example) we want to study the element r QUS (K) QG satisfying (9). However, is not in general unique, even if it exists. We explain a way to render it unique by projecting onto a certain eigenspace of r QUS (K) as a QG-module: QG Still in the set-up of C(K/k, S, r), we assume H(S, r) is satised and V , W are chosen as above. Then rS () r for every G and we dene eS,r :=

rS ()=r

and eS,>r =

rS ()>r

e = 1 eS,r
Thus eS,r is an idempotent, and the unique element of CG satisfying (eS,r ) = Set s = |S| r + 1 so that S = {v1 ,. , vr } {vr+1 ,. , vs } = V V Write Dv for Dv (K/k) G. (So Dv = {1} if v = v1 ,. , vr ). Let NDv :=
1 if rS () = r 0 if rS () > r

g so that

1 N #Dv Dv
is an idempotent of QG and = 1 if (Dv ) = {1} 0 if (Dv ) = {1}

1 NDv #Dv

Lemma 3.1 1

vV 1 N #Dv Dv 1 N #Dv Dv

if s > r + 1 (so rS (0 ) > r) + e0 if s = r + 1 (so rS (0 ) = r)

eS,r =

In particular, eS,r lies in QG, hence also eS,>r. Proof Exercise using (10) and denition of rS (). Thus for any QG-module A we can dene a QG-submodule A
A[S,r] = eS,r A = {a A : eS,r a = a} = ker(eS,>r |A) Remarks The map a eS,r a projects A onto A[S,r] eS,r QG = QG[S,r] and eS,>r QG are rings with identities the idempotents eS,r and eS,>r respectively. Moreover QG is a product of rings QG[S,r] eS,>r QG and acts on A[S,r] via its projection on QG[S,r]. If f : A B is a QG-hom. then the restriction of f to A[S,r] denes a QG-homomorphism f [S,r] : A[S,r] B [S,r]. In this way, [S,r] denes an exact functor from the category of QGmodules to the category of QG[S,r] modules. Suppose A is embedded in a CG-module A. Then A[S,r] is the intersection of A with the for those s.t. rS () = r. Alternatively, sum of the -eigenspaces e A a A[S,r] e a = 0 in A, for all s.t. rS () > r Proposition 3.1 K/k,S (0) lies in RG[S,r] 15
Proof Suppose rS () > r for some G then (working in CG): e K/k,S (0) = (K/k,S (0))e = LK/k,S (0, )e = 0 by Thm. 2.2. P

(r) (r) (r)

Proposition 3.2 QUS (K)[S,r] is isomorphic to (QG[S,r] )r over QG. Proof Since QG-modules are dened up to iso. by their character which is invariant under C , it suces show that CUS (K)[S,r] (CG[S,r] )r over CG. But for any G, Cor. 3.2 = [S,r] gives dimC (e CUS (K) ) = r if rS () = r, otherwise = 0. The same is obviously true of dimC (e (CG[S,r] )r ) so the result follows. P Remark Similar reasoning shows more generally that QUS (K) is isomorphic over QG to ker(QSK Q) because the isomorphism holds over RG after applying Q R, by (8) Denition (Warning: this is nonstandard!) We shall say that a solution of C(K/k, S, r) is canonical i it lies in
[S,r] r QG QUS (K) r QG QUS (K)

r [S,r] ). QG (QUS (K)

Proposition 3.3 If is any solution of C(K/k, S, r) then eS,r is a canonical solution. Proof RK/k,W (eS,r ) = eS,r RK/k,W () = eS,r K/k,S (0) = K/k,S (0) by Prop. 3.1

(r) (r)

We shall next show that a canonical solution is unique. The RG-injection LS : RUS (K) RSK gives rise to LS

: RUS (K)[S,r] RSK

[S,r] K
If w VK divides v V then Dv xes w and it follows from Lemma 3.1 that RV

= eS,r RVK =

if s > r + 1 w|vr+1 w if s = r + 1 (so V = {vr+1 })
Now recall that LS,V = V LS : RUS (K) RVK , so LS,V = V Proposition 3.4 (i) If s > r + 1 then LS,V = LS. (ii) If s > r + 1 and US (K) is such that 1 QUS (K)[S,r] then in fact UV (K). (iii) In any case LS,V : RUS (K)[S,r] RVK
[S,r] [S,r] [S,r] [S,r] [S,r] [S,r]

