Analogues of Brauer-Siegel theorem in arithmetic geometry

@article{Hindry2019AnaloguesOB,
  title={Analogues of Brauer-Siegel theorem in
 arithmetic geometry},
  author={Marc Hindry},
  journal={Arithmetic Geometry: Computation and
                    Applications},
  year={2019}
}
  • M. Hindry
  • Published 2019
  • Mathematics
  • Arithmetic Geometry: Computation and Applications
We will explain analogies between the classical Brauer-Siegel theorem, a statement relating asymptotically the class number, regulator of units and discriminant of a number field, and similar statement involving arithmetic invariants of algebraic varieties over a finite or global field. We present precisely the analogy for surfaces over a finite field and for abelian varieties over a global field (i.e. a number field or the function field of a cuve over a finite field), surveying some recent… 
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References

SHOWING 1-5 OF 5 REFERENCES
On the Brauer–Siegel ratio for abelian varieties over function fields
Hindry has proposed an analogue of the classical Brauer-Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell-Weil group
A Brauer–Siegel theorem for Fermat surfaces over finite fields
TLDR
An analogue of the Brauer-Siegel theorem for Fermat surfaces over a finite field is proved and the product grows like $q^{p_g(F_d)}$ when $d$ tends to infinity.
An Analogue of the Brauer–Siegel Theorem for Abelian Varieties in Positive Characteristic
Consider a family of abelian varieties Ai of fixed dimension defined over the function field of a curve over a finite field. We assume finiteness of the Shafarevic-Tate group of Ai. We ask then when
Analogues of Brauer-Siegel theorem in arithmetic geometry. Arithmetic geometry: computation and applications, Contemp
  • 2019
Analogues of Brauer - Siegel theorem in arithmetic geometry . Arithmetic geometry : computation and applications
  • Contemp . Math .