arXiv subjects

Jeanine Van Order

Publications and source records attributed to Jeanine Van Order.

14 recordsLinked to original sources

arXiv2026

$L$-functions of elliptic curves in ring class extensions of real quadratic fields via regularized theta liftings

We derive new integral presentations for central derivative values of $L$-functions of elliptic curves $E/{\bf{Q}}$ twisted by ring class characters of a real quadratic field $K$ in terms of automorphic Green's functions for certain Hirzebruch-Zagier-like arithmetic divisors on the product of modular curves $X_0(N) \times X_0(N)$ along real geodesic cycles. We also relate these sums to Birch-Swinnerton-Dyer constants and periods.

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arXiv2026

Arithmetic Hirzebruch-Zagier divisors and central derivative values of Rankin-Selberg $L$-functions

Let $E$ be an elliptic curve parametrized by a newform $ϕ\in S_2(Γ_0(N))$. Let $k$ be a quadratic field of discriminant $d_k$ prime to $N$. Given a character $χ$ of the ideal class group of $k$ with theta series $θ(χ)$, we relate the central derivative values $Λ'(E/k, χ, 1) = Λ'(1/2, ϕ\times θ(χ))$ of the $χ$-twisted $L$-function of $E/k$ to arithmetic heights of Hirzebruch-Zagier divisors on $X_0(N)\times X_0(N)$ when $k$ is imaginary quadratic, and to sums of Green's functions of Hirzebruch-Zagier divisors along real geodesic cycles of $X_0(N) \times X_0(N)$ determined by ideal classes when $k$ is real quadratic. More generally, we refine the higher Gross-Zagier formulae for CM cycles on spin Shimura varieties of any dimension this way. This gives two arithmetic height formulae for $Λ'(E/k, χ, 1)$ when $k$ is imaginary quadratic, as well as a distinct proof of the Gross-Zagier formula, with implied relations between the arithmetic heights of Heegner divisors on $X_0(N)$ and Hirzebruch-Zagier divisors on $X_0(N) \times X_0(N)$. We also explain connections to the refined conjecture of Birch-Swinnerton-Dyer, and to Fourier coefficients of half-integral weight forms.

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arXiv2023

Nonvanishing of self-dual $L$-values via spectral decomposition of shifted convolution sums

We obtain nonvanishing estimates for central values of certain self-dual Rankin-Selberg $L$-functions on $\operatorname{GL}_2({\bf{A}}_F) \times \operatorname{GL}_2({\bf{A}}_F)$, and more generally $\operatorname{GL}_r({\bf{A}}_F) \times \operatorname{GL}_2({\bf{A}}_F)$ for $r \geq 2$ an integer over $F$ a totally real number field, contingent on the best known approximations towards the generalized Lindelöf hypothesis for $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms in the level aspect, as well as the best known approximations to the generalized Ramanujan conjecture hypothesis for $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms. We proceed by developing a spectral approach to the shifted convolution problem for coefficients of $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms, accessing he higher-rank case through the classical projection operator $\mathbb P^r_1$ and the way it respects Fourier-Whittaker expansions. In the course of deriving our results, we supply the required nonvanishing hypothesis for recent work of Darmon-Rotger to bound Mordell-Weil ranks of elliptic curves in number fields cut out by tensor products of two odd, two-dimensional Artin representations whose product of determinants is trivial. This in particular allows us to deduce bounds (on average) for Mordell-Weil ranks of elliptic curves in ring class extensions of real quadratic fields which had not been accessible previously.

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arXiv2023

Class group twists and Galois averages of $\operatorname{GL}_n$-automorphic $L$-functions

Fix $n \geq 2$ an integer, and let $F$ be a totally real number field. We derive estimates for the finite parts of the $L$-functions of irreducible cuspidal $\operatorname{GL}_n({\bf{A}}_F)$-automorphic representations twisted by class group characters or ring class characters of a totally imaginary quadratic extensions $K$ of $F$, evaluated at central values $s=1/2$ or more generally values $s \in {\bf{C}}$ within the strip $\frac{1}{2} - \frac{1}{n^2 + 1} < \Re(s) < 1$. Assuming the generalized Ramanujan conjecture at infinity, we obtain estimates for all arguments in the critical strip $0 < \Re(s) < 1$. We also derive finer nonvanishing estimates for central values $s=1/2$ twisted by ring class characters of $K$. When the dimension $n \leq 4$ is small, these give us nonvanishing estimates depending on the best known approximations towards the generalized Lindelöf hypothesis for $\operatorname{GL}_2({\bf{A}}_F)$-automorphic forms in the level aspect, and in particular unconditional nonvanishing for $n \leq 3$ (with the case of $n=3$ being new). We derive such estimates via certain exact integral representations for the moments, and in particular for new developments of bounds on the shifted convolution problem in this context. In the setting where the cuspidal representation is cohomological of even rank $n \geq 2$, we also explain how to view these estimates in terms of recent rationality theorems towards Deligne's conjecture for automorphic motives over CM fields.

