Classification of Abelian-Surface L-Functions of Small Conductor

Definitions and mathematical statement

Let an abelian surface over ${\mathbb{Q}}$ mean an abelian variety of dimension $2$ defined over ${\mathbb{Q}}$. For such a surface $A$, let $N(A)$ denote its arithmetic conductor and let $L(A,s)$ denote its Hasse--Weil $L$-function, including the Euler factors at primes of bad reduction. Set \[ \mathcal L_{\leq 1000}=\{L(A,s): A/{\mathbb{Q}}\text{ is an abelian surface and }N(A)\leq 1000\}. \] For this draft, distinct $L$-functions are counted as distinct full Euler products, rather than as distinct models of abelian surfaces. The supplied record does not explicitly specify this equivalence convention.

Proposed enumeration and classification. Establish the source's proposed count \[ \#\mathcal L_{\leq 1000}=456 \] and the completeness of its proposed classification into the following numbers of entries:

• $360$ arising from products $E_1\times E_2$ of elliptic curves over ${\mathbb{Q}}$, with \[ L(E_1\times E_2,s)=L(E_1,s)L(E_2,s); \]
• $17$ arising from weight-$2$ newforms with quadratic Fourier-coefficient fields, meaning coefficient fields $K_f={\mathbb{Q}}(a_n(f):n\geq1)$ satisfying $[K_f:{\mathbb{Q}}]=2$;
• $2$ arising from Weil restrictions $\operatorname{Res}_{K/{\mathbb{Q}}}E$, where $K/{\mathbb{Q}}$ is a quadratic extension and $E/K$ is an elliptic curve; here Weil restriction is characterized by \[ (\operatorname{Res}_{K/{\mathbb{Q}}}E)(R)=E(R\otimes_{{\mathbb{Q}}}K) \] for every ${\mathbb{Q}}$-algebra $R$;
• $77$ arising from surfaces described in the source as generic, with at least $67$ of these asserted to be Jacobians.

A genus-$2$ Jacobian over ${\mathbb{Q}}$ means $\operatorname{Jac}(C)$ for a smooth projective geometrically connected curve $C/{\mathbb{Q}}$ of genus $2$. At the level of the set $\mathcal L_{\leq1000}$, a possible reading of the last assertion is that at least $67$ distinct entries in the generic category have such Jacobian representatives; the source does not specify whether this or a stronger assertion about particular surface representatives is intended.

Unresolved qualifications. The supplied curation describes the count as provisional and conditional on the paramodular conjecture, without specifying its version or the precise conditional implication. This qualification is retained; no unconditional classification is asserted here. The proposed list itself is not supplied. Moreover, generic is not defined, the precise newform-to-surface correspondence and its character restrictions are not stated, and no rule distinguishing potentially overlapping origins is given. Although the four counts sum to $456$, a precise disjoint classification requires these missing specifications. Consequently, the displayed enumeration and category descriptions record the proposed target, while its complete formal formulation remains unresolved.

Short English statement

The proposed problem is to prove that abelian surfaces over ${\mathbb{Q}}$ of conductor at most $1000$ yield exactly $456$ distinct $L$-functions, with the stated counts by origin and at least $67$ Jacobian entries among the $77$ generic entries. The supplied assessment makes this provisional and conditional on the paramodular conjecture. Missing classification conventions and the absent explicit list require human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005980$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as $2024$.
• Supplied source citation: Explicit Methods in Number Theory (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/40, pp. 2309--2368. https://doi.org/10.4171/owr/2024/40. The supplied excerpt labels the enumeration a conjecture.
• Supplied literature source: Booker--Sutherland, L-functions from nothing. https://math.mit.edu/~drew/JA2025.pdf.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and describes the count of $456$ as provisional and conditional on the paramodular conjecture, with completeness and proof of the classification not established in the source literature. This is supplied dated curation, not an independently verified literature assessment. The record requests independent verification of time-sensitive status and cited claims.