Let $E/K$ be a non-isotrivial elliptic curve over a function field $K$ of characteristic $p$, with field of constant $F_q$, with semistable reduction. Its Hasse-Weil L-function $L(s)$ is a polynomial with integral coefficients $P(t)$ with $t=q^{-s}$ of degree $a$. Using the functional equation of $L$, on can show that there exists a polynomial with integral coefficients too, $p$-primitive, denoted $Q(t)$ and some rational number $b\geq 0$ such that $q^{a/2}P(t/q)=q^b.Q(t)$. Does anyone know an other definition/geometric-arithmetic interpretation of this $b$? Thanks!

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    $\begingroup$ The Newton-above-Hodge theorem will give a lower bound for $b$ as half the dimension of the $(1,1)$ part of the associated de Rham cohomology, which at least away from bad primes can be written as an explicit sum over the singular fibers of the curve. Beyond that, it's a measure of the supersingularity of the elliptic surface. Conjecturally (Tate/BSD), $b=a/2$ if and only if the elliptic surface has, over $\overline{\mathbb F}_q$, rank $a$. $\endgroup$ – Will Sawin Mar 1 at 14:31

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