Let $E/k$ be an elliptic curve with CM by $\Oc$. We determine a formula for (a generalization of) the arithmetic local constant of [{\bf5}] at almost all primes of good reduction. We apply this formula to the CM curves defined over $\q$ and are able to describe extensions $F/\q$ over which the $\Oc$-rank of $E$ grows.

## References

S. Chetty,

*Arithmetic local constants for abelian varieties with complex multiplication*, http://www.csbsju.edu/Mathematics/sunil-chetty.htm (preprint). 06862553 10.7169/facm/2016.55.1.5 euclid.facm/1474301230S. Chetty,*Arithmetic local constants for abelian varieties with complex multiplication*, http://www.csbsju.edu/Mathematics/sunil-chetty.htm (preprint). 06862553 10.7169/facm/2016.55.1.5 euclid.facm/1474301230H. Cohen,

*Advanced topics in computational number theory*, Grad. Texts Math. \bf193, Springer, New York, 2000. 0977.11056H. Cohen,*Advanced topics in computational number theory*, Grad. Texts Math. \bf193, Springer, New York, 2000. 0977.11056S. Lang,

*Elliptic functions*, second edition, Grad. Texts Math. \bf112, Springer, New York, 1987. 0615.14018S. Lang,*Elliptic functions*, second edition, Grad. Texts Math. \bf112, Springer, New York, 1987. 0615.14018B. Mazur and K. Rubin,

*Finding large Selmer rank via an arithmetic theory of local constants*, Ann. Math. \bf166 (2007), 581–614. 1219.11084 10.4007/annals.2007.166.579B. Mazur and K. Rubin,*Finding large Selmer rank via an arithmetic theory of local constants*, Ann. Math. \bf166 (2007), 581–614. 1219.11084 10.4007/annals.2007.166.579J.S. Milne,

*Abelian varieties*, in*Arithmetic geometry*, G. Cornell and J. Silverman, eds., Springer-Verlag, New York, 1986, available at http://www.jmilne.org/math/. 0604.14028J.S. Milne,*Abelian varieties*, in*Arithmetic geometry*, G. Cornell and J. Silverman, eds., Springer-Verlag, New York, 1986, available at http://www.jmilne.org/math/. 0604.14028G. Shimura,

*Introduction to the arithmetic theory of automorphic functions*, Princeton University Press, Princeton, 1971. 0221.10029G. Shimura,*Introduction to the arithmetic theory of automorphic functions*, Princeton University Press, Princeton, 1971. 0221.10029A. Silverberg,

*Group order formulas for reductions of CM elliptic curves*, in*Arithmetic, geometry, cryptography and coding theory*, Contemp. Math. \bf521, American Mathematical Society, 2010. 1227.11079A. Silverberg,*Group order formulas for reductions of CM elliptic curves*, in*Arithmetic, geometry, cryptography and coding theory*, Contemp. Math. \bf521, American Mathematical Society, 2010. 1227.11079J. Silverman,

*Arithmetic of elliptic curves*, Grad. Texts Math. \bf106, Springer, New York, 1986. 0585.14026J. Silverman,*Arithmetic of elliptic curves*, Grad. Texts Math. \bf106, Springer, New York, 1986. 0585.14026