Congruent numbers, elliptic curves, and the passage from the local to the global

Abstract

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

Simple Methods for Finding Actual Congruent Number Triangles

“…The result that is almost the "poster-child" for congruent numbers is Don Zagier's solution for N = 157, which is always given to show the sizes of the numbers involved. Nobody describes how it was computed and, in fact, some get it wrong, ascribing the solution to the use of Heegner points -a subject on which Zagier was one of the first experts, see Dalawat [6]. In a reply, on 10 Nov. 2015, to a question on the web-site MathOverflow, Noam…”

“…The most well-known such triangle is probably the (3,4,5) triangle, introduced at school. It has area 6. A natural question is what other integers can occur as the area of a right-angled triangle with integer sides?…”