Paramodularity of Antisymmetric Borcherds Products

Definitions and mathematical statement

Let $N\in\{713,893\}$. Write $K(N)$ for the genus-two paramodular group associated with a polarization of type $(1,N)$, and let $M_2(K(N))$ denote its space of holomorphic scalar-valued modular forms of weight $2$. Let $f_N\in M_2(K(N))$ denote the particular antisymmetric Borcherds product referred to in the supplied source, and let $A_N/{\mathbb{Q}}$ denote the corresponding source-designated abelian surface of conductor $N$. An abelian surface is an abelian variety of dimension $2$. If the source's term "rank-one" means Mordell--Weil rank, its assertion is $\operatorname{rank}_{{\mathbb{Z}}} A_N({\mathbb{Q}})=1$; the record does not explicitly define this use of rank.

Intended conjectural statement. For each $N\in\{713,893\}$, the particular form $f_N$ corresponds to the particular surface $A_N$ in the sense predicted by the Paramodular Conjecture. Under the customary interpretation by matching degree-four $L$-functions, the intended target is \[ L_{\mathrm{spin}}(f_N,s)=L(A_N,s), \qquad N=713,893, \] with the spinor $L$-function placed in the arithmetic normalization appropriate to the Hasse--Weil $L$-function of $A_N$. In particular, this would require equality of the degree-four spinor Hecke polynomial of $f_N$ and the Hasse--Weil local polynomial of $A_N$ at every prime $p\nmid N$, together with the corresponding agreement of local factors at primes dividing $N$.

Unresolved specification. The record supplies neither the Borcherds input data or Fourier expansions identifying $f_N$, nor equations or other unique identifiers for $A_N$. It also omits the precise antisymmetry convention, the eigenform and cuspidality data needed to specify the intended spinor $L$-functions, and the normalization and bad-prime conventions for the displayed equality. Thus the equality records the expected meaning of the requested correspondence, subject to these missing specifications; it is not a fully instantiated conjecture. The quantifier concerns the particular source-designated pairs, not every weight-two antisymmetric Borcherds product or every rank-one surface of these conductors.

Short English statement

For each of the levels $713$ and $893$, the source asks whether its particular weight-two antisymmetric paramodular Borcherds product corresponds to its known rank-one abelian surface over ${\mathbb{Q}}$, as predicted by the Paramodular Conjecture. The intended correspondence is expressed by matching spinor and Hasse--Weil $L$-functions, but the supplied record does not identify the forms and surfaces sufficiently to specify those equalities completely.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003140$, source problem code catalogue entry, title "Paramodularity of Antisymmetric Borcherds Products," in the Oberwolfach Reports Open Problems collection. The record gives proposed year $2016$ and names no individual proposer.
• Supplied report citation: Moduli spaces and Modular forms (2017), Oberwolfach Reports, pp. 1259--1318, DOI 10.4171/owr/2016/23. https://doi.org/10.4171/owr/2016/23. The same link is listed among the literature sources as "original OWR report." The citation year $2017$ differs from the record's proposed year and year tag $2016$; this discrepancy is retained unresolved.
• Supplied literature source: Poor--Gritsenko--Yuen, Antisymmetric Paramodular Forms. https://arxiv.org/abs/1609.04146.
• Supplied dated literature curation, checked 2026-08-21: the record labels the problem open and states that the listed levels remain conjecturally modular for the corresponding rank-one abelian surfaces, with the constructed forms providing evidence rather than a proof of the Paramodular Conjecture. This is a supplied dated assessment, not an independently verified literature conclusion; the supplied background itself requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked. Its verification note says that "the listed" was replaced by the two source-specified weight-two levels and corresponding rank-one abelian surfaces. These are supplied provenance claims; no source visit or independent verification is asserted here.