[Paper Review] Sturm bounds for Siegel modular forms of degree 2 and odd weights

Toshiyuki Kikuta, Sho Takemori|arXiv (Cornell University)|Aug 7, 2015

Advanced Algebra and Geometry4 references3 citations

TL;DR

This paper establishes sharp Sturm bounds for Siegel modular forms of degree 2 and odd weights modulo any prime $p$, correcting gaps in prior work for $p=2,3$ and extending results to odd weights. It proves that vanishing of Fourier coefficients up to a bound $b_k = \lfloor (k-5)/10 \rfloor$ implies vanishing modulo $\mathfrak{p}^\nu$, with sharpness confirmed for algebraic Fourier coefficients.

ABSTRACT

We correct the proof of the theorem in the previous paper presented by the first named author, which concerns Sturm bounds for Siegel modular forms of degree $2$ and of even weights modulo a prime number dividing $2\cdot 3$. We give also Sturm bounds for them of odd weights for any prime numbers, and we prove their sharpness. The results cover the case where Fourier coefficients are algebraic numbers.

Motivation & Objective

• To correct flaws in the proof of Sturm bounds for Siegel modular forms of degree 2 and even weights modulo $p=2,3$.
• To establish sharp Sturm bounds for Siegel modular forms of degree 2 and odd weights modulo any prime $p$.
• To prove the sharpness of these bounds, which is essential for verifying congruences via numerical experiments.
• To extend the theory to modular forms with algebraic Fourier coefficients, ensuring applicability in arithmetic contexts.

Proposed method

• Introduce a new proof technique to correct the flawed argument in [7] for $p=2,3$ and even weights.
• Define the Sturm bound $b_k$ as $\lfloor k/10 \rfloor$ for even $k$, and $\lfloor (k-5)/10 \rfloor$ for odd $k$, based on the weight and prime modulus.
• Use induction on the weight $k$, leveraging known modular forms $X_{10}, X_{16}, X_{35}, Y_{12}$ and their $p$-adic valuations.
• Apply the $W$-map and $W''$-map to analyze $p$-adic valuations of Fourier coefficients and derive congruence conditions.
• Use the fact that $f \equiv 0 \mod{\mathfrak{p}^\nu}$ if all coefficients $a_f(m,r,n) \equiv 0 \mod{\mathfrak{p}^\nu}$ for $0 \leq m,n \leq b_k$ and $4mn - r^2 \geq 0$.
• Verify sharpness by constructing explicit modular forms $f_k$ for $k=35,39,41,43,47$ whose leading terms match the bound.

Experimental results

Research questions

• RQ1 What is the correct and sharp Sturm bound for Siegel modular forms of degree 2 and odd weights modulo any prime $p$?
• RQ2 How can the proof gap in Theorem 2.1 of [7] for $p=2,3$ and even weights be corrected?
• RQ3 Can the Sturm bound for odd weights be proven to be sharp, and what is its precise form?
• RQ4 How do $p$-adic valuations of Fourier coefficients relate to the vanishing of modular forms modulo prime powers?
• RQ5 What is the minimal set of Fourier coefficients needed to detect vanishing modulo $\mathfrak{p}^\nu$ for modular forms with algebraic coefficients?

Key findings

• The Sturm bound for odd weights is given by $b_k = \lfloor (k-5)/10 \rfloor$, and it is sharp for all primes $p$.
• For even weights and $p=2,3$, the corrected Sturm bound is $b_k = \lfloor k/10 \rfloor$, and it is sharp.
• The sharpness of the bounds is confirmed by explicit modular forms $f_k$ for $k=35,39,41,43,47$ whose leading terms achieve the bound.
• The proof relies on a new method using $W$-maps and $p$-adic valuation arguments, correcting the flawed approach in [7].
• The results hold for modular forms with algebraic Fourier coefficients, extending applicability to arithmetic geometry and congruence studies.
• The bound $b_k$ satisfies the recurrence $b_k = b_{k-8} + 1$ for $k \geq 45$, which supports the inductive proof structure.