A rational Diophantine *m*-tuple a is set of *m* non-zero rational numbers with the property that the product of any two of them
increased by 1 is a perfect square. The problem of extension of a rational Diophantine triple {*a*,*b*,c} to a quadruple
leads naturally to studding the elliptic curve
*y*^{2} = (*ax*+1)(*bx*+1)(*cx*+1).**Q** and **Q**(*t*) are obtained by this method. The course will deal with the mentioned connections of elliptic curves
with rational Diophantine *m*-tuples, and also with integer Diophantine *m*-tuples, where questions of possible torsion groups
and integer points on such curves arise.

Topics to be covered in this course include: a review of the basic results on integer and rational Diophantine m-tuples;
introduction to elliptic curves over the field of rational numbers; Mordell-Weil group of elliptic curves induced by
rational Diophantine triples; construction of infinite families of rational Diophantine quintuples and sextuples;
general methods for calculating rank and construction of elliptic curves of large rank; application of rational Diophantine triples
in the construction of high-rank curves over **Q** and **Q**(*t*) with torsion groups
**Z**/2**Z** × **Z**/2**Z**, **Z**/2**Z** × **Z**/4**Z**,
**Z**/2**Z** × **Z**/6**Z** and **Z**/2**Z** × **Z**/8**Z**;
elliptic curves with rank 0; possible torsion groups of elliptic curves induced by integer Diophantine triples;
integer points on some families of elliptic curves; *D*(*q*)-*m*-tuples and elliptic curves;
strong Diophantine triples; elliptic curves induced by rational Diophantine quadruples.
Some additional related topics will be covered through student seminar lectures.

It will be assumed that the students are familiar with the basic notions and results from number theory, at the level covered in the undergraduate course Number Theory.

- A. Dujella:
*Teorija brojeva*, Skolska knjiga, Zagreb, 2019. - A. Dujella:
*Number Theory*, Skolska knjiga, Zagreb, 2021. - A. Filipin, Z. Franusic:
*Diofantovi skupovi*, University of Zagreb, lecture notes, 2020. - J. H. Silverman, J. Tate:
*Rational Points on Elliptic Curves*, Springer-Verlag, Berlin, 1992. - L. C. Washington:
*Elliptic Curves: Number Theory and Cryptography*, CRC Press, Boca Raton, 2008. - Selected recent papers from
https://web.math.pmf.unizg.hr/~duje/dtuples.html

**Lecture notes**

(in pdf format; in Croatian)

Seminar on Number Theory and Algebra (University of Zagreb)

Number Theory - Undergraduate course

Elliptic Curves in Cryptography - Graduate course

Diophantinem-tuples page

High rank elliptic curves with prescribed torsion

Infinite families of elliptic curves with high rank and prescribed torsion

History of elliptic curves rank records

High rank elliptic curves with prescribed torsion over quadratic fields

PARI/GP home page

PARI/GP in browser

SageMathCell

MAGMA Calculator

Elliptic curves in GeoGebra (Sime Suljic)

Number Theory Web

Number theory groups and seminars

Recommended readings for graduate students in number theory