Suppose $\mathcal{E}$ is an elliptic curve defined over $\mathbb{Q}$,

Then Hasse's theorem states that for any large characteristic $p$ there exists an algebraic number $\lambda_p$ of modulus $\sqrt{p}$ with

$$ \sharp({\mathcal{E}/\mathbb{F}_p}) = p -\lambda_p -\overline{\lambda_p} +1. $$

Can we construct an elliptic curve $\mathcal{E}'$ such that for large enough primes $p$ we have $$ \sharp({\mathcal{E}'/\mathbb{F}_p}) = p +\lambda_p +\overline{\lambda_p} +1 \ \ ? $$

If not in general, are there any conditions which allow such a construction ? (e.g. $\mathcal{E}$ has CM.)

  • 3
    $\begingroup$ I think multiplicity-one theorem forbids such a construction. Wiles proves that $\lambda_p+\bar{\lambda_p}$ comes from the coeffecient of some modular form of weight 2, and multiplicity-one theorem asserts that coefficients from two modular (eigen)forms satisfy $a_p^2=b_p^2$ for almost all prime $p$ if and only if $a(p)=b(p)\chi(p)$, where $\chi$ is some quadratic character. Your construct forces $\chi(p)=-1$ for almost all prime $p$, which is not possible. $\endgroup$ – zy_ Nov 6 '14 at 15:12
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    $\begingroup$ This is not quite classical multiplicity-one (which is for $GL_2$). It is closely related to multiplicity-one for $SL_2$. For a proof of this statement see D. Ramakrishnan's paper "Recovering modular forms from squares", math.caltech.edu/papers/sq.ps (published in 2000 as an appendix to a paper by Duke and somebody). $\endgroup$ – David Loeffler Nov 6 '14 at 15:38

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