Nontriviality of Rankin-Selberg <i>L</i>-functions and CM points

Cornut, Christophe; Vatsal, Vinayak

doi:10.1017/cbo9780511721267.005

Cited by 68 publications

(95 citation statements)

References 24 publications

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“…This perspective also explains why our most complete results are restricted to abelian varieties with maximal real endomorphism ring: the techniques of [21], which reduce questions about arbitrary free modules over a ring to sublattices of its field of fractions, rely on that ring satisfying the Gorenstein property. This property holds for all quadratic orders and for any order with maximal real suborder (Lemma 4.4), but not for a general order (even in a quartic field).…”

Section: Proof Strategy: -Adic Lattices and Tate's Theoremmentioning

confidence: 99%

Isogeny graphs of ordinary abelian varieties

2017

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Fix a prime number . Graphs of isogenies of degree a power of are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called l-isogenies, resolving that, in arbitrary dimension, their structure is similar, but not identical, to the "volcanoes" occurring as graphs of isogenies of elliptic curves. Specializing to the case of principally polarizable abelian surfaces, we then exploit this structure to describe graphs of a particular class of isogenies known as ( , )-isogenies: those whose kernels are maximal isotropic subgroups of the -torsion for the Weil pairing. We use these two results to write an algorithm giving a path of computable isogenies from an arbitrary absolutely simple ordinary abelian surface towards one with maximal endomorphism ring, which has immediate consequences for the CM-method in genus 2, for computing explicit isogenies, and for the random self-reducibility of the discrete logarithm problem in genus 2 cryptography.

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Section: Proof Strategy: -Adic Lattices and Tate's Theoremmentioning

confidence: 99%

Isogeny graphs of ordinary abelian varieties

2017

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“…Based on [CV07] we should expect that this is caused by systematic collections of isogenies (coming from Hecke relations), and in our case we should expect 2-isogenies to play a role. The results here have a similar flavor to those of [BM04, pp.…”

Section: Resultsmentioning

confidence: 76%

“…(2) From the work of Cornut-Vatsal [CV05,CV07] and Jetchev-Kane [JK11] we have that the reductions of the j-invariants of elliptic curves with CM by O are equidistributed among the supersingular values in F p 2 (as we vary the conductors O subject to certain congruence conditions). Moreover, for each p and all but finitely many O where p is inert, the map from elliptic curves with CM by O to supersingular j-invariants in F p 2 is surjective.…”

Section: Introductionmentioning

confidence: 99%

On the $j$-invariants of CM-elliptic curves defined over $\mathbb{Z}_p$

Fiori¹

2017

Funct. Approx. Comment. Math.

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Abstract. We characterize the possible reductions modulo p of the j-invariants of supersingular elliptic curves which admit complex multiplication by a (potentially non-maximal) order O where the curve itself is defined over Zp. In particular, we show that the collection of possible j-invariants as well as some aspects of the distribution depends on which primes divide the discriminant and conductor of the order O.

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“…A few years ago, Vatsal and Cornut [Vat02,Vat03,CV04] used period relations and equidistribution in a similar way to obtain somewhat stronger non-vanishing results for Rankin/Selberg L-functions but associated to anti-cyclotomic 4 characters of a fixed imaginary quadratic field. This is in constrast with the present case where one can allow the quadratic field to vary.…”

Section: Equidistribution and Non-vanishing Of L-functionsmentioning

confidence: 99%