: RUS (K)[S,r] RVK

is an isomorphism.
Proof If s > r + 1 then (11) shows that RSK = RVK on which V is the identity, [S,r] proving (i). For (ii), 1 QUS (K)[S,r] implies LS (1 ) lies in RSK RVK by the [S,r] above. Hence UV (K). For (iii), Prop. 3.2 implies RUS (K)[S,r] (RG[S,r] )r RVK by = = [S,r] [S,r] the splitting assumption. So it suces to prove that LS,V is injective. But LS , hence LS , [S,r] [S,r] [S,r] [S,r] P is injective and im(LS ) ker V = (RSK )0 RV K = {0}, by (11). Taking exterior powers in (iii), we nd that r LS,V maps r (RUS (K)[S,r] ) = ( r RUS (K))[S,r] RG RG RG [S,r] r r [S,r] [S,r] isomorphically onto RG (RVK ) = ( RG RVK ) = RG (w1 . wr ) and so Corollary 3.3 RK/k,W restricts to an isomorphism Consequently: Corollary 3.4 There exists at most one canonical solution of C(K/k, S, r). Remark: To summarise: If is any solution of C(K/k, S, r), then eS,r is the unique canonical solution. It is a remarkable (?) fact that in the cases where C(K/k, S, r) is proven, the naturally occurring solution is also usually the canonical solution. Remark: Since QG and QG[S,r] are products of elds, it follows from Prop. 3.2 that every element ( r QUS (K))[S,r] = r [S,r] (QUS (K)[S,r] ) can be written as x1 . xr , with QG QG 1 [S,r] xi QUS (K) i. We can write xi as ni i with ni Z1 and 1i QUS (K)[S,r] (which implies i UV (K) if s > r + 1, by Prop. 3.4 (ii)). In any case, putting n = n1. nr Z1 , we have 1 = () . (1 r ) n Of course, this expression is not unique (even though is). Nevertheless, the condition (9) that be a canonical solution of C(K/k, S, r) can now be written explicitly as follows

(r) K/k,S (0) r [S,r] RG RUS (K) [S,r]

RG[S,r]

det = det(wi (1 j ))r i,j=1 = n n

log |gj |wi g 1

gG i,j=1

References

[B-P-S-S] D. Burns, C. Popescu, J. Sands, D. Solomon, Starks Conjectures: Recent Work and New Directions, (Proceedings of Baltimore conference, 2002) Contemporary Mathematics, vol. 358, Amer. Math. Soc., 2004. [Bu] [Fr] D. Burns, Congruences between Derivatives of Abelian L-Functions at s = 0, preprint, 2004. A.Frhlich (ed.), Algebraic number elds (L-functions and Galois properties), o Proceedings of a Symposium, London, Academic Press , 1977. 17
[F-T] [Ha] [Ko] [La] [Po] [R-S] [Ru] [Se]
A. Frhlich and M. Taylor, Algebraic Number Theory. Cambridge University o Press , 1991. D. Hayes Lecture notes on Starks conjectures, http://www.math.umass.edu/dhayes/lecs.html available at
N. Koblitzp-Adic Numbers, p-Adic Numbers and Zeta-Functions, SpringerVerlag, New York, Berlin, Heidelberg, 1977 (GTM 58) S. Lang, Algebraic number theory 2nd ed, Springer-Verlag, New York, London, 1994, (GTM 110) C. Popescu, Base Change for Stark-Type Conjectures over Z J. Reine und Angew. Math., 542, 2002, p. 85-111. X.-F. Roblot and D. Solomon, Verifying a p-adic Abelian Stark Conjecture at s = 1, Journal of Number Theory, 107, (2004), p. 168-206. K. Rubin, A Stark Conjecture over Z for Abelian L-functions with Multiple Zeros Ann. Inst. Fourier 46, 1996, p. 33-62. J-P. Serre, Reprsentations Linaires de Groupes Finis Paris, Hermann, 1971 e e or the English translation Linear Representations of Finite Groups GTM 42, Springer Verlag, 1977 T. Shintani, On Certain Ray-Class Invariants of Real Quadrtaic Fields, J. Math. Soc. Japan, 30, No. 1, (1978), p. 139-167. D. Solomon, Twisted Zeta-Functions and Abelian Stark Conjectures, Journal of Number Theory, 94, No. 1, (2002), p. 10-48. D. Solomon, On p-Adic Abelian Stark Conjectures at s = 1, Annales de LInstitut Fourier 52, No. 2, (2002), p. 379-417. D. Solomon, On Twisted Zeta-Functions at s = 0 (and Partial Zeta-Functions at s = 1), preprint, 2004. H. Stark, L-Functions at s = 1 I,II,III,IV, Advances in Mathematics, 7, (1971), p. 301-343, 17, (1975), p. 60-92, 22, (1976), p. 64-84, 35, (1980), p. 197-235. J. Tate, Les Conjectures de Stark sur les Fonctions L dArtin en s = 0, Birkhaser, Boston, 1984. u L. Washington, Introduction to Cyclotomic Fields, 2nd Ed., Graduate Texts in Math. 83, Springer-Verlag, New York, 1996.
[Sh] [So1] [So2] [So3] [St] [Tate] [Wa]