We compute the Chow group of 0-cycles on a rational surface defined over a finite extension K of the field Q p of p-adic numbers (p a prime) when it is split by an unramified extension of K. We use intersection theory to define a specialisation map so we need to assume that the surface admits a regular proper integral model. A family of examples is worked out to illustrate the method.

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KeywordsCongruent numbers, elliptic curves, Hasse principle, Birch and Swinnerton-Dyer conjecture, Shafarevich-Tate conjecture. Chandan Singh Dalawatwas born in Bhitwara and now lives in Chhatnag. He would like to see more young people from diverse backgrounds take up mathematics and hopes T h e a n c ie n t u n so lv ed p ro b le m o f co n g ru e n t n u mbe r s h a s bee n red u ced to o n e o f th e m a jo r q u e stio n s o f co n te m p o ra ry a rith m e tic : th e¯n ite n e ss o f th e n u m be r o f c u rv e s o v e r Q w h ich beco m e iso m o rp h ic a t e v e ry p la ce to a g iv e n c u rv e . W e g iv e a n e le m e n ta ry in tro d u c tio n to co n g ru e n t n u m be rs a n d th e ir co n jec tu ra l c h a ra c te risa tio n , d isc u ss lo ca l-to -g lo ba l issu e s lea d in g to th e¯n iten e ss p ro b le m , a n d list a fe w re su lts a n d co n jectu re s in th e a rith m e tic th eo r y o f ellip tic c u rv e s.T h e a rea ® o f a rig h t tria n g le w ith sid es a ;b;c (so th a t a 2 + b 2 = c 2 ) is g iv en b y 2 ® = a b. If a ;b;c a re ra tion a l, th en so is ® . C on v ersely, w h ich ra tio n a l n u m b ers ® a rise a s th e a rea of a ra tion a l righ t tria n gle a ;b;c ? T h is p ro b lem o f ch a ra cterisin g \con g ru en t n u m b ers" { a rea s of ra tio n a l righ t tria n g les { is p erh a p s th e old est u n so lv ed p ro b lem in a ll o f M a th em a tics. It d a tes b a ck to m o re th a n a th o u sa n d yea rs a n d h a s b een va rio u sly a ttrib u ted to th e A ra b s, th e C h in ese, a n d th e In d ia n s.T h ree ex cellen t a cco u n ts o f th e p rob lem a re ava ila b le o n th e W eb : R ight trian gles an d elliptic cu rves b y K a rl R u b in , L e problμ em e des nom bres con gru en ts b y P ierre C olm ez, w h ich a lso a p p ea rs in th e O ctob er 2 0 0 6 issu e of th e G azette des m ath ¶ em aticien s, a n d F ra n z L em m erm eyer's tra n sla tio n C on gru en t n u m bers, elliptic cu rves, an d m odu la r form s of a n a rticle in F ren ch b y G u y H enn ia rt. A m o re elem en ta ry in tro d u ctio n is p rov id ed b y th e n o tes of a lectu re in H on g K o n g by Joh n C o a tes, w h ich h ave a p p ea red in th e A u g u st 2 0 0 5 issu e of th e Q u aterly jo u rn al of pu re an d applied m athem atics. A

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For a local field F with finite residue field of characteristic p, we describe completely the structure of the filtered) and G = Gal(K|F). As an application, we give an elementary proof of Serre's mass formula in degree p. We also determine the compositum C of all degree-p separable extensions with solvable galoisian closure over an arbitrary base field, and show that C is K( p E q −c(E) = n.