Mathematicians unlock hidden geometric structure in mysterious number sequences
Researchers have discovered that cryptic mathematical constants called multiple L-values—long thought to be unconnected to geometry—actually emerge naturally from the deformations of modular forms. This finding bridges two previously separate mathematical worlds, potentially opening new computational pathways for cryptography and number-theoretic algorithms that underpin digital security.
Originaltitel: Formal deformations of modular forms and multiple L-values
We relate analytically defined deformations of modular curves and modular forms from the literature to motivic periods via cohomological descriptions of deformation theory. Leveraging cohomological vanishing results, we prove the existence and essential uniqueness of deformations, which we make constructive via established Lie algebraic arguments and a notion of formal Lie deformations. Further, we construct a canonical and a totally holomorphic universal family of deformations of modular forms of all weights, which we obtain from the canonical cocycle associated with periods on the moduli space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℳ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:math> . Our uniqueness statement shows that non-critical multiple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi mathvariant="normal">L</mml:mi> </mml:math> -values, which appear in our deformations but are a priori non-geometric, are genuinely linked to deformations. Our work thus suggests a new geometric perspective on them.