FOR A PRIME P AND FIXED INTEGERS A AND B, LET N (P) BE THE NUMBER OF SOLUTIONS (X, Y) OF THE CUBIC Y2=X3+AX+B OVER THE FINITE FIELD FP AND LET A (P) =P − N (P). IN 1976, SERGE LANG AND HALE TROTTER FORMULATED A conjecture REGARDING THE DISTRIBUTION OF PRIMES P FOR WHICH A (P) =A FOR A FIXED INTEGER A. THIS conjecture IS WIDELY OPEN. IN THIS TALK WE GIVE AN EXPOSITION OF THIS conjecture AND DESCRIBE SOME OF THE WORK DONE IN THIS TOPIC OVER THE LAST FEW DECADES. WE ALSO REPORT ON OUR RECENT JOINT WORK WITH JAMES PARKS (KTH ROYAL INSTITUTE OF TECHNOLOGY-SWEDEN) ON A VERSION OF THIS conjecture FOR TWO CUBICS (ELLIPTIC CURVES).

LET E: Y2 = X3 + AX + B BE AN ELLIPTIC CURVE. SILVERMAN’S conjecture EXPRESSES THAT IF E IS AN ELLIPTIC CURVE, THEN THERE ARE INFINITELY MANY PRIMES P FOR WHICH E(P): Y2 = X3 +AP2X+BP3 HAS RANK 0, AND THERE ARE INFINITELY MANY PRIMES L FOR WHICH E(L) HAS POSITIVE RANK. IN THIS PAPER, WE INVESTIGATE SILVERMAN’S conjecture FOR A SPECIAL ELLIPTIC CURVE Y2 = X3 − 635318657X.

LET K BE A FIELD AND S=K [X1, ..., XN] BE THE POLYNOMIAL RING IN N VARIABLES OVER THE FIELD K. IN 1982, STANLEY DEFINED WHAT IS NOW CALLED THE STANLEY DEPTH OF A MULTIGRADED S-MODULE. HE conjectureD THAT STANLEY DEPTH IS AN UPPER FOR THE DEPTH OF THE MODULE. THIS conjecture HAS BEEN RECENTLY DISPROVED BY DUVAL ET AL., [2]. IN THIS TALK, WE DESCRIBE THEIR COUNTEREXAMPLE. WE ALSO PRESENT THE RECENT DEVELOPMENTS IN THIS TOPIC.