Geodesic
NORMAL FUNCTIONS FOR ALGEBRAICALLY TRIVIAL CYCLES ARE ALGEBRAIC FOR ARITHMETIC REASONS
Forum of Mathematics, Sigma, Tome 7 (2019)

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For families of smooth complex projective varieties, we show that normal functions arising from algebraically trivial cycle classes are algebraic and defined over the field of definition of the family. In particular, the zero loci of those functions are algebraic and defined over such a field of definition. This proves a conjecture of Charles. 
@article{10_1017_fms_2019_34,
     author = {JEFFREY D. ACHTER and SEBASTIAN CASALAINA-MARTIN and CHARLES VIAL},
     title = {NORMAL {FUNCTIONS} {FOR} {ALGEBRAICALLY} {TRIVIAL} {CYCLES} {ARE} {ALGEBRAIC} {FOR} {ARITHMETIC} {REASONS}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
     volume = {7},
     year = {2019},
     doi = {10.1017/fms.2019.34},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2019.34/}
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JEFFREY D. ACHTER; SEBASTIAN CASALAINA-MARTIN; CHARLES VIAL. NORMAL FUNCTIONS FOR ALGEBRAICALLY TRIVIAL CYCLES ARE ALGEBRAIC FOR ARITHMETIC REASONS. Forum of Mathematics, Sigma, Tome 7 (2019). doi: 10.1017/fms.2019.34

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