Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p
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Reduction of an elliptic curve over the rational numbers to an elliptic curve over a finite field mod p by
Ciro Santilli 32 Updated 2024-09-07 Created 1970-01-01
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This construction taks as input:and it produces an elliptic curve over a finite field of order as output.
- elliptic curve over the rational numbers
- a prime number
The constructions is used in the Birch and Swinnerton-Dyer conjecture.
To do it, we just convert the coefficients and from the Equation "Definition of the elliptic curves" from rational numbers to elements of the finite field.
For example, suppose we have and we are using .
For the denominator , we just use the multiplicative inverse, e.g. supposing we have
where because , related: math.stackexchange.com/questions/1204034/elliptic-curve-reduction-modulo-p