Brauer-Siegel for arithmetic tori and lower bounds for Galois orbits of special points

By Jacob Tsimerman. 2011

50 citations recorded by the source corpus.

In \cite{S}, Shyr derived an analogue of Dirichlet's class number formula for arithmetic Tori. We use this formula to derive a Brauer-Siegel formula for Tori, relating the Discriminant of a torus to the product of its regulator and class number. We apply this formula to derive asymptotics and lower bounds for Galois orbits of CM points in the Siegel modular variety $A_{g,1}$. Specifically, we ask that the sizes of these orbits grows like a power of Discriminant of the underlying endomorphism algebra. We prove this unconditionally in the case $g\leq 5$, and for all $g$ under the Generalized Riemann Hypothesis for CM fields. Along the way we derive a general transfer principle for torsion in ideal class groups of number fields.

Subfields: Geometry and Topology, Mathematical Physics, Pharmacology

View Paper: https://www.semanticscholar.org/paper/dc95200ecf2ec190f45ef011bb7d921e7dcced0c

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Citations in the corpus

Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves (2017, methodology): “In the case n = 3, we can do even better by appealing to the work of Helfgott and Venkatesh [9] on integral points on elliptic curves, and via the arguments of [14] these improvements are carried over to the case n = 4.”
Ax-Lindemann for $\mathcal{A}_g$ (2014, methodology): “These have been established by the second author [37] unconditionally for g ≤ 6, and for all g on GRH. (The former for g ≤ 3 and the latter were shown independently by Ullmo-Yafaev [41].)”
On the distribution of the Hodge locus (2021, unknown): “We note here that, for 1 < g ≤ 6, it is known by combining the PilaZannier strategy with the Ax–Lindemann conjecture and the Galois bound appearing in [51] (which is easier than the Galois bound needed in Ag, for g arbitrary).”
O-minimality and certain atypical intersections (2014, background): “For special points of Shimura varieties the best results known are those of Tsimerman [45]: lower bounds of the above form for the size of the Galois orbit for special points in the coarse moduli space of Abelian varieties of dimension g, Ag, for g ≤ 6 (or on GRH for all g; see also [47]).”
Towards the Andr\'e-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points (2013, background): “Ullmo and Pila-Tsimerman explained separately in [61] [49] how to deduce the André-Oort conjecture from the Ax-LindemannWeierstraß theorem for pure Shimura varieties.”