Rigorous computation of the endomorphism ring of a Jacobian

@article{Costa2019RigorousCO,
  title={Rigorous computation of the endomorphism ring of a Jacobian},
  author={Edgar Costa and Nicolas Mascot and J. Sijsling and J. Voight},
  journal={Math. Comput.},
  year={2019},
  volume={88},
  pages={1303-1339}
}
We describe several improvements to algorithms for the rigorous computation of the endomorphism ring of the Jacobian of a curve defined over a number field. 
Numerical computation of endomorphism rings
We give practical numerical methods to compute the period matrix of a plane algebraic curve (not necessarily smooth). We show how automorphisms and isomorphisms of such curves, as well as theExpand
A PRYM VARIETY WITH EVERYWHERE GOOD REDUCTION
We compute an equation for a modular abelian surface A that has everywhere good reduction over the quadratic field K = Q( √ 61) and that does not admit a principal polarization over K.
Identifying central endomorphisms of an abelian variety via Frobenius endomorphisms
Assuming the Mumford-Tate conjecture, we show that the center of the endomorphism ring of an abelian variety defined over a number field can be recovered from an appropriate intersection of theExpand
A database of nonhyperelliptic genus-3 curves over ℚ
We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant.
Separation of periods of quartic surfaces.
We give an effective lower bound on the distance between two distinct periods of a given quartic surface defined over the algebraic numbers. The main ingredient is the determination of height boundsExpand
Computing isogenies from modular equations between Jacobians of genus 2 curves
Let k be a field of large enough characteristic. We present an algorithm solving the following problem: given two genus 2 curves over k with isogenous Jacobians, compute an isogeny between themExpand
Lifts of Hilbert modular forms and application to modularity of abelian varieties
In this paper, we prove the existence of certain lifts of Hilbert cusp forms to general odd spin groups. We then use those lifts to provide evidence for a conjecture of Gross on the modularity ofExpand
Computing period matrices and the Abel-Jacobi map of superelliptic curves
TLDR
An algorithm for the computation of period matrices and the Abel-Jacobi map of complex superelliptic curves based on rigorous numerical integration of differentials between Weierstrass points relies on Gauss method and Double-Exponential method. Expand
Plane quartics over $\mathbb {Q}$ with complex multiplication
We give examples of smooth plane quartics over QQQ with complex multiplication over Q¯¯¯¯Q¯\overline{Q} by a maximal order with primitive CM type. We describe the required algorithms as we go, theseExpand
Counterexamples to a Conjecture of Ahmadi and Shparlinski
Ahmadi-Shparlinski conjectured that every ordinary, geometrically simple Jacobian over a finite field has maximal angle rank. Using the L-Functions and Modular Forms Database, we provide twoExpand
...
1
2
3
4
...

References

SHOWING 1-10 OF 41 REFERENCES
Computing functions on Jacobians and their quotients
We show how to efficiently compute functions on jacobian varieties and their quotients. We deduce a quasi-optimal algorithm to compute $(l,l)$ isogenies between jacobians of genus two curves.
Computational Aspects of Curves of Genus at Least 2
  • B. Poonen
  • Mathematics, Computer Science
  • ANTS
  • 1996
This survey discusses algorithms and explicit calculations for curves of genus at least 2 and their Jacobians, mainly over number fields and finite fields. Miscellaneous examples and a list ofExpand
Proving that a genus 2 curve has complex multiplication
TLDR
This paper proves the conjecture that genus 2 curves defined over the rationals should have complex multiplication by computing an explicit representation of a rational map defining complex multiplication. Expand
Examples of genus two CM curves defined over the rationals
TLDR
A systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field finds 19 non-isomorphic such curves. Expand
A database of genus 2 curves over the rational numbers
We describe the construction of a database of genus 2 curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated L-function. ThisExpand
Computing the geometric endomorphism ring of a genus-2 Jacobian
  • D. Lombardo
  • Mathematics, Computer Science
  • Math. Comput.
  • 2019
TLDR
This work uses an algorithm to confirm that the description of the structure of the geometric endomorphism ring of $\operatorname{Jac}(C)$ given in the LMFDB ($L$-functions and modular forms database) is correct for all the genus 2 curves currently listed in it. Expand
Real multiplication through explicit correspondences
We describe a method to compute equations for real multiplica- tion on the divisors of genus two curves via algebraic correspondences. We implement our method for various examples drawn from theExpand
Lifts of Hilbert modular forms and application to modularity of abelian varieties
In this paper, we prove the existence of certain lifts of Hilbert cusp forms to general odd spin groups. We then use those lifts to provide evidence for a conjecture of Gross on the modularity ofExpand
Poonen's question concerning isogenies between Smart's genus 2 curves
TLDR
A method for proving that two explicitly given genus two curves have isogenous jacobians is described and applied to the list of genus 2 curves with good reduction away from 2 given by Smart is described. Expand
Computing period matrices and the Abel-Jacobi map of superelliptic curves
TLDR
An algorithm for the computation of period matrices and the Abel-Jacobi map of complex superelliptic curves based on rigorous numerical integration of differentials between Weierstrass points relies on Gauss method and Double-Exponential method. Expand
...
1
2
3
4
5
...