$2$-Selmer groups of hyperelliptic curves with marked points

We show that the class of the locus of hyperelliptic curves with $\ell$ marked Weierstrass points, $m$ marked conjugate pairs of points, and $n$ free marked points is rigid and extremal in the cone of effective codimension-($\ell + m$) classes on $\overline{\mathcal{M}}_{2,\ell+2m+n}$. This generalizes work of Chen and Tarasca and establishes an infinite family of rigid and extremal classes in arbitrarily-high codimension. Comment: Published version

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The Rationality of the Moduli Space of Two-Pointed Ineffective Spin Hyperelliptic Curves

The Quarterly Journal of Mathematics ◽

10.1093/qmath/haab006 ◽

2021 ◽

Author(s):

Francesco Zucconi

Keyword(s):

Moduli Space ◽

Hyperelliptic Curves ◽

Marked Points

Abstract By the geometry of the 3-fold quadric, we show that the coarse moduli space of genus g ineffective spin hyperelliptic curves with two marked points is a rational variety for every g ≥ 2.

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Rational points on hyperelliptic curves having a marked non-Weierstrass point

Compositio Mathematica ◽

10.1112/s0010437x17007515 ◽

2017 ◽

Vol 154 (1) ◽

pp. 188-222

Author(s):

Arul Shankar ◽

Xiaoheng Wang

Keyword(s):

Weierstrass Point ◽

Hyperelliptic Curves ◽

Rational Points ◽

Selmer Groups ◽

Automorphic Representations ◽

The Family ◽

Odd Degree ◽

Fundamental Research ◽

Average Size ◽

Fixed Genus

In this paper, we consider the family of hyperelliptic curves over$\mathbb{Q}$having a fixed genus$n$and a marked rational non-Weierstrass point. We show that when$n\geqslant 9$, a positive proportion of these curves have exactly two rational points, and that this proportion tends to one as$n$tends to infinity. We study rational points on these curves by first obtaining results on the 2-Selmer groups of their Jacobians. In this direction, we prove that the average size of the 2-Selmer groups of the Jacobians of curves in our family is bounded above by 6, which implies a bound of$5/2$on the average rank of these Jacobians. Our results are natural extensions of Poonen and Stoll [Most odd degree hyperelliptic curves have only one rational point, Ann. of Math. (2)180(2014), 1137–1166] and Bhargava and Gross [The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point, inAutomorphic representations and$L$-functions, Tata Inst. Fundam. Res. Stud. Math., vol. 22 (Tata Institute of Fundamental Research, Mumbai, 2013), 23–91], where the analogous results are proved for the family of hyperelliptic curves with a marked rational Weierstrass point.

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Variation of Tamagawa numbers of semistable abelian varieties in field extensions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004118000075 ◽

2018 ◽

Vol 166 (3) ◽

pp. 487-521

Author(s):

L. ALEXANDER BETTS ◽

VLADIMIR DOKCHITSER ◽

V. DOKCHITSER ◽

A. MORGAN

Keyword(s):

Abelian Varieties ◽

Hyperelliptic Curves ◽

Complete Classification ◽

Local Fields ◽

Selmer Groups ◽

Genus 2 ◽

Parity Conjecture ◽

Tamagawa Numbers ◽

Principally Polarised Abelian Varieties

AbstractWe investigate the behaviour of Tamagawa numbers of semistable principally polarised abelian varieties in extensions of local fields. In particular, we give a simple formula for the change of Tamagawa numbers in totally ramified extensions and one that computes Tamagawa numbers up to rational squares in general extensions. As an application, we extend some of the existing results on thep-parity conjecture for Selmer groups of abelian varieties by allowing more general local behaviour. We also give a complete classification of the behaviour of Tamagawa numbers for semistable 2-dimensional principally polarised abelian varieties that is similar to the well-known one for elliptic curves. The appendix explains how to use this classification for Jacobians of genus 2 hyperelliptic curves given by equations of the formy2=f(x), under some simplifying hypotheses.

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Iwasawa Theory of Elliptic Modular Forms Over Imaginary Quadratic Fields at Non-ordinary Primes

International Mathematics Research Notices ◽

10.1093/imrn/rnz117 ◽

2019 ◽

Cited By ~ 2

Author(s):

Kâzım Büyükboduk ◽

Antonio Lei

Keyword(s):

Modular Form ◽

Modular Forms ◽

Quadratic Field ◽

Euler System ◽

Base Change ◽

Iwasawa Theory ◽

Imaginary Quadratic Field ◽

Selmer Groups ◽

Imaginary Quadratic Fields ◽

Elliptic Modular Form

AbstractThis article is a continuation of our previous work [7] on the Iwasawa theory of an elliptic modular form over an imaginary quadratic field $K$, where the modular form in question was assumed to be ordinary at a fixed odd prime $p$. We formulate integral Iwasawa main conjectures at non-ordinary primes $p$ for suitable twists of the base change of a newform $f$ to an imaginary quadratic field $K$ where $p$ splits, over the cyclotomic ${\mathbb{Z}}_p$-extension, the anticyclotomic ${\mathbb{Z}}_p$-extensions (in both the definite and the indefinite cases) as well as the ${\mathbb{Z}}_p^2$-extension of $K$. In order to do so, we define Kobayashi–Sprung-style signed Coleman maps, which we use to introduce doubly signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson–Flach elements (out of the collection of unbounded Beilinson–Flach elements of Loeffler–Zerbes), which we use to define doubly signed $p$-adic $L$-functions. The main conjecture then relates these two sets of objects. Furthermore, we show that the integral Beilinson–Flach elements form a locally restricted Euler system, which in turn allow us to deduce (under certain technical assumptions) one inclusion in each one of the four main conjectures we formulate here (which may be turned into equalities in favorable circumstances).

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Coleman integration for even-degree models of hyperelliptic curves

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157015000029 ◽

2015 ◽

Vol 18 (1) ◽

pp. 258-265 ◽

Cited By ~ 2

Author(s):

Jennifer S. Balakrishnan

Keyword(s):

Number Theory ◽

Computer Science ◽

Hyperelliptic Curves ◽

Open Book ◽

Line Integral ◽

Book Series ◽

Numerical Examples ◽

Lecture Notes ◽

Integration Algorithms ◽

Mathematical Sciences

The Coleman integral is a $p$-adic line integral that encapsulates various quantities of number theoretic interest. Building on the work of Harrison [J. Symbolic Comput. 47 (2012) no. 1, 89–101], we extend the Coleman integration algorithms in Balakrishnan et al. [Algorithmic number theory, Lecture Notes in Computer Science 6197 (Springer, 2010) 16–31] and Balakrishnan [ANTS-X: Proceedings of the Tenth Algorithmic Number Theory Symposium, Open Book Series 1 (Mathematical Sciences Publishers, 2013) 41–61] to even-degree models of hyperelliptic curves. We illustrate our methods with numerical examples computed in Sage.

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