gpt 6 astra pro attempt on Math-163

Uniform Subconvexity for Standard Automorphic L-Functions on GL(n)

Definitions and mathematical statement

For a unitary cuspidal $\pi$ on $\operatorname{GL}_n({\mathbb{A}}_F)$, its analytic conductor $C(\pi)$ combines the finite conductor and the sizes of its archimedean gamma-factor parameters, in the cited normalization. The target is \[ \forall n,F\ \exists\delta_{n,F}>0,\ A_{n,F}>0\ \forall\pi:\qquad |L(\tfrac12,\pi)|\le A_{n,F}C(\pi)^{1/4-\delta_{n,F}}. \] The $L$-function is the finite part. All finite and archimedean parameters vary simultaneously, including unitary imaginary twists; neither a fixed representation family nor a single conductor aspect suffices. Any positive saving is allowed.

Formulation and concept references: [S1].

Short English statement

Can one beat the convexity exponent uniformly for every standard automorphic $L$-function of each fixed degree over each number field?

Research attempt

Current frontier.

The formulation source proves a $t$-aspect result for fixed representations over ${\mathbb{Q}}$, and a spectral theorem under uniform growth of archimedean parameters, with a polynomial finite-conductor loss. Its Section 1.2 explicitly distinguishes these from the universal-family target and records the logarithmic general improvement and the full degree-two theorem. [S1] Hu--Nelson's 2025 Theorem 1.2 allows varying ramified sets but retains specified local uniform-depth hypotheses and an uncontrolled-conductor factor. [E1] Berta's February 2026 manuscript treats prime-conductor twists of a fixed degree-three representation over a number field, not every parameter at once. [E2]

Attack route.

Try either to combine complementary aspect estimates or to control short additive correlations of the actual approximate-functional-equation coefficients. We calculate exactly when complementary estimates cover every conductor ratio, then derive a quantitative sufficient correlation estimate. The actual varying weights and full conductor remain visible throughout.

Attempt: exact aspect coverage.

Let $C=QP$, with finite and archimedean conductor factors $Q,P\ge1$. Suppose, hypothetically on the same family with uniform constants, that \[ |L|\ll Q^{1/4+a}P^{1/4-d_1},\qquad |L|\ll Q^{1/4-d_2}P^{1/4+b}, \quad a,b\ge0, \] where $d_1,d_2>0$. With $x=\log Q/\log C$, the exponents relative to $C^{1/4}$ are $ax-d_1(1-x)$ and $-d_2x+b(1-x)$. The maximum of their minimum occurs at their intersection and equals \begin{equation}\label{wsref-0-33-30-30-30-36-37-35-32-3a-63-6f-76-65-72-61-67-65} \frac{ab-d_1d_2}{a+b+d_1+d_2}. \end{equation} Consequently these bounds give a positive uniform saving exactly if $ab<d_1d_2$. Otherwise the interval \[ \frac{d_1}{a+d_1}\le x\le\frac{b}{b+d_2} \] is uncovered, including a zero-saving boundary point when equality holds. This describes what the bounds imply, not the true size of $L$ there. For example $a=b=1/4$, $d_1=d_2=1/100$ leaves a large middle interval. Taking a minimum or geometric mean does not repair it. The finite factor in the source's Theorem 2 is not a harmless constant when finite conductor varies. [S1] A new bound must reduce those losses or cover the middle regime, besides removing local parameter restrictions.

A finite weighted correlation inequality.

Let $b_j$ be supported on $N$ consecutive integers and zero elsewhere. Set \[ S=\sum_jb_j,\quad Q_0=\sum_j|b_j|^2,\quad R_h=\sum_jb_{j+h}\overline{b_j}. \] For $1\le H\le N$, \begin{equation}\label{wsref-0-33-30-30-30-36-37-35-32-3a-76-64-63} |S|^2\le\frac{N+H-1}{H^2} \left(HQ_0+2\sum_{h=1}^{H-1}(H-h)\Re R_h\right). \end{equation} Indeed $HS=\sum_m\sum_{h=1}^H b_{m+h}$ and at most $N+H-1$ outer summands are nonzero. Cauchy--Schwarz and expansion of the finite square give the formula. Its bracket is nonnegative; no diagonal term has been discarded.

For a fixed $0<\eta\le1/2$, put $H=\lceil N^\eta\rceil$. Suppose \begin{equation}\label{wsref-0-33-30-30-30-36-37-35-32-3a-63-6f-72-72} Q_0\ll_\epsilon NC^\epsilon,\qquad \sum_{h=1}^{H-1}(H-h)\Re R_h \ll_\epsilon H^2N^{1-\eta}C^\epsilon. \end{equation} For $N\ge2$, $H\le N$, and (30006752:vdc) yields \[ |S|\ll_\epsilon C^{\epsilon/2}N^{1-\eta/2}. \] The case $N=1$ follows from the first estimate. Only an upper bound on the weighted real sum is needed; bounding every absolute correlation is stronger and could waste cancellation among shifts.

