Details
Original language | English |
---|---|
Pages (from-to) | 459-509 |
Number of pages | 51 |
Journal | Journal de Theorie des Nombres de Bordeaux |
Volume | 33 |
Issue number | 2 |
Publication status | Published - 10 Sept 2021 |
Externally published | Yes |
Abstract
We provide a new, elementary proof of the multiplicative independence of pairwise distinct GL +2 (Q)-translates of the modular j-function, a result due originally to Pila and Tsimerman. We are thereby able to generalise this result to a wider class of modular functions. We show that this class includes a set comprising modular functions which arise naturally as Borcherds lifts of certain weakly holomorphic modular forms. For f a modular function belonging to this class, we deduce, for each n ≥ 1, the finiteness of n-tuples of distinct f-special points that are multiplicatively dependent and minimal for this property. This generalises a theorem of Pila and Tsimerman on singular moduli. We then show how these results relate to the Zilber–Pink conjecture for subvarieties of the mixed Shimura variety Y (1) n × G nm and prove some special cases of this conjecture.
ASJC Scopus subject areas
- Mathematics(all)
- Algebra and Number Theory
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In: Journal de Theorie des Nombres de Bordeaux, Vol. 33, No. 2, 10.09.2021, p. 459-509.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Multiplicative independence of modular functions
AU - Fowler, Guy
N1 - Funding information: Manuscrit reçu le 27 mai 2020, révisé le 16 mars 2021, accepté le 19 avril 2021. Mathematics Subject Classification. 11F03, 11G15. Mots-clefs. Modular functions, multiplicative independence, Zilber–Pink conjecture. This work was supported by an EPSRC doctoral scholarship.
PY - 2021/9/10
Y1 - 2021/9/10
N2 - We provide a new, elementary proof of the multiplicative independence of pairwise distinct GL +2 (Q)-translates of the modular j-function, a result due originally to Pila and Tsimerman. We are thereby able to generalise this result to a wider class of modular functions. We show that this class includes a set comprising modular functions which arise naturally as Borcherds lifts of certain weakly holomorphic modular forms. For f a modular function belonging to this class, we deduce, for each n ≥ 1, the finiteness of n-tuples of distinct f-special points that are multiplicatively dependent and minimal for this property. This generalises a theorem of Pila and Tsimerman on singular moduli. We then show how these results relate to the Zilber–Pink conjecture for subvarieties of the mixed Shimura variety Y (1) n × G nm and prove some special cases of this conjecture.
AB - We provide a new, elementary proof of the multiplicative independence of pairwise distinct GL +2 (Q)-translates of the modular j-function, a result due originally to Pila and Tsimerman. We are thereby able to generalise this result to a wider class of modular functions. We show that this class includes a set comprising modular functions which arise naturally as Borcherds lifts of certain weakly holomorphic modular forms. For f a modular function belonging to this class, we deduce, for each n ≥ 1, the finiteness of n-tuples of distinct f-special points that are multiplicatively dependent and minimal for this property. This generalises a theorem of Pila and Tsimerman on singular moduli. We then show how these results relate to the Zilber–Pink conjecture for subvarieties of the mixed Shimura variety Y (1) n × G nm and prove some special cases of this conjecture.
UR - http://www.scopus.com/inward/record.url?scp=85116563013&partnerID=8YFLogxK
U2 - 10.5802/jtnb.1167
DO - 10.5802/jtnb.1167
M3 - Article
VL - 33
SP - 459
EP - 509
JO - Journal de Theorie des Nombres de Bordeaux
JF - Journal de Theorie des Nombres de Bordeaux
SN - 1246-7405
IS - 2
ER -