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Rank bias for elliptic curves mod $p$

### Kimball Martin and Thomas Pharis

Vol. 15 (2022), No. 4, 709–726
##### Abstract

We conjecture that, for a fixed prime $p$, rational elliptic curves with higher rank tend to have more points mod $p$. We show that there is an analogous bias for modular forms with respect to root numbers, and conjecture that the order of the rank bias for elliptic curves is greater than that of the root number bias for modular forms.

##### Keywords
elliptic curves, ranks, counting points mod $p$, modular forms, root numbers, Fourier coefficients
##### Mathematical Subject Classification
Primary: 11F11, 11F30, 11G05