arXiv subjects

David Loeffler

Publications and source records attributed to David Loeffler.

At least 19 recordsLinked to original sources

arXiv2025

Ultra-Kolyvagin systems and non-ordinary Selmer groups

We develop a machine for bounding Selmer groups of Galois representations via Euler systems in "non-ordinary" settings, using Pottharst's definition of Selmer groups via Robba-ring $(\varphi, \Gamma)$-modules. Our approach relies on Sweeting's interpretation of Kolyvagin derivative classes via non-principal ultrafilters. We apply these results to prove new cases of the cyclotomic Iwasawa main conjecture for non-ordinary Rankin--Selberg convolutions.

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arXiv2025

The Asai--Flach Euler system in $p$-adic families

We show that the Euler system for the Asai representation corresponding to a Hilbert modular eigenform over a real quadratic field, constructed by Lei, Loeffler and Zerbes (2018), can be interpolated $p$-adically as the Hilbert modular form varies in a Hida family. This work is used as an important input in recent work of Grossi, Loeffler and Zerbes (2025) on the proof of the Bloch--Kato conjecture in analytic rank zero for the Asai representation.

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arXiv2025

$p$-adic Asai and twisted triple product $L$-functions for finite slope families

We define a two-variable $p$-adic Asai $L$-function for a finite-slope family of Hilbert modular forms over a real quadratic field (with one component of the weight, and the cyclotomic twist variable, varying independently); and a two-variable ``twisted triple product'' $L$-function, interpolating the central $L$-value of the tensor product of such a family with a family of elliptic modular forms. The former construction generalizes a construction due to Grossi, Zerbes and the second author for ordinary families; the latter is a counterpart of the twisted triple product $L$-function of arXiv:2401.13230, but differs in that it interpolates classical $L$-values in a different range of weights, in which the dominant weight comes from the Hilbert modular form. Our construction relies on a ``nearly-overconvergent'' version of higher Coleman theory for Hilbert modular surfaces.

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arXiv2025

Formalizing zeta and L-functions in Lean

The Riemann zeta function, and more generally the L-functions of Dirichlet characters, are among the central objects of study in number theory. We report on a project to formalize the theory of these objects in Lean's "Mathlib" library, including a proof of Dirichlet's theorem on primes in arithmetic progressions and a formal statement of the Riemann hypothesis

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arXiv2025

Poles of p-adic Asai L-functions and distinguished representations

We give a criterion in terms of p-adic Asai L-functions for a cuspidal automorphic representation of GL(2) over a real quadratic field to be a distinguished representation, providing a p-adic counterpart of a well-known theorem of Flicker for the complex Asai L-function.

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arXiv2024

A universal Euler system for GSp(4)

In our earlier work with Christopher Skinner (J. Eur. Math. Soc 24 (2022), no. 2; DOI 10.4171/JEMS/1124; Arxiv 1706.00201), we constructed Euler systems for the 4-dimensional spin Galois representations corresponding to automorphic forms for GSp(4). This construction depended on various arbitrary choices of local test data. In this paper, we use multiplicity-one results for smooth representations to determine how these Euler system classes depend on the choice of test data, showing that all of these classes lie in a 1-dimensional space and are explicit multiples (given by local zeta-integrals) of a "universal" class independent of the choice of test data.

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arXiv2024

Asai-Flach classes, p-adic L-functions and the Bloch-Kato conjecture for GO(4)

We prove the Bloch-Kato conjecture for critical values of Asai L-functions of p-ordinary Hilbert modular forms over quadratic fields (with p split); and one inclusion in the Iwasawa main conjecture for these L-functions (up to a power of p). Along the way, we also prove a version of the p-adic Eichler-Shimura comparison isomorphism for Hida families of Hilbert modular forms.

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arXiv2024

P-adic Rankin-Selberg L-functions in universal deformation families and functional equations

We construct a $p$-adic Rankin-Selberg $L$-function associated to the product of two families of modular forms, where the first is an ordinary (Hida) family, and the second an arbitrary universal-deformation family (without any ordinarity condition at $p$). This gives a function on a 4-dimensional base space - strictly larger than the ordinary eigenvariety, which is 3-dimensional in this case. We prove our $p$-adic $L$-function interpolates all critical values of the Rankin-Selberg $L$-functions for the classical specialisations of our family, and derive a functional equation for our $p$-adic $L$-function.

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arXiv2023

An Euler system for the adjoint of a modular form

We construct an Euler system for the adjoint Galois representation of a modular form, using motivic cohomology classes arising from Hilbert modular surfaces. We use this Euler system to give an upper bound for the Selmer group of the adjoint representation over the cyclotomic Zp-extension, which agrees with the predictions of the Iwasawa main conjecture up to powers of p.

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arXiv2023

Asai-Flach classes and p-adic L-functions

We prove a formula for the Bloch-Kato logarithm of the bottom class in the Asai-Flach Euler system associated to a quadratic Hilbert modular form. We show that this can be expressed as a value, outside the interpolation range, of the p-adic Asai L-function constructed in the prequel paper arXiv:2307.07004.

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arXiv2023

P-adic Asai L-functions for quadratic Hilbert eigenforms

We construct p-adic Asai L-functions for cuspidal automorphic representations of GL2 / F, where F is a real quadratic field in which p splits. Our method relies on higher Hida theory for Hilbert modular surfaces with Iwahori level at one prime above p.

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arXiv2023

On $p$-adic $L$-functions for $\text{GSp}_4 \times \text{GL}_2$

We use higher Coleman theory to construct a new $p$-adic $L$-function for $\text{GSp}_4 \times \text{GL}_2$. While previous works by the first author, Pilloni, Skinner and Zerbes had considered the $p$-adic variation of classes in the $H^2$ of Shimura varieties for $\text{GSp}_4$, in this note we explore the interpolation of classes in the $H^1$, which allows us to access to a different range of weights. Further, we show an interpolation property in terms of complex $L$-values using the algebraicity results established in previous work by the authors.

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arXiv2023

Algebraicity of $L$-values for $\text{GSp}_4 \times \text{GL}_2$ and $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$

We prove algebraicity results for critical $L$-values attached to the group $\text{GSp}_4 \times \text{GL}_2$, and for Gan--Gross--Prasad periods which are conjecturally related to central $L$-values for $\text{GSp}_4 \times \text{GL}_2 \times \text{GL}_2$. Our result for $\text{GSp}_4 \times \text{GL}_2$ gives a new proof (by a very different method) of a recent result of Morimoto, and will be used in a sequel paper to construct a new $p$-adic $L$-function for $\text{GSp}_4 \times \text{GL}_2$. The results for Gross--Prasad periods appear to be new. A key aspect is the computation of certain archimedean zeta integrals, whose $p$-adic counterparts are also studied in this note.

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arXiv2022

Plectic structures in p-adic de Rham cohomology

Given a Hilbert modular form for a totally real field $F$, and a prime $p$ split completely in $F$, the $f$-eigenspace in $p$-adic de Rham cohomology of the Hilbert modular variety has a family of partial filtrations and partial Frobenius maps, indexed by the primes of $F$ above $p$. The general plectic conjectures of Nekovar and Scholl suggest a "plectic comparison isomorphism" comparing these structures to etale cohomology. We prove this conjecture in the case $[F : \mathbf{Q}] = 2$ under some mild assumptions; and for general $F$ we prove a weaker statement which is strong evidence for the conjecture, showing that plectic Hodge filtration has a canonical splitting given by intersecting with simultaneous eigenspaces for the partial Frobenii. (In memory of Jan Nekovar)

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arXiv2022

Higher Hida theory and p-adic L-functions for GSp(4)

We use the "higher Hida theory" recently introduced by the second author to p-adically interpolate periods of non-holomorphic automorphic forms for GSp(4), contributing to coherent cohomology of Siegel threefolds in positive degrees. We apply this new method to construct p-adic L-functions associated to the degree 4 (spin) L-function of automorphic representations of GSp(4), and the degree 8 L-function of GSp(4) x GL(2).

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arXiv2022

Eisenstein degeneration of Euler systems

We discuss the theory of Coleman families interpolating critical-slope Eisenstein series. We apply it to study degeneration phenomena at the level of Euler systems. In particular, this allows us to prove relations between Kato elements, Beilinson--Flach classes and diagonal cycles, and also between Heegner cycles and elliptic units. We expect that this method could be extended to construct new instances of Euler systems.

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arXiv2021

P-adic L-functions for GL(3)

Let $\Pi$ be a regular algebraic cuspidal automorphic representation (RACAR) of $\mathrm{GL}_3(\mathbb{A}_{\mathbb{Q}})$. When $\Pi$ is $p$-nearly-ordinary for the maximal standard parabolic with Levi $\mathrm{GL}_1 \times \mathrm{GL}_2$, we construct a $p$-adic $L$-function for $\Pi$. More precisely, we construct a (single) bounded measure $L_p(\Pi)$ on $\mathbb{Z}_p^\times$ attached to $\Pi$, and show it interpolates all the critical values $L(\Pi\times\eta,-j)$ at $p$ in the left-half of the critical strip for $\Pi$ (for varying $\eta$ and $j$). This proves conjectures of Coates-Perrin-Riou and Panchishkin in this case. We also prove a corresponding result in the right half of the critical strip, assuming near-ordinarity for the other maximal standard parabolic. Our construction uses the theory of spherical varieties to build a "Betti Euler system", a norm-compatible system of classes in the Betti cohomology of a locally symmetric space for $\mathrm{GL}_3$. We work in arbitrary cohomological weight, allow arbitrary ramification at $p$ along the Levi factor of the standard parabolic, and make no self-duality assumption. We thus give the first constructions of $p$-adic $L$-functions for RACARs of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$ of 'general type' (i.e., those that do not arise as functorial lifts) for any $n > 2$.

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arXiv2021

On the Birch-Swinnerton-Dyer conjecture for modular abelian surfaces

Let $A$ be a modular abelian surface over $Q$ which either has trivial geometric endomorphism ring, or arises as the restriction of scalars of an elliptic curve over an imaginary quadratic field which is modular and is not a $Q$-curve. In the former case, assume that there exists an odd Dirichlet character $\chi$ such that $L(A,\chi,1)\neq 0$. We prove the following implication: if $L(A, 1) \ne 0$, and the $p$-adic eigenvariety for $GSp_4$ is smooth at the point corresponding to $A$ (and some auxiliary technical hypotheses hold), then $A(Q)$ is finite, as predicted by the Birch--Swinnerton-Dyer conjecture, and the $p$-part of the Tate--Shafarevich group is also finite. We also prove one inclusion of the cyclotomic Iwasawa Main Conjecture for $A$. Moreover, we also prove analogous results for cohomological automorphic representations of $GSp_4$, removing many of the restrictive hypotheses in our earlier work [2003.05960]; for cohomological representations we do not need to assume smoothness of the eigenvariety, since it is automatic in this case. The main ingredient in the proof is the Euler system attached to the spin representations of genus $2$ Siegel modular forms constructed in our earlier work with Skinner.

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