On the Tate constant

B. Dwork

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Let $D (s, f, g)$ be the Rankin product $L$ -function for two Hilbert cusp forms $f$ and $g$ . This $L$ -function is in fact the standard $L$ -function of an automorphic representation of the algebraic group $G L (2) \times G L (2)$ defined over a totally real field. Under the ordinarity assumption at a given prime $p$ for $f$ and $g$ , we shall construct a $p$ -adic analytic function of several variables which interpolates the algebraic part of $D (m, f, g)$ for critical integers $m$ , regarding all the ingredients $m$ , $f$ and $g$ as variables.

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