2 results
Faltings height and Néron–Tate height of a theta divisor
- Part of
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- Journal:
- Compositio Mathematica / Volume 158 / Issue 1 / January 2022
- Published online by Cambridge University Press:
- 24 January 2022, pp. 1-32
- Print publication:
- January 2022
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- Article
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We prove a formula, which, given a principally polarized abelian variety
$(A,\lambda )$ over the field of algebraic numbers, relates the stable Faltings height of 
$A$ with the Néron–Tate height of a symmetric theta divisor on 
$A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our formula can be expressed as the tropical moments of the tropicalizations of 
$(A,\lambda )$.
The Chowla–Selberg formula for abelian CM fields and Faltings heights
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- Journal:
- Compositio Mathematica / Volume 152 / Issue 3 / March 2016
- Published online by Cambridge University Press:
- 24 September 2015, pp. 445-476
- Print publication:
- March 2016
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In this paper we establish a Chowla–Selberg formula for abelian CM fields. This is an identity which relates values of a Hilbert modular function at CM points to values of Euler’s gamma function
${\rm\Gamma}$ and an analogous function 
${\rm\Gamma}_{2}$ at rational numbers. We combine this identity with work of Colmez to relate the CM values of the Hilbert modular function to Faltings heights of CM abelian varieties. We also give explicit formulas for products of exponentials of Faltings heights, allowing us to study some of their arithmetic properties using the Lang–Rohrlich conjecture.