Conditional application retaining the true weights.

Fix $n,F$. The uniform approximate functional equation supplies two original/dual sums truncated at $C^{1/2+\tau}$ with rapidly decaying weights depending on the actual gamma parameters, plus a rapidly decreasing error. This is the uniform truncation input in Harcos's Theorem 1 and Corollary 1, with its erratum, also recorded in his later dissertation, Section 1.2. [E3] No replacement by a parameter-independent cutoff is made.

Let $C=\max\{2,C(\pi)\}$ and aggregate Dirichlet coefficients $a_\pi(m)$ by ideal norm when $F\ne{\mathbb{Q}}$. On a dyadic interval $N\le m<2N$, set \[ b_{\pi,N}(m)=\mathbf1_{N\le m<2N}\mathbf1_{m\le C^{1/2+\tau}} (N/m)^{1/2}a_\pi(m)V_\pi(m/\sqrt C), \] using the genuine weight; do likewise for the dual sum. The precise missing hypothesis is (30006752:corr) for these sequences, every $\epsilon>0$, all allowed $\pi$ and all $N$, with one $\eta=\eta(n,F)>0$ and uniform constants. The coefficient-square estimate is explicitly part of the hypothesis, not an unproved Ramanujan assertion silently used below.

Each dyadic contribution is $N^{-1/2}S$, bounded by $C^{\epsilon/2}N^{(1-\eta)/2}$. Since that exponent of $N$ is positive, the dyadic sum is bounded geometrically by its largest scale. Choose $\tau=\eta/16$, $\epsilon=\eta/32$. The final exponent is \[ \frac\epsilon2+\left(\frac12+\tau\right)\frac{1-\eta}{2} =\frac14-\frac\eta4+\frac{\eta(1-\eta)}{32}+\frac\eta{64} \le\frac14-\frac{13\eta}{64}<\frac14-\frac\eta8. \] Thus the correlation hypothesis implies the exact requested type of bound, for example with $\delta_{n,F}=\eta/8$.

For pole characters in degree one, the contour shift retains its residue terms rather than pretending the $L$-function is entire. Such characters are norm twists of $\zeta_F$; with a Gaussian auxiliary factor their residue contributions are $O_F((1+|t|)^B e^{-t^2+O_F(|t|)})=O_F(1)$, and do not affect the deduction. Alternatively these terms may simply be retained in the analytic input. The two-sum formula without residue terms is used only in the entire case.

Stress test.

For $b_j=1$ on $[1,N]$, $Q_0=N$ but $R_h=N-h$. With $H=o(N)$ the weighted sum is asymptotic to $NH^2/2$, exceeding the proposed bound by a power. Thus the diagonal estimate and functional-equation length do not generate cancellation. Random mean-zero unit-modulus coefficients have expected correlation zero, but are not automorphic evidence. A unitary twist introduces phases $m^{-it}$ and does not guarantee uniform cancellation in every short interval.

Keeping the true weight inside $b$ avoids assuming uniform partial summation near degenerate archimedean parameters. Norm aggregation over $F$ is also retained. The script checks the finite correlation identity, exact inequalities for Gaussian-integer sequences and the aspect-coverage algebra, not the unproved automorphic hypotheses.

Outcome.

Outcome: Conditional reduction. The attempt derives a two-aspect coverage threshold and a short-correlation sufficient condition with explicit exponent bookkeeping. Proving that condition uniformly in all ramification and archimedean regimes would yield the target. Its arithmetic inputs remain unproved; no new universal subconvexity theorem or illicit extension of a fixed-family result is claimed.

Sources

• S1 Bounds for standard L-functions. https://ultronozm.github.io/math/standard.html. Catalogue formulation source.
• E1 Y. Hu and P. D. Nelson, The subconvexity bound for standard L-function in level aspect, arXiv:2503.12310v1, Sections 1.2--1.4, Theorem 1.2 and Corollary 1.3. https://arxiv.org/html/2503.12310v1.
• E2 F. Berta, Subconvexity Problem on $\operatorname{GL}_3$ over number fields: the twist aspect, arXiv:2602.20095v1, 23 February 2026, stated main result. https://arxiv.org/abs/2602.20095.
• E3 G. Harcos, Uniform approximate functional equation for principal L-functions, IMRN 2002, 923--932, Theorem 1 and Corollary 1; erratum, IMRN 2004, 659--660. https://arxiv.org/abs/math/0111312. See also the author's Subconvex Bounds for Automorphic L-functions and Applications, 2011 dissertation, Section 1.2, whose bibliography includes the erratum. https://www.renyi.hu/~gharcos/ertekezes.pdf.