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arXiv2023

Integral presentations of the shifted convolution problem and subconvexity estimates for $\operatorname{GL}_n$-automorphic $L$-functions

Fix $n \geq 2$ an integer, and $F$ be a totally real number field. We reduce the shifted convolution problem for $L$-function coefficients of $\operatorname{GL}_n({\bf{A}}_F)$-automorphic forms to the better-understood setting of $\operatorname{GL}_2({\bf{A}}_F)$. The key idea behind this reduction is to use the classical projection operator $\mathbb P^n_1 φ$ together with properties of its Fourier-Whittaker expansion. This allows us to derive novel integral presentations for the shifted convolution problem as Fourier-Whittaker coefficients of certain $L^2$-automorphic forms on the mirabolic subgroup $P_2({\bf{A}}_F)$ of $\operatorname{GL}_2({\bf{A}}_F)$ or its two-fold metaplectic cover $\overline{P}_2({\bf{A}}_F)$. We then construct liftings of these mirabolic forms to $\operatorname{GL}_2({\bf{A}}_F)$ and its two-fold metaplectic cover $\overline{G}({\bf{A}}_F)$ to justify expanding the underlying forms into linear combinations of Poincaré series. Decomposing each of the Poincaré series spectrally then allows us to derive completely new bounds for the shifted convolution problem in dimensions $n \geq 3$. As an application, we derive a uniform subconvexity bound for $\operatorname{GL}_n({\bf{A}}_F)$-automorphic $L$-functions twisted by Hecke characters. This uniform level-aspect subconvexity estimate appears to the the first of its kind for dimensions $n \geq 3$.

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arXiv2021

Rankin-Selberg L-functions in cyclotomic towers, III

Let $π$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times \operatorname{GL}_2$ Rankin-Selberg $L$-functions associated to $π$ times representations induced from Hecke characters of $K$ which are ramified only at a given prime ideal $\mathfrak{p}$ of $F$. More specifically, we use spectral decompositions of shifted convolution sums and relations to Fourier-Whittaker coefficients of genuine and non-genuine metaplectic forms to obtain nonvanishing estimates, averaging over primitive ring class characters of a given exact order. When $π$ corresponds to a holomorphic Hilbert modular form of arithmetic weight $k \geq 2$, we then derive finer results from the rationality theorems of Shimura, together with the existence of suitable $\mathfrak{p}$-adic $L$-functions. This allows us to generalize the theorems of Rohrlich, Vatsal, and Cornut-Vatsal to this setting. Finally, in a self-contained appendix, we explain how to use these results to deduce bounds for Mordell-Weil ranks of the associated $\operatorname{GL}_2$-type abelian varieties via existing Iwasawa main conjecture divisibilities.

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arXiv2021

Dirichlet twists of $\operatorname{GL}_n$-automorphic $L$-functions and hyper-Kloosterman Dirichlet series

We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series associated to $\operatorname{GL}_n$-automorphic $L$-functions.

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arXiv2019

Rankin-Selberg L-functions in cyclotomic towers, I

We formulate and for the most part prove a conjecture in the style of Mazur-Greenberg for the nonvanishing of central values of Rankin-Selberg $L$-functions attached to elliptic curves in abelian extensions of imaginary quadratic fields. This in particular generalizes the theorem of Rohrlich on $L$-functions of elliptic curves in cyclotomic towers to the setting of abelian extensions of imaginary quadratic fields, corresponding to families of degree-four $L$-functions given by $\operatorname{GL}(2)\times\operatorname{GL}(2)$ Rankin-Selberg $L$-functions. It also generalizes the theorems of Rohrlich, Greenberg, Vatsal, and Cornut for $L$-functions of elliptic curves in $\operatorname{Z}_p^2$-extensions of imaginary quadratic fields.

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arXiv2019

Rankin-Selberg L-functions in cyclotomic towers, II

Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an elliptic curve in the ${\bf{Z}}_p^2$-extension is finitely generated modulo Heegner points. The novelty of the approach here is to use the existence of a suitable $p$-adic $L$-function to reduce the problem to a minimal nonvanishing criterion, which should be applicable to a broader class of problems than considered here.

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arXiv2019

p-adic interpolation of automorphic periods for GL(2)

We give a new and representation theoretic construction of $p$-adic interpolation series for central values of self-dual Rankin-Selberg $L$-functions for $\operatorname{GL}_2$ in dihedral towers of CM fields, using expressions of these central values as automorphic periods. The main novelty of this construction, apart from the level of generality in which it works, is that it is completely local. We give the construction here for a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real field corresponding to a $\mathfrak{p}$-ordinary Hilbert modular forms of parallel weight two and trivial character, although a similar approach can be taken in any setting where the underlying $\operatorname{GL}_2$-representation can be chosen to take values in a discrete valuation ring. A certain choice of vectors allows us to establish a precise interpolation formula thanks to theorems of Martin-Whitehouse and File-Martin-Pitale. Such interpolation formulae had been conjectured by Bertolini-Darmon in antecedent works. Our construction also gives a conceptual framework for the nonvanishing theorems of Cornut-Vatsal in that it describes the underlying theta elements. To highlight this latter point, we describe how the construction extends in the parallel weight two setting to give a $p$-adic interpolation series for central derivative values when the root number is generically equal to $-1$, in which case the formula of Yuan-Zhang-Zhang can be used to give an interpolation formula in terms of heights of CM points on quaternionic Shimura curves.

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arXiv2014

On the dihedral Euler characteristics of Selmer groups of abelian varieties

This note shows how to use the framework of Euler characteristic formulae to study Selmer groups of abelian varieties in certain dihedral or anticyclotomic extensions of CM fields via Iwasawa main conjectures, and in particular how to verify the p-part of the refined Birch and Swinnerton-Dyer conjecture in this setting. When the Selmer group is cotorsion with respect to the associated Iwasawa algebra, we obtain the p-part of formula predicted by the refined Birch and Swinnerton-Dyer conjecture. When the Selmer group is not cotorsion with respect to the associated Iwasawa algebra, we give a conjectural description of the Euler characteristic of the cotorsion submodule, and explain how to deduce inequalities from the associated main conjecture divisibilities of Perrin-Riou and Howard.

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arXiv2012

On the quaternionic p-adic L-functions associated to Hilbert modular eigenforms

We construct p-adic L-functions associated to cuspidal Hilbert modular eigenforms of parallel weight two in certain dihedral or anticyclotomic extensions via the Jacquet-Langlands correspondence, generalizing works of Bertolini-Darmon, Vatsal and others. The construction given here is adelic, which allows us to deduce a precise interpolation formula from a Waldspurger type formula, as well as a formula for the dihedral mu-invariant. We also make a note of Howard's nonvanishing criterion for these p-adic L-functions, which can be used to reduce the associated Iwasawa main conjecture to a certain nontriviality criterion for families of p-adic L-functions.

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arXiv2012

On the dihedral main conjectures of Iwasawa theory for Hilbert modular eigenforms

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini-Darmon, Longo, Nekovar, Pollack-Weston and others. The construction has direct applications to Iwasawa main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated p-adic L-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over CM fields via the technique of Skinner and Urban.

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arXiv2011

Some remarks on the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication

We establish several results towards the two-variable main conjecture of Iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields, namely (i) the existence of an appropriate p-adic L-function, building on works of Hida and Perrin-Riou, (ii) the basic structure theory of the dual Selmer group, following works of Coates, Hachimori-Venjakob, et al., and (iii) the implications of dihedral or anticyclotomic main conjectures with basechange. The result of (i) is deduced from the construction of Hida and Perrin-Riou, which in particular is seen to give a bounded distribution. The result of (ii) allows us to deduce a corank formula for the p-primary part of the Tate-Shafarevich group of an elliptic curve in the Z_p^2-extension of an imaginary quadratic field. Finally, (iii) allows us to deduce a criterion for one divisibility of the two-variable main conjecture in terms of specializations to cyclotomic characters, following a suggestion of Greenberg, as well as a refinement via basechange.

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