arXiv subjects

Ananth N. Shankar

Publications and source records attributed to Ananth N. Shankar.

21 records · Page 2Linked to original sources

arXiv2017

Exceptional splitting of reductions of abelian surfaces

Heuristics based on the Sato--Tate conjecture suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces having real multiplication. Similar to Charles' theorem on exceptional isogeny of reductions of a given pair of elliptic curves and Elkies' theorem on supersingular reductions of a given elliptic curve, our theorem shows that a density-zero set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.

math.NT↗

arXiv2016

2-Selmer groups of hyperelliptic curves with two marked points

We consider the family of hyperelliptic curves over $\Q$ of fixed genus along with a marked rational Weierstrass point and a marked rational non-Weierstrass point. When these curves are ordered by height, we prove that the average Mordell-Weil rank of their Jacobians is bounded above by 5/2. We prove this by showing that the average rank of the 2-Selmer groups is bounded above by 6. We also prove that the average size of the $ϕ$-Selmer groups of a family of isogenies associated to this family is exactly 2.

math.NT↗

arXiv2016

The $p$-curvature conjecture and monodromy about simple closed loops

The Grothendieck-Katz $p$-curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$-curvature vanishes modulo $p$, for almost all primes $p$. We prove that if the variety is a generic curve, then every simple closed loop on the curve has finite monodromy.

math.NT